Download Mergers of massive main sequence binaries

Mergers of massive main sequence binaries
Bachelor research report
February 2010
Adrian Hamers
Under supervision of
O.R. Pols
S.E. de Mink
Universiteit Utrecht
Faculteit Bètawetenschappen
Departement Natuur­ en sterrenkunde
Sterrenkundig instituut
Contents
On the cover page: Image of the star-forming region
N 11B in the Large Magellanic Cloud (LMC) taken by Hubble’s Wide Field Planetary Camera 2 using filters that isolate
light emitted by oxygen and hydrogen gas, taken in 19991 .
Abstract
In this bachelor research project, we investigated mergers between close main-sequence binaries with the aim of determining whether they result in blue stragglers. We define blue
stragglers to be hot and luminous stars that cannot be explained with canonical single star evolution models2 . Using
a pre-calculated grid of binary models, we determined the
composition of merger products for a wide range of binaries
with the assumption of no mass loss and homogeneous mixing during the merging process. We find that there are two
types of mergers which result from different phases of mass
transfer. Both of these types have relatively low hydrogen
content which leads to high luminosities. We then used the
data of this analysis to simulate a cluster of a large number of
stars with a binary percentage of 50% which formed instantaneously. The result of this simulation is that merger stars of
close massive main-sequence binaries are dominant among the
most luminous stars in the main-sequence band of the simulated cluster, however we have made assumptions that may
greatly affect results. In addition, blue stragglers may also be
formed in different processes that do not involve the merging
of massive main-sequence binaries. We could improve on this
by using stellar evolution models to model the merger products instead of using homology relations, assuming different
cluster formation processes and including different blue straggler scenarios.
1 Introduction
3
2 Mass transfer in binary systems
2.1 Roche lobe overflow . . . . . . . . . . . . . . .
2.2 Sub-cases of case A evolution . . . . . . . . . .
3
4
5
3 Mergers in binary systems
3.1 Assumptions made . . . . . . . . . . . . . . . .
3.2 Qualitative properties . . . . . . . . . . . . . .
3.3 Quantitative properties: grid analysis . . . . .
6
6
6
6
4 Cluster simulation
4.1 Method . . . . . . . . . .
4.1.1 Cluster generation
4.1.2 Cluster population
4.2 Results . . . . . . . . . . .
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5 Discussion
11
5.1 Mass loss and angular momentum loss . . . . . 11
5.2 Homology relations . . . . . . . . . . . . . . . . 11
5.3 Cluster simulation . . . . . . . . . . . . . . . . 11
1 Source:
6 Conclusion
12
7 Acknowledgements
12
8 References
12
9 Appendix: flow diagram
14
http://hubblesite.org/newscenter/archive/releases/2004/22/.
these we mean evolution models with no implementation
of non-standard physical processes such as rotational mixing.
2 By
2
log10 (L / L⊙) 
to as merger blue stragglers, to produce more luminous stars
than the channel which leads to accretion blue stragglers.
Also, as N 11 is a young star cluster, its stars are expected
to have a relatively high binary fraction. Therefore, in
this research we will focus on merger blue stragglers. More
precisely, mergers between core-hydrogen burning stars in
close massive binaries are investigated. The key questions are:
• Which binary configurations lead to merger stars on the
main sequence?
• What is the relation between binary configuration and
merger star composition and luminosity?
• What is the relative importance of mergers of close
massive binaries among the massive stars in a stellar
population, such as a (young) star cluster?
log10 (Teff / K) 
The third question addresses the issue that in young
clusters, blue stragglers are hard to distinguish from very
massive, single (i.e. ’normal’) and young stars, as the latter
appear in roughly the same position in the HR-diagram.
Whereas the relative importance of mergers of close massive
binaries with respect to single massive stars cannot easily
be determined from observations, this determination can be
done by a simulation of a cluster of stars, as we shall do
in this research. We remark that this question relates to
questions raised by Hunter et al. (2008) as to the possibly
considerable binary fraction in the LMC needed to explain
the observed low nitrogen surface abundances among massive
main-sequence stars.
Figure 1: Hertzsprung-Russel (HR) diagram of LMC cluster N 11
by Evans et al. (2006) (edited image), whose data originated from
the VLT-Flames survey. N 11 is a young cluster with an age of a few
Myr. The region of stars which we define to be blue stragglers has
been indicated. Dotted lines indicate calculated evolution tracks.
1
Introduction
Looking at regions of cluster N 11 in the Large Magellanic
Cloud (LMC) (cover page), one immediately notices the
presence of many bright blue- and white-colored stars. These
stars are among the hottest and most massive stars known
in the Universe. In the Hertzsprung-Russel (HR) diagram of
this cluster (Fig. 1), they belong to the stars which lie in the
region of stars which we will refer to as blue stragglers. In the
context of low-mass stars, blue stragglers are known as stars
which have luminosities and effective temperatures that are
significantly greater for their age than predicted by standard
single-star evolution models. In analogy, we define blue
stragglers in the context of high-mass stars as very hot and
luminous stars that cannot be explained with canonical single
star evolution models, by which we mean star evolution models which do not incorporate effects such as rotational mixing.
In Sect. 2 we will begin with a brief overview of the physical processes that occur in binary systems before a possible
merging event. To tackle the second question quantitatively,
we analyzed a grid of binary SMC models computed by De
Mink et al. (2007) which allowed us to calculate the hydrogen fraction of the merger star, which we shall discuss in
Sect. 3.3. We considered the third question by simulating a
cluster of stars with SMC metallicity (Sect. 4) which formed
instantaneously, i.e. in a starburst, where we generated single stars and binaries. We processed the binaries with the
potential to merge on the main sequence and thus we could
distinguish between single stars, binaries that quite certainly
do not merge on their main sequence (which we will refer to
as unevolved binaries) and mergers that originate from close
main-sequence binaries. This allowed us to make statements
of the importance of these mergers among other stars with
comparable luminosity, i.e. massive single stars and massive
unevolved binaries.
The origin of these blue stragglers is not yet properly
understood as they could be very massive single stars which
had an unusual evolutionary history or they could also have
a binary origin. One of the binary explanations involves the
collision of single or binary stars as pointed out by Lombardi
et al. (2002), which can be modeled with hydrodynamic
simulations. Other explanations are given by Han et al.
(2009). One possibility is that a binary component accretes
mass from its companion and gets rejuvenated, increasing
the main-sequence life-time of the component. We will refer
to the resulting blue stragglers as accretion blue stragglers.
An other possibility is that a binary merges to a single star
after a common envelope situation.
2
Mass transfer in binary systems
Since interacting binaries are the focus of this research, we
discuss some relevant aspects of mass transfer in this section:
Roche lobe overflow (Sect. 2.1) and two important binary
evolution sub-cases (Sect. 2.2).
We expect the latter channel, whose remnants we will refer
3
z 
z 
Field point
Field point
Roche lobe surface
Roche Lobe surface
r
r
rp
rp
rs r
s
p
Figure 3: Depiction of the possible binary configurations5 .
L
L11
ss
p
a

 yy
Eggleton (1983):
RL,p =

xx 
Figure 2: Contour plot of surfaces of constant Roche potential
for a typical binary system4 with mass ratio 2. For the Roche-lobe
surface, which corresponds to the inner thick solid line, the point
of intersection L1 has been indicated.
P2
4π 2
.
=
3
a
G(Mp + Ms )
The theory of Roche lobe overflow describes mass transfer
in binary systems as explained by Verbunt (2008). For a binary system with primary3 mass Mp , secondary mass Ms and
period P , consider the geometry of surfaces of constant potential (Fig. 2), which is determined by the mass distribution
and the period. The potential per unit mass, Φ(rp , rs , rω ), is
given by:
GMp
GMs
ω 2 rω2
−
−
,
rp
rs
2
(3)
A particle within the Roche lobe of a star is attached to
this particular star because the gravitational attraction of
this star exceeds the gravitational attraction of the other star
and the centrifugal force. At the inner Lagrangian point L1 ,
the forces exactly balance such that a particle around this
point can migrate to the other star. Hence if the radius of
a star in a binary exceeds the Roche-lobe radius, in other
words, if a star fills its Roche lobe, then mass transfer can
occur through the nozzle around the inner Lagrangian point
L1 , where hydrostatic equilibrium is no longer possible. This
leads to three possible binary configurations which are also
observed (Fig. 3):
Roche lobe overflow
Φ(rp , rs , rω ) = −
(2)
which is an increasing function of q. The Roche-lobe radius
of the secondary star can be obtained by using Eq. 2 with
q 7→ 1q . It is useful to mention the relation between the
separation a and period P , which is Kepler’s Third Law:
Axis of rotation
Axis of rotation
2.1
0.49 a
,
0.6 + q −2/3 ln(1 + q 1/3 )
(1)
• Detached binary: both stars do not fill their Roche lobe
and thus there is no mass transfer. Binary evolution
closely resembles single star evolution.
with rp and rs the distances from the field point (the point of
interest) to the centers of mass of the primary and secondary
star respectively, rω the (shortest) distance of the field point
to the axis of rotation, ω = 2π
the angular frequency of
P
rotation and G the gravitational constant. Thus the forces
we consider are the gravitational attraction of both stars
and the outward centrifugal repulsion. There exists a certain
equipotential surface which consists of two surfaces that each
enclose one of the stars. These two surfaces intersect at a
point called the inner Lagrangian point L1 and each enclose
a certain volume; the radius of the sphere with the same
corresponding volume is called the Roche-lobe radius RL .
• Semi-detached binary: one of the stars fills its Roche
lobe and may transfer mass to its companion.
• Contact binary: both stars fill their Roche lobes and may
transfer mass and heat to each other (common-envelope
situation). This may result in a merger star.
Mass transfer greatly affects binary evolution. It is of
importance to be able to specify the evolutionary state at
which it occurs, hence the following evolutionary cases are
discerned:
Thus in a binary system there exist two Roche-lobe radii,
one for the primary and one for the secondary star. In general
these radii depend on the separation a of the centers of mass
of the stars (Fig. 2) and on the mass ratio q, here defined
as the ratio of the mass of the primary and secondary star,
M
i.e. q ≡ Mps . An approximate equation for all mass ratios
accurate to better than 1% for the primary star was given by
• Case A: mass transfer occurs during core-hydrogen burning.
• Case B: mass transfer occurs during hydrogen shell burning.
• Case C: mass transfer occurs after central helium depletion.
3 We define the primary star as the initially most massive star
and the secondary star as the initially least massive star.
4 Unedited image extracted on 01-12-09 from Internet page
https://www.e-education.psu.edu/astro801/content/l6 p5.html.
5 Unedited
image extracted on 11-01-10 from Internet
page
http://zebu.uoregon.edu/∼imamura/122/lecture9/typeIaSN.html.
4
Mp,i = 10 M⊙; Ms,i = 5.6 M⊙; Pi = 1.7 d
It must be noted that three assumptions are made in the
Roche geometry displayed in Fig. 2. First, the gravitational
fields of both stars are assumed to be those of point masses.
This is a reasonable assumption, even when stars are filling
their Roche lobes, because in most stars the mass is very
centrally located. Second, the binary orbit is assumed to be
circular. This assumption is not generally valid, but since
the circularization timescale depends on (R/a)8 as shown by
Zahn (1977), it is small compared to the stellar expansion
timescale when R is close to RL , i.e. when the binary components are filling their Roche lobes. Last, the stellar rotation is
assumed to be synchronized with the orbital motion, since the
Roche geometry only applies to matter that co-rotates with
the orbit. As with the second assumption, this assumption is
not generally valid, but since the synchronization timescale
depends on (R/a)6 , also shown by Zahn (1977), it is again
small compared to the stellar expansion timescale if R is close
to RL .
M/M⊙ 
(a) Case AR mass transfer.
Mp,i = 10 M⊙; Ms,i = 8.9 M⊙; Pi = 1.1 d
log10 (R/R⊙) 
2.2
Primary
Secondary
Roche Lobe
log10 (R/R⊙) 
Close binaries, i.e. binaries with short periods (lower than
about 8 days), interact at a relatively early stage of their
evolution, i.e. on the main sequence. This is because short
periods imply small separations a (Eq. 3), which imply small
Roche-lobe radii (Eq. 2). Thus the binary components fill
their Roche lobes at a relatively early stage in their evolution
because stars expand on the main sequence. Hence case A
mass transfer applies in close binaries.
Sub-cases of case A evolution
Further distinctions of case A binary evolution can be made
as illustrated in detail by Nelson & Eggleton (2001). We are
only interested in sub-cases that lead to contact binaries with
both components still on the main sequence, hence we only
consider the two sub-cases which Nelson & Eggleton (2001)
refer to as case AR and case AS. For each of these sub-cases a
contact binary is eventually formed, but in different phases of
mass transfer in case A systems. These phases are referred to
as the rapid and slow phase, which are driven by expansion
on the thermal timescale of the primary and expansion on
the nuclear timescale of the primary respectively.
M/M⊙ 
(b) Case AS mass transfer.
Figure 4: Radius as function of mass for (a) typical AR systems
and (b) typical AS systems6 . The index i stands for initial; d
stands for day.
In the rapid phase of mass transfer, the initial phase of
mass transfer in case A systems, mass is transferred from
the more massive primary to the less massive secondary,
which causes the orbit to shrink. The shrinking orbit implies
a decrease of the Roche-lobe radius (Eq. 2), such that
the equilibrium radius of the primary exceeds the primary
Roche-lobe radius and the primary is brought out of thermal
equilibrium. Hence mass transfer occurs at the relatively
short thermal timescale of the primary. If the mass ratio is
moderate or large - these corresponding binaries are referred
to as case AR -, then the secondary has, by definition, a
(much) smaller mass than the primary, which implies that
its thermal timescale is (much) longer. This means that
for these mass ratios, the secondary is also brought out of
thermal equilibrium and quickly expands (point B in Fig.
4(a)). Soon a contact binary is formed (point C in Fig.
4(a)). Binaries with small mass ratios and small periods which are referred to as case AS binaries as discussed below
- remain semi-detached during this phase. This is because
for these binary systems, the secondary is able to maintain
its thermal equilibrium since the mass and hence thermal
timescale of the secondary are comparable to that of the
primary.
After the primary has transferred so much mass that it has
become the least massive component, the orbit widens again
and hence the primary restores its thermal equilibrium. Mass
transfer is continued in the slow phase, where the primary
transfers mass to the secondary on its own nuclear timescale.
6 Images
extracted
on
01-12-09
from
lecture
notes
used
for
a
binary
stars
course
by
O.R.
Pols, p.
22 Fig.
8.4, which can be found at
http://www.astro.uu.nl/∼pols/education/binaries/lnotes/Binaries
2007.pdf.
5
If the initial mass ratio and initial period are small (case
AS systems), the secondary remains in thermal equilibrium
and expands as it accretes mass from the primary (point B
in Fig. 4(b)). Then the secondary overtakes the primary
(point C in Fig. 4(b)) and eventually a contact binary is
formed. From this it is clear that in case AR systems, mass
transfer can quickly lead to a contact binary, i.e. on the
thermal timescale of the primary (hence the name AR, where
R stands for rapid), while in case AS systems it can take
considerably more time before a contact binary is formed, i.e.
on the nuclear timescale of the primary (hence the name AS,
where S stands for slow). It will turn out that this difference
leads to different compositions of the merger product if the
contact binary were to merge to a single star, as we shall
discuss below in Sect. 3.
3
3.1
mergers have lower hydrogen abundances than components
in binaries that have not merged. We therefore also expect
that both case AS and case AR mergers are more luminous
than single stars with the same mass and binary stars with
comparable mass that do not merge on the main sequence.
3.3
In order to quantify the effect of case AR and case AS mergers, we analyzed a grid of binary models computed by De
Mink et al. (2007). The code used for this grid is an updated version of the STARS/TWIN evolution code which implements convective mixing and overshooting and spin-orbit
interaction by tides. Heat transfer between the binary components is not treated and mass transfer is assumed to be
fully conservative, i.e. no mass is lost during mass transfer.
For more details we refer to Sect. 2 of De Mink’s paper. The
binaries in this grid have SMC metallicity, i.e. Z = 0.004 and
are evolved up to the end of their main sequence. In the grid
there are three parameters: initial primary mass Mp,i , initial
M
mass ratio qi = Mp,i
and initial period Pi which are spaced
s,i
at equal logarithmic intervals:
8
0.10, ... 1.70;
< log10 (Mp,i /M ) = 0.05,
log10 (qi )
= 0.050, 0.075, ... 0.350;
:
log10 (Pi /PZAMS ) = 0.050, 0.10, ... 0.750,
Mergers in binary systems
Assumptions made
There still exists much uncertainty about the merging process of a binary system as merging binaries are only rarely
observed in practice. Therefore we are forced to make the
necessary assumptions. First, we assume that main-sequence
mergers of close binary systems can only originate from contact binaries whose components are both still main-sequence
stars. This, in conjunction with Sect. 2.2, rules out all subcases of case A evolution except case AR and case AS systems. Furthermore, we assume that the merging process is
quick, i.e. occurs on the dynamical timescale of one of the
components and that the stars merge to a single star without
any loss of mass into the interstellar medium. Also, we assume that the merger star has a homogeneous composition,
constant opacity, ideal gas pressure and that it is in radiative
equilibrium. These quite severe assumptions for the merger
star are necessary to justify the use of a homology relation
(Eq. 6), which greatly reduces computing time in the cluster
simulation in Sect. 4 compared to the use of single stellar
evolution models with modified composition.
3.2
Quantitative properties: grid analysis
(4)
hence in total there are over 6000 models. The unit PZAMS is
an approximation of the orbital period at which the initially
more massive component would fill its Roche lobe on the zeroage main sequence for a system with equal masses and was
given approximately by Nelson & Eggleton (2001):
PZAMS /d ∼
=
0.19 (Mp,i /M ) + 0.47 (Mp,i /M )2.33
, (5)
1 + 1.18 (Mp,i /M )2
with day abbreviated as d. For the given mass ranges in Eq.
4, the minimum and maximum values of PZAMS are 0.333 d
and 1.452 d respectively.
Qualitative properties
Using the assumptions made in Sect. 3.1, we can make statements about the merger product composition based on the
properties of case AR and case AS binary systems discussed
in Sect. 2.2. Case AS binaries merge at a much later moment in their evolution than case AR binaries. This implies
that the components of AS systems have lower core-hydrogen
abundances at the time of merging than the components of
AR systems, because stars steadily burn core-hydrogen on the
main sequence. With the assumption of no mass loss, this
implies that the case AS mergers, i.e. mergers that originate
from case AS binary systems, have (much) lower hydrogen
abundances than case AR mergers at the moment of their
formation. This in turn implies (much) larger helium abundances for AS mergers compared to AR mergers and thus
higher mean molecular weights. These higher mean molecular weights imply higher luminosities, for which we refer to
Eq. 6. Thus we expect case AS mergers to be more luminous than case AR mergers. In general, case AS and case AR
Using a self-written code in Fortran 77, we determined
which binary configurations lead to contact binaries on the
main sequence. This was done by comparing the radii of both
binary components to their Roche-lobe radii; we defined the
time of contact of both stars to be the maximum of the times
at which both stars fill their Roche lobes. Since we assumed
that a binary merges if it is a contact binary and that this
happens on a short, i.e. dynamical timescale (Sect. 3.1), we
let the time of merging be equal to the time of contact. We
determined the amount of hydrogen in the merger product
by adding the amounts of hydrogen of both components
at the time of contact, which required our assumption of
no mass loss during the merging process. Thus we could
calculate the hydrogen fraction X of the merger product
using X = Mhydrogen,merger /Mtotal, merger .
We show results of this grid analysis in Fig. 5(a) for
Mp,i = 12.6M . In general the hydrogen fraction decreases
6
the relative size of the convective core is greater for higher
masses, hence a high-mass binary component can burn a
larger fraction of its hydrogen fuel before a merging event
than a low-mass binary component. We see in Fig. 5(b)
that the case AS mergers have a lower hydrogen fraction
than their AR counterparts, as is the case for lower initial
primary masses. The difference between the lowest hydrogen
fractions for both cases is about 0.2.
Mp, i = 12.6 M⊙
AR
AS
Hydrogen fraction log10 (Pi / Pzams) →
→
(a)
Thus, as we have shown qualitatively in Sect. 3.2, we have
also shown quantitatively that mergers of close binaries can
lead to merger stars with rich helium content. More specifically, case AS mergers contain significantly more helium than
case AR mergers as expected. In order to relate these helium abundances to luminosity quantitatively, we recall the
following homology relation given by Kippenhahn & Weigert
(1990),
log10 (qi) →
Mp, i = 50.1 M⊙
L ∝ M 3 µ4 ,
log10 (Pi / Pzams) →
→
(b)
with L luminosity, M mass and µ the mean molecular weight.
This relation is an analytic scaling relation which applies to
main-sequence stars and only gives rough estimates. We note
that our assumptions for the merger product of homogeneous
composition, constant opacity, ideal gas pressure and radiative equilibrium (which we made in Sect. 3.1), are necessary
to justify the use of this homology relation. The mean molecular weight µ is also given by Kippenhahn & Weigert (1990):
Hydrogen fraction AR
AS
(6)
µ=
log10 (qi) →
1
,
2X + 43 Y + 21 Z
(7)
with X the hydrogen fraction, Y the helium fraction and Z
the metallicity. Taking the metallicity Z constant, we can
rewrite Eq. 7 as follows with the relation X + Y + Z = 1
(which is true by the definitions of X, Y and Z):
Figure 5: Plot of results of the grid analysis for (a) Mp,i =
12.6M and (b) Mp,i = 50.1M . Each rectangle corresponds to a
binary evolution model. On the x-axis is the initial mass ratio qi
in log scale and on the y-axis the initial period Pi , also in log scale.
The period has been plotted in units of PZAMS , which is defined
in Eq. 5. The color shading indicates the hydrogen fraction of the
resulting merger product; black signifies that this particular binary
does not merge on the main sequence and green signifies that this
particular binary model was missing from the binary grid. The
regions of binary cases AR and AS have been indicated.
µ=
1
.
2 − 54 Y − 23 Z
(8)
Hence we see that, taking the metallicity Z constant, greater
helium fractions Y imply greater mean molecular weights µ
according to Eq. 8, which in turn imply higher luminosities
L according to Eq. 6. Thus case AS mergers are more luminous than case AR mergers when comparing stars of the
same mass, a property that is thus fully attributed to their
greater mean molecular weights. Also we see that merger
stars are more luminous than single stars and binaries that
do not merge on the main sequence with comparable mass,
as their mean molecular weights are always greater. In Sect.
4 we will apply the results of this grid analysis to a cluster
simulation to be able to make statements about the relative
importance of these mergers among other massive stars (stars
with high luminosity).
with initial period. This is because as the initial period
increases, the initial separation a increases (Eq. 3), the
initial Roche-lobe radii increase (Eq. 2) and the time of
merger increases, hence the binary components have burnt
more hydrogen before a contact situation. This results in a
low hydrogen fraction in the merger product. Furthermore,
it is clear that case AS mergers indeed contain significantly
less hydrogen and thus more helium than their case AR
counterparts. For some AS mergers the hydrogen fraction
can be as low as about 0.4 while the AR mergers do not have
a fraction lower than about 0.5.
4
Cluster simulation
In the introduction we have given a motivation for a simulation of a cluster of stars to see whether the most luminous
main-sequence stars originate from mergers of close binary
systems. In Sect. 4.1 we clarify the methods used in this
simulation and in Sect. 4.2 we present our results.
Next, we show a comparable diagram in Fig. 5(b) for
Mp,i = 50.1M . Here the hydrogen fraction for higher
initial periods is lower compared to the same situation at
lower initial primary masses (Fig. 5(a)). This is because
7
4.1
Method
4.1.1
1
Cluster generation
f (m) 
The cluster simulation consists of 2 · 105 stars with a binary
percentage of 50% which all formed at the same time, an
assumption known as a starburst. We assumed the current
age of the cluster t0 to be known. Furthermore, we let the
ranges of initial parameters Mp,i , qi and Pi in this simulation
be the same as in the binary grid we analyzed (Eq. 4), with
the exception that we extended the period to about 103
days (more precisely: log10 (Pi /PZAMS ) = 3.000). This is to
also account for binaries that do not interact on the main
sequence.
O
We generated the initial star parameters using Monte
Carlo methods and assumed the following distribution
functions. For the initial mass distribution, we assumed
that the probability density is proportional to the mass to
the power -2.3 as found to be valid by Kroupa (2001) for
stars with masses greater than 1.0 M . Furthermore, for the
initial mass ratio distribution (applicable only to binaries),
we assumed that the probability density is constant with
respect to the inverse of the mass ratio, which Goldberg et
al. (2003) showed to be approximately valid for 1 < q < 2.2.
Lastly, for the initial period distribution (binaries only), we
assumed that the probability density is proportional to the
log of the period as found by Mazeh et al. (2007). In symbols:
8
>
<
>
:
dN
;
∝ m−2.3
i
dmi
dN
= C;
dqi0
dN
∝ log10 (Pi ),
dPi
f (m) ≡ x. This inversion yields:
1
ˆ
˜ 1−α
m = m1−α
(1 − x) + xm1−α
.
1
2
For the initial mass ratio distribution, let the range of q
be determined by q1 and q2 where q2 > q1 . The normalized
probability density is given by
˛
˛
dN
dN ˛˛ dq 0 ˛˛
1 dN
C
ρ(q) =
=
= 2 0 = 2
dq
dq 0 ˛ dq ˛
q dq
q
q1 q2 1
(13)
=
,
q2 − q1 q 2
(9)
f (m) ≡
m1
m1−α − m1−α
1
.
ρ(m ) dm = 1−α
m2 − m1−α
1
0
(12)
Thus if a number of N initial masses must be generated with
probability density ρ(m) and range m1 to m2 , we can execute
a simple do-loop N times to generate a random number x
which we can use in conjunction with Eq. 12 to generate a
mass m(x). Hence we refer to Eq. 12 as a generating function.
with q 0 = 1/q; we took the absolute value of the derivative
dq 0 /dq to ensure that the probability density is always
positive. Hence the function f (q), defined similarly as with
the initial mass distribution, is given by:
f (q) =
(1 − α)m−α
dN
≡ ρ(m) = 1−α
,
(10)
dm
m2 − m1−α
1
Rm
with α = 2.3, such that m12 ρ(m) dm = 1. Note that m21−α −
m1−α
< 0 and that (1 − α) < 0 such that ρ(m) > 0 for all
1
m1 < m < m2 . Now we define the function f (m) to be the
integral of ρ(m) of m1 to a value m where m1 < m < m2 :
0
m2
Figure 6: Sketch of the function f (m) as function of m, defined
In the remainder of this section, we shall elaborate on
the technical details of the initial parameter generation. For
the mass distribution, we let7 m1 and m2 be the lowest and
highest occurring initial masses respectively. The normalized
probability density is then given by
m
m 
in Eq. 11.
with N the number of stars, mi the initial primary mass,
qi0 = 1/qi and C a constant.
Z
m1
q2 q − q1
.
q q2 − q1
(14)
Inversion of this function (defining x ≡ f −1 (q)) yields the
generating function:
q=
q1 q2
.
q1 x + q2 (1 − x)
(15)
For the initial period distribution, we generated the periods
in units of P̃ = log10 (P/PZAMS ) where PZAMS was defined in
Eq. 5. Hence the probability density with respect to P̃ is
linear, i.e.
(11)
We have sketched the function f (m) in Fig. 6. Note that
0 ≤ f (m) ≤ 1 for all m1 < m < m2 . Now we associate a
random number x between 0 and 1 with f (m), i.e. we obtain
m from the inverse function of f (m), f −1 (m), where now
ρ(P̃ ) = C̃,
(16)
with C̃ a constant. This leads to the generating function
P̃ = P̃1 (1 − x) + xP̃2 ,
(17)
where P̃1 and P̃2 refer to the lowest and highest periods respectively.
7 To avoid cumbersome notation, we shall omit the index i for
initial in the rest of this section.
8
100000
0
t0
nuc
t
Pi > 8 d
Binaries
Pi < 8 d
Cluster population
15092
Merging on MS
8592
birth of cluster
No merging on MS
76316
15092
Mergers
nuc + tm
tm
23684
8592
Unevolved binaries
merging event
single stars; unresolved binaries
current cluster age
t 
mergers
100000
Single stars
Figure 8: Timeline to illustrate the criteria for the main-sequence
band.
Figure 7: Overview of the cluster population. We indicated the
with X the hydrogen fraction, µ the mean molecular weight
and τnuc, SMC (M ) the nuclear timescale for a single star
with SMC metallicity for particular mass M . We calculated
τnuc, SMC (M ) for a few values of M using the Window to
the Stars interface developed by Izzard & Glebbeek (2006).
To obtain τnuc, SMC (M ) for other masses, we used a linear
interpolation.
absolute numbers of objects within the groups.
4.1.2
Cluster population
We divided the simulated cluster population into three
groups: binary stars with initial periods lower than about
8 days (more precisely: log10 (Pi /PZAMS ) = 0.750), binary
stars with initial periods greater than about 8 days and
single stars. We analyzed the first group, which can lead
to mergers on the main sequence, with De Mink’s grid as
discussed in Sect. 3.3. For this group we used Eq. 7 to
obtain the mean molecular weights and neglected the metals,
i.e. we put8 Z = 0 and used Eq. 6 to obtain the luminosity
of the merger product9 . The second group, which we refer
to as pre-interacting main-sequence binaries or unevolved
binaries, we did not analyze with De Mink’s grid. Instead,
we treated its binaries essentially as single stars. We defined
the luminosity of an unevolved binary to be the sum of
luminosities of both components calculated using Eq. 6. We
also used Eq. 6 for the third group, which consists of single
stars. The reader might wonder why we did not use single
stellar evolution models for the single stars and unevolved
binaries to obtain more accurate results. This is because
we needed to ensure that our comparisons are reliable in
the sense that we used the same method to model all three
groups. We note that we included the binaries of the first
group that do not merge on the main sequence in the second
group. In Fig. 7 we illustrate the cluster population, where
we also include the absolute numbers of objects within the
groups.
We used this nuclear timescale to determine which stars
in the cluster lie in the main-sequence band. The required
criterion depends on the group type; as an illustration, consider the following time line in Fig. 8. At t = 0 the cluster
is formed in an infinitesimal burst of stars. For a single star
or unevolved binary, if the current age of the cluster is t0 and
if the nuclear timescale τnuc is larger than t0 , then this star
is on the main sequence. For a merger star, we must modify
this criterion: the merger star merged at a certain time, say
tm . Thus it is only on the main sequence if t0 lies between tm
and τnuc + tm . The reason why we added tm to τnuc is that
the nuclear timescale must be relative to t = 0 and not to
the time of merger tm . Therefore, expressing these criteria in
inequalities, we have that for main-sequence stars:

t0 < τnuc ,
(G2 , G3 );
(19)
tm < t0 < τnuc + tm ,
(G1 ),
with G1 the group of merger stars, G2 the group of unevolved
binaries and G3 the group of single stars.
4.2
Using the procedure discussed in Sect. 4.1.2, we could
calculate a luminosity for each group member. In Fig. 9,
we show for all three group types the luminosity function,
i.e. the number of objects formed in a small interval of
luminosity ∆ log L/L = 0.1 as function of luminosity. It
is important to note that this figure does not represent
data which could be observed in practice because it does
not contain any time information, i.e. in reality, all the
mergers shown would never exist simultaneously. The aim
of the figure however is to give an impression of the cluster
population and demonstrate the potential importance of
mergers. It is clear from Fig. 9 that for all groups, the bulk
of the stars have low luminosities. This is a result of the
mass distribution: there is a relatively high probability for
a star being a low-mass star (Eq. 9) and low-mass stars
have low luminosities (Eq. 6). There are relatively few
We calculated a nuclear timescale τnuc for the stars of all
groups. The physical meaning of the formula we used is
that the (main sequence) nuclear timescale is the amount of
available nuclear fuel for hydrogen burning divided by the
luminosity. Thus:
τnuc ∝ Xµ−4 τnuc, SMC (M ),
Results
(18)
8 We neglected metals because it proved to be complicated to
obtain compositions of the merger product other than hydrogen
and helium abundances. We expect the error made in neglecting
metals to be only small however.
9 The reference value of µ used in Eq. 6 was taken to be about
0.6, which is approximately the current solar value. For the purposes of this research however (comparing the different groups),
the actual value of this number is not important.
9
(a)
t0 = 5 Myr
Single stars
Unevolved binaries
Merger stars
log10 (N) 
log10 (N) 
Single stars
Unevolved binaries
Merger stars
log10 (L/L⊙) 
log10 (L/L⊙) 
(b)
Figure 9: Luminosity function, i.e. number of objects formed as
t0 = 500 Myr
function of luminosity, of the simulated cluster. All objects have
been counted in bins of bin width ∆ log L/L = 0.1.
log10 (N) 
Single stars
Unevolved binaries
Merger stars
merger stars, this is because of the many criteria for the
existence of these objects: they must originate from a binary
with periods less than about 8 days and with the right
combination of mass ratio and period, i.e. they may not lie
in the black areas in figures like Fig. 5(a) and Fig. 5(b). At
luminosities greater than about 105.3 L , only the merger
stars remain. This is firstly due to the fact that compared
to single stars and unevolved binaries, the merger stars have
a larger mean molecular weight and thus higher luminosity
(Eq. 6). Secondly, when comparing single stars and mergers
of the same mass, the mergers always have larger mass due
to the companion star and hence also larger luminosity.
log10 (L/L⊙) 
Figure 10: Luminosity function, i.e. number of objects as func-
To be able to say more about the relative importance of
the merger stars in the main-sequence band of the cluster
population, we must consider the nuclear timescales, i.e.
crudely evolve the simulated stars. In Fig. 10(a) we plotted
a luminosity function for all three groups for a cluster age
of t0 = 5 Myr, i.e., we only plotted the objects that still
lie in the main-sequence band 5 Myr after the formation of
the cluster, which formed in a starburst by assumption. In
contrast to Fig. 9, this is a figure we expect to be observable
in princible. In Fig. 10(a), there are no merger stars at
luminosities lower than about 103.8 L . This is due to the
fact that for this very young cluster age, only the binaries
with very massive components, with high luminosities,
merge. This is because the more massive a binary, the
faster its (main sequence) evolution and hence the faster
a merger may be formed and appear in the main-sequence
band. From the figure it is clear that the merger stars
are dominant among the high luminosities. All stars with
luminosities greater than about 104.7 L are expected to be
merger stars, although they are not as abundant in quantity
as the lower-mass stars.
tion of luminosity of the simulated cluster for a cluster age of (a)
5 Myr and (b) 500 Myr. All objects have been counted in bins of
bin width ∆ log L/L = 0.1.
We show the same diagram for a cluster age of 500 Myr in
Fig. 10(b). There is a shift for all groups towards lower lumi-
nosities. This is because a high cluster age implies that only
older stars remain in the main-sequence band, which are stars
with lower masses and hence lower luminosities. We see that
interestingly at t0 = 500 Myr the merger-band consists of
two bumps. The reason for lies in the difference between case
AR and case AS mergers. The first bump, centered around
a luminosity of about 101.5 L , is due to case AR mergers;
their number decreases with increasing luminosities. This is
because a higher luminosity corresponds to a greater mean
molecular weight which corresponds to a lower hydrogen fraction. This, according to Figs. 5(a) and 5(b), corresponds to a
higher period (for case AR mergers). Thus there is a decrease
in number with increasing luminosities as there are less stars
with high periods than there are stars with low periods (Eq.
9). The second bump appears at luminosities greater than
about 102.2 L . For these luminosities, the case AS mergers,
stars with higher luminosities than case AR mergers as we
have shown before in Sect. 3.3, appear and ensure that the
10
number of mergers again increases with increasing luminosity.
As the binaries that produce AS mergers do not have a very
wide range of periods (Figs. 5(a) and 5(b)), there is no visible
decrease of number with increasing luminosity in Fig. 10(b)
until no stars remain at all. We see that in Fig. 10(b), as in
Fig. 10(a), merger stars are expected to be dominant among
the brightest stars in the main-sequence band. The overlap
between the other groups in luminosity is even smaller which
makes their presence even more pronounced.
5
6.5
log10 (L/L⊙) 
6.0
Discussion
4.0
3.5
1
1.2
1.4
1.6
1.8
2.0
log10 (M/M⊙) 
Figure 11: Thick red curve: plot of luminosity as function of
mass, made using Window to the Stars, developed by Izzard &
Glebbeek (2006). Dashed line: luminosity as function of mass
according to Eq. 6.
Mass loss and angular momentum
loss
By analyzing a grid of binary models that implements conservative mass transfer (Sect. 3.3), we have implicitly made
the assumption that no mass is lost before a contact binary
is formed. From observations it is known that this is not
necessarily the case however (as shown by e.g. De Mink et
al. (2007)). Grids with non-conservative mass transfer were
also available but we did not consider them. Furthermore,
we assumed the merging process itself to be short-lasting and
without mass loss. Although the assumption of no mass loss
does appear to be severe, hydrodynamic simulations of stellar
collisions have shown that only a modest fraction of mass is
lost as pointed out by Lombardi et al. (2002). It is reasonable
to assume that mass loss in a merging event of binary components is less severe than mass loss in a collision of stars, as
the latter situation is much more violent. Furthermore, in the
hydrodynamic simulations there remains the issue of angular
momentum. It appears that a substantial amount of angular momentum would need to be lost for the remnant star to
rotate at a rotational velocity smaller than or equal to the
critical rotational velocity. Mass loss is the main mechanism
to achieve this loss of angular momentum. This issue also
exists in the merging within a binary system, which we confirmed using data from the binary grid. It appeared that up
to 50% of angular momentum would need to be lost for noncritical rotation of the merger product. Thus it would not
be unreasonable to implement mass loss in the calculation
of merger compositions. The uncertainties involved however
have forced us to implement the assumption of no mass loss
during the merging event in this research.
5.2
5.0
4.5
It is clear that in this research we have made many assumptions and choices that could potentially greatly influence the
results we obtained. In the following section we shall elaborate on these.
5.1
5.5
evolution models.
The dashed line shows the relation
according to Eq. 6, i.e. L ∝ M 3 . We see that the assumed
M 3 -dependence is only approximate: it fails for masses
greater than about 25 M . This is because in high-mass
stars, radiative pressure is considerable, which is not taken
into account in the homology relation since ideal gas pressure
is assumed. We could correct this problem in future research
by interpolating data from stellar evolution models.
An other important assumption that we made, is that the
luminosity remains constant during the main sequence. In
reality however, the luminosity increases slightly on the main
sequence because of the slight increase of mean molecular
weight. As this increase of mean molecular weight during the
main sequence is relatively greater for more massive stars, this
implies that high-mass stars of all group types in Figs. 10(a)
and 10(b) are shifted towards higher luminosities. This would
mean that mergers are less important among the massive stars
than is suggested by these figures. To improve on this, we
could assume the luminosity to increase linearly with time
on the main sequence. A better solution would be to use
stellar evolution models for the merger stars (and hence also
the single stars and unresolved binaries to keep comparisons
fair), although unfortunately this is quite time consuming for
a large cluster population. The latter solution would of course
also remove the objection that homology relations are not
accurate for very high masses and it would remove the need
to interpolate data from existing models as we did e.g. in Eq.
18.
Homology relations
5.3
We frequently used homology relation Eq. 6 to calculate
luminosities. Although the dependence of mean molecular
weight in this relation is quite accurate, even for inhomogeneous stars, as shown by Poelarends (2002), the mass
dependence is not. As an illustration, consider Fig. 11,
which shows the luminosity as function of mass. The thick
red curve shows the relation according to accurate stellar
Cluster simulation
In this research we have assumed that the simulated cluster
formed instantaneously, i.e. in a starburst. It must be
said however that this assumption is not very realistic. In
reality, the stars in a cluster are formed in bursts that can
be short-lasting, i.e. ∼ 5 Myr, or also long-lasting, i.e. ∼ 100
Myr, as pointed out by McQuinn et al. (2009). As a result,
11
high-mass stars could also lie in the main-sequence band for
high cluster ages as they need not have formed at t = 0,
as we assumed here. This would make the difference in
luminosity between mergers, i.e. potential blue stragglers
and massive single stars or unevolved binaries much smaller
than suggested in this research.
Furthermore, we based the choice of initial parameter
ranges in the cluster simulation largely on the initial
parameter ranges of the binary grid we used, which were
given in Eq. 4. Although we increased the maximum initial
period from about 8 to about 103 days, in reality larger
initial periods of course do occur in binary systems. Also
it must be noted that the assumed period distribution is
only valid for 2 to about 10 days as shown by Mazeh et al.
(2007). According to Goldberg et al. (2003) the distribution
does vary from being flat in log P for periods higher
than about 10 days. The same objection of limited parameter ranges applies to the initial masses and initial mass ratios.
become contact binaries quickly and slowly respectively
after formation of the binary. As the result of merging
after an appreciable amount of hydrogen has been burnt,
case AR and case AS mergers contain significantly lower
hydrogen abundances than single stars on the main sequence.
Furthermore, AS mergers contain less hydrogen than their
AR counterparts. Homology relations then imply that these
lower hydrogen abundances, i.e. higher mean molecular
weights, lead to more luminous stars.
In this research, we could easily extend the maximum
initial period because high-initial-period binaries are not
expected to merge on the main sequence and hence do not
need to be analyzed with help of the binary grid. In other
words, they will certainly not become merger stars as defined
in Sect. 4.1.2. This does not apply to the initial mass and
initial mass ratio however, as binaries with these parameters
outside the grid could very well still merge on the main
sequence and hence become merger stars. It is possible
however to make a crude assumption for the mass ratios.
In the grid used, we considered mass ratios smaller than
about 2.2, i.e. only about 50% of all binaries with respect
to the mass ratio. As it is reasonable to state from Figs.
5(a) and 5(b) that binaries with these higher mass ratios
and sufficiently low periods are also case AR binaries, we
expect these binaries also to become contact binaries and
thus merge and have properties comparable to the case AR
mergers present in the grid.
Furthermore, from a simulation of a cluster of stars in
Sect. 4, we have shown that these AR and AS mergers are
very important if compared with single stars and unevolved
binaries.
This is because they dominate the brightest
regions in the main-sequence band for young clusters, as
well as older clusters. For young clusters, we expect that
all stars that are on the main sequence with luminosities
greater than about 105 L (Fig. 10(a)), originate from
mergers of close massive binaries. This would imply that
all blue stragglers in the HR-diagram of N 11 (Fig. 1),
which is an example of a young cluster, are mergers of these
binaries. This rather bold statement depends severely on
our assumptions made however. One of the main objections
is that we have ignored other possible binary interactions
that do not lead to a merging event but could produce blue
stragglers, e.g. accretion blue stragglers, or explanations
that do not even require a binary explanation, such as
stellar collisions. Furthermore we have made many other
assumptions, such as instantaneous star formation (starburst) which may significantly change our conclusions as well.
To finish up, although in this research we have gained more
insight in the possible consequences of mergers of massive
main-sequence binaries, more effort needs to be made before
the results from this research can be compared to empirical
data.
7
Finally, as we have already mentioned in the introduction,
mergers of massive binaries are only one of the plausible explanations for the existence of blue stragglers. Other scenarios, such as accretion blue stragglers (mass transfer in binaries
without a merging event) or stellar collisions, may also play
a crucial role. To examine the relative importance of these
groups, we would also have to model these in the cluster simulation. Then by experimenting with different assumptions for
all groups, i.e. changing their relative frequency and comparing the resulting luminosity distribution with observational
data, we could in principle obtain more information of the actual relative importance of mergers of massive main-sequence
binaries.
I would like to thank my supervisors Onno Pols and Selma
de Mink for their excellent guidance in this bachelor research
project. I hope that they will be able to make use of the
results of this research in the future. Furthermore I would
like to thank Joke Claeys for taking me to the Stellar Mergers
workshop in Leiden, her assistance in using Window to the
Stars and the calculation of nuclear timescales from stellar
evolution models. Last but not least, I would like to thank
the participants of the stellar evolution and hydrodynamics
weekly group meetings who provided valuable comments.
8
6
Acknowledgements
Conclusion
References
Eggleton, P.P. ApJ, 1983, 268, 368
From Sect. 2.2, it has become clear that mass transfer in
relatively close main-sequence binaries may lead to contact
binary situations which may lead to merger stars. We have
identified two types of binaries that may lead to contact
binaries on the main sequence: case AR and case AS, which
Evans, C. J., Lennon, D. J., Smartt, S., J. & Trundle,
C. 2006, A&A, 456, 635
Goldberg, D., Mazeh, T., Latham, D. 2003, ApJ, 591,
399, 404
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Han, Z., Chen, X., Zhang, F., Podsiadlowksi, Ph. 2009,
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Hunter, I., Brot, I., Lennon, D.J., Langer, N. et al. 2008,
ApJL, 676, 29
Izzard, R., Glebbeek, E. 2006, NA, 12-2, 161-163
Kippenhahn, R., Weigert, A. 1990, Stellar Structure and
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Kroupa, P. 2001, MNRAS, 322, 242
Lombardi, J.C., Warren, J.S., Rasio, F.A., Sills, A.,
Warren, A.R. 2002, ApJ, 568, 939
Mazeh, T., Tamuz, O., North, P. 2007, IAU, 240, 234
McQuinn, K. B. W., Skillman, E. D., Cannon, J. M.,
Dakanton, J. J., Dolphin, A., Stark, D., Weisz, D. 2009,
ApJ, 695, 565
Mink, S.E. 2005, Msc thesis Efficiency of mass transfer
in close massive binaries, 10
Mink, S.E., Pols, O.R., Hilditch, R.W. 2007, A&A, 467,
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Nelson, C.A., Eggleton, P.P. ApJ, 552, 194-199
Poelarends, A.J. 2002, Msc thesis
Schaerer, D., Meynet, G., Maeder, A., & Schaller, G.
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13
9
Appendix: flow diagram
For the interested readers, we provide a flow diagram of the technical details of the structure of the grid analysis and the
cluster simulation in Fig. 12.
(convervative)
list.f
GRID
Key
mergercompgridlist.txt
Text files
M, Q, P
⋮ ⋮ ⋮
mergercomp.f
Text files (important)
mergercompgridoutput.txt
populationgenerator.f
populationlist.txt
M, Q, P
⋮ ⋮ ⋮
Fortran 77 programmes
M, Q, P, mtot, mH, mHe, m , tm ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
⋮ ⋮
*
N
P < 750
P > 750
awk scripts
population analysis.f
group1unevolved.txt
M, Q, P
⋮ ⋮ ⋮
single.txt
unevolved binaries.txt
M, Q, P
⋮ ⋮ ⋮
M, Q, P
⋮ ⋮ ⋮
finaloutput.txt / mergers.txt
mtot, mH, mHe, mN*, tm ⋮ ⋮ ⋮
⋮ ⋮
/bigG/
Lbinmergers.awk
Lbinsingle.awk
Lbinunevolved.awk
DL01vt0single.txt
DL01vt0unevolved.txt
DL01vt0merger.txt
L, t0, Ns
⋮ ⋮⋮
L, t0, Nu
⋮ ⋮⋮
L, t0, Nm
⋮ ⋮⋮
DL01vt0joined.txt
L, t0, Ns, Nu, Nm
⋮ ⋮⋮ ⋮ ⋮
/outputs/
Figure 12: Flow diagram of the technical details of the structure of the grid analysis and the cluster simulation. The large dashed
boxes indicate that different folder locations were used. Capital letters used such as M indicate that the unit is in log units and possibly
multiplied by a factor, i.e. 100 (for the masses) or 1000 (for the mass ratios and the periods). Small letters for the masses m indicate
that these are in conventional units, i.e. M . The astrix * in the nitrogen masses indicates that these numbers are uncertain and thus
should not be relied upon. The symbols t0 and tm stand for cluster age and time of merging respectively. The value of the size of the
bins of luminosity in the .awk-files can be modified (deltalogL), as well as the ranges of t0 (t0begin, t0end). For more details and/or
elaboration, contact the author at [email protected].
14