1. Introduction
|
|
The self-enrichment of galactic halo globular: a clue to their formation ? |
* Maître de Recherches au Fonds Natrional de la
Recherche Scientifique (Belgium)
** Chercheur qualifié au Fonds Natrional de la
Recherche Scientifique (Belgium)
| Abstract |
We present a model of globular cluster self-enrichment. In the
protogalaxy, cold and dense clouds embedded in the hot
protogalactic medium are assumed to be the progenitors of galactic halo
globular clusters.
The massive stars of a first generation of metal-free stars, born in
the central areas of the proto-globular cluster clouds, explode as
Type II supernovae.
The associated blast waves trigger the expansion of a supershell, sweeping
all the material of the cloud, and the heavy elements released by these
massive stars enrich the supershell. A second generation of stars is
born in these compressed and enriched layers of gas. These stars
can recollapse and form a globular cluster.
This work aims at revising the most often encountered argument against
self-enrichment, namely the presumed ability of a small number of
supernovae to disrupt a proto-globular cluster cloud.
We describe a model of the dynamics of the supershell and of its
progressive chemical enrichment.
We show that the minimal mass of the primordial cluster cloud
required to avoid disruption by several tens of Type II supernovae is
compatible with the masses usually assumed for
proto-globular cluster clouds. Furthermore,
the corresponding self-enrichment level is
in agreement with halo globular cluster metallicities.
Key words: globular clusters: general - Galaxy: evolution -
supernovæ: general - ISM: bubbles - population III
| Table of contents |
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1. Introduction
|
The study of the chemical composition and dynamics of the galactic halo
components, field metal-poor stars and globular clusters (hereafter GCs),
provides a
natural way to trace the early phases of the galactic evolution. In an
attempt to get some new insights on the early galactic nucleosynthesis,
accurate relative abundances have been obtained from the analysis of
high resolution and high signal-to-noise spectra for a sample of 21
mildly metal-poor stars (Jehin et al. 1998, 1999).
The correlations between the relative
abundances of 16 elements have been studied, with a special emphasis on the
neutron capture ones. This analysis reveals the existence of two
sub-populations of field metal-poor stars, namely Pop IIa and
Pop IIb. They differ by the behaviour of the s-process
elements versus the 
and
r-process elements.
To explain such correlations, Jehin et al. (1998, 1999) have suggested
a scenario for the formation of metal-poor stars which closely
relates the origin of these stars to the formation and the evolution of
galactic globular clusters.
At present, there is no widely accepted theory of globular cluster
formation. According to some scenarios,
GC formation represents the high-mass tail of star cluster
formation. Bound stellar clusters form in the dense cores of much
larger star-forming clouds with an efficiency of the order
10-3 to 10-2 (Larson, 1993).
If GCs form in a similar way, the total
mass of the protoglobular clouds should therefore be two or three orders of
magnitude greater than the current GC masses, leading to a total mass of
108 
. Harris and Pudritz (1994) have
investigated the GC formation in such clouds which they call SGMC
(Super Giant Molecular Clouds). The physical conditions in these SGMC
have been further explored by McLaughlin and Pudritz (1996a).
Another type of scenarios rely on a
heating-cooling balance to preserve a given temperature
(of the order of 104K)
and thus a characteristic Jeans Mass at the protogalactic epoch.
In this context, Fall and Rees (1985) propose that GCs would form in the
collapsing gas of the protogalaxy. During this collapse, a thermal
instability triggers the development of a two-phase structure, namely cold
clouds in pressure equilibrium with a hot and diffused medium.
They assume that the temperature of the cold clouds remains at
104K since the cooling rate drops sharply at this temperature in
a primordial gas. This assumption leads to a characteristic mass of order
106 for the cold clouds.
However, this temperature, and therefore the characterictic mass,
is preserved only if there is a flux of UV or X-ray radiation able
to prevent any H2 formation, the main coolant in a metal-free
gas below 104K.
Provided that this condition is fullfilled, and since the characteristic
mass is of the order of GC masses (although a bit larger, but see
section 5.2.1),
Fall and Rees identify the cold clouds with the progenitors
of GCs. Several formation scenarios have included their key idea :
cloud-cloud collisions (Murray and Lin 1992), self-enrichment model
(Brown, Burkert and Truran 1991, 1995).
According to the scenario suggested by Jehin et al. (1998, 1999),
GCs may have undergone a Type II supernovae phase in their early history.
This scenario appears therefore to be linked with the self-enrichment
model developed by Brown et al. within the context of the Fall
and Rees theory.
Following Jehin et al., thick disk and field halo stars were born in
globular clusters from which they escaped either during an early disruption
of the proto-globular cluster (Pop IIa) or through a later disruption
or evaporation process of the cluster (Pop IIb). The basic idea is that
the chemical evolution of the GCs can be described in two phases.
During phase I, a first generation of metal-free stars form in the
central regions of proto-globular cluster clouds (hereafter PGCC). The
corresponding massive stars
evolve, end their lives as Type II supernovae (hereafter SNeII) and
eject , r-process and possibly a
small amount of light s-process elements into the interstellar
medium. A second generation
of stars form out of this enriched ISM. If the PGCC get disrupted,
those stars form Pop IIa. If it
survives and forms a globular cluster, we get to the second phase where
intermediate mass stars reach the AGB stage of stellar evolution, ejecting
s elements into the ISM through stellar
winds or superwind events. The matter released in the ISM by AGB stars
will be accreted by lower mass stars, enriching their external layers
in s elements. During the subsequent dynamical evolution of the
globular
cluster, some of the surface-enriched low-mass stars evaporate from the
cluster, become field halo stars and form PopIIb.
Others studies have already underlined the two star generations concept :
Cayrel (1986) and Brown et al. (1991, 1995) were pioneers in this
field. Zhang and Ma (1993) have demonstrated that no single star formation
can fit the observations of GCs chemical properties. They show that there
must be
two distinct stages of star formation : a self-enrichment stage (where
the currently
observed metallicity is produced by a first generation of stars) and a
starburst stage (formation of the second generation stars).
These two generations scenarios were marginal for a long time.
Indeed, the major criticism of such globular cluster self-enrichment model
is based
on the comparison between the energy released by a few supernova explosions
and the PGCC gravitational binding energy : they are of the same order of
magnitude. It might seem, therefore, that proto-globular cluster clouds
(within the context of the Fall and Rees theory)
cannot survive a supernova explosion phase and are disrupted (Meylan and
Heggie, 1997). Nevertheless,
a significant part of the energy released by a supernova
explosion is lost by radiative cooling (Falle, 1981) and the kinetic
energy fraction interacting with the ISM must be reconsidered.
While Brown et al. (1991, 1995) have mostly focused on the
computations of
the supershell behaviour through hydrodynamical computations, we
revisit some of the ideas that have been used against the hypothesis
of GC self-enrichment. In this first paper, we tackle
the questions of the supernova energetics and of the narowness of GC
red giant branch.
Dopita and Smith (1986) have already addressed the first point from a
purely dynamical point of view. In their model,
they assume the simultaneity of central supernova explosions and they use
the Kompaneets (1960) approximation to describe the resulting blast
wave motion. During this progression from the central regions to the edge
of the PGCC, all the material encountered by the blast wave is swept up
into a dense shell.
They demonstrate that, when the shell emerges from the cloud, its kinetic
energy, based on the number
of supernovae that have exploded, is compatible with the gravitational
binding energy of a cloud whose mass is more or less
107 .
When the kinetic energy of the emerging shell is larger than the
binding energy of the initial cloud, this cloud is assumed to be disrupted
by the SNeII. There is therefore a relation
between the cloud mass and the maximum number of supernovae
it can sustain without being disrupted. However, a
107 cloud
is more massive than the PGCCs considered within the Fall and Rees theory.
We derive here a similar relation based on
the supershell description (Castor, McCray, Weaver, 1975)
of the central supernova explosions.
Contrary to the Kompaneets approximation, this theory allows us to take
into account the existence of a mass spectrum for the massive supernova
progenitors and, therefore, the spacing in time of the explosions.
In addition to the above dynamical constraint, we also establish a chemical
one.
For a given mass of primordial gas, we compute the maximum number of
supernovae
the PGCC can sustain and the corresponding self-enrichment level
at the end of the supernova phase. We show that the metallicity
reached is compatible with the metallicity observed in
galactic halo globular clusters.
The paper is organized as follows. In section 2,
we review the observations
gathered by Jehin et al.² (1998, 1999) and the scenario proposed
to explain them.
In section 3, we describe the PGCCs, the first
generation of metal-free stars, and the supershell propagation
inside PGCCs due to SNeII explosions. In section 4,
we show that the disruption criterion proposed by Dopita and Smith (1986),
here computed with the supershell theory, gives the correct
globular cluster metallicities. In section 5,
we discuss the sensitivity of our model to the first generation
IMF parameter values and we examine the implications of an important
observational constraint, the RGB narrowness noticed in most globular
cluster Color Magnitude Diagrams (CMDs).
Finally, we present our conclusions in section 6.
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2. The EASE scenario
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2.1. Observational results
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Jehin et al. (1998, 1999) selected a sample of 21 unevolved metal-poor stars with roughly one tenth of the solar metallicity. This corresponds to the transition between the halo and the disk. All stars are dwarfs or subgiants, at roughly solar effective temperature and covering a narrow metallicity range. High quality data have been obtained and a careful spectroscopic analysis was carried out. The scatter in element abundances reflects genuine cosmic scatter and not observational uncertainties. Abundances of iron-peak elements (V, Cr, Fe, Ni), elements (Mg, Ca, Ti), light s-process elements (Sr, Y, Zr), heavy s-process elements (Ba, La, Ce), an r-process element (Eu) and mixed r-,s-process elements (Nd, Sm) have been determined. Among these data, Jehin et al. (1998, 1999) have found correlations between abundance ratios at a given metallicity. If some elements are correlated, they are likely to have been processed in the same astrophysical sites, giving fruitful information about nucleosynthesis. The following results were obtained (Jehin et al. 1999) :
element Ti : most
points are located on a single straight line
with a slope 
1 ending with a
clumping at the maximum value of [Ti/Fe] -0.6.
The two-branches diagram describes a relation that applies only to
metal-poor stars. Higher metallicity stars do not fit the two-branches
diagram .
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2.2. Interpretation: two-phases globular cluster
evolution
|
We associate the two branches of the diagram with two distinct chemical
evolution phases of globular clusters, namely a SNII phase
(phase I) and an AGB wind phase (phase II).
Phase 1
We assume the formation of a first generation of stars in the central
regions, the densest ones, of a
proto-globular primordial gas cloud. The most massive stars of this first
generation evolve and become supernovae, ejecting
and r-process
elements into the surrounding ISM. These supernova explosions also
trigger the formation of an expanding shell, sweeping all the PGCC material
encountered during its expansion and decelerated by the surrounding ISM.
In this supershell, the supernova ejecta mix with the ambient ISM,
enriching it in and r-process
elements. The shell constitutes a dense medium since it contains all the
PGCC gas in a very thin layer (a few tenths of parsecs) (Weaver et al.,
1977). This favours the birth of a second generation of
stars (triggered star formation) with a higher star formation
efficiency (hereafter SFE) than the first
one (spontaneous star formation).
There are today several observational evidence of triggered star formation
(Fich, 1986; Walter et al., 1998; Blum et al., 1999).
For instance, within the violent interstellar medium of the nearby dwarf
galaxy IC2574,
Walter et al. (1998) have studied a supergiant HI shell obviously
produced by the combined effects of stellar winds and supernova explosions.
A major star formation event (equivalent to our first generation) has
likely taken
place at the center of this supershell and the most massive stars have
released energy into the ISM. This one, swept by the blast waves
associated with the first supernova explosions, accumulates in the form of
an expanding shell surrounding a prominent HI hole.
On the rim of the HI shell, H
emissions reveal the existence of
star forming regions (equivalent to our second generation).
Actually, these regions of HI accumulated by the sweeping
of the shell have reached densities high enough for a secondary star
formation to start via gravitational fragmentation.
In our scenario, the second generation of stars formed in this dense
and enriched shell makes up the proto-globular cluster. If the shell
doesn't recontract, these stars
form PopIIa and they appear somewhere on the slowly varying branch
depending on the time at which the second generation formation has
occurred. When all stars more massive than
9
have exploded as supernovae,
the and r element synthesis
stops, leading to a typical value of
[/Fe].
Our scenario requires this typical value of
[/Fe] to be the maximum
value observed in the two-branches diagram: the end of the supernova
phase must correspond to the bottom of the vertical branch.
If the proto-globular cluster survives the supernova phase,
the shell of stars will recontract and form a globular cluster.
Phase 2
After the birth of the second generation, intermediate mass stars evolve
until they reach the asymptotic giant branch where they enrich their
envelope in s-process elements through dredge-up phases during
thermal pulses. These enriched envelopes are ejected into the ISM through
stellar winds. Lower mass stars in the globular cluster can accrete this
gas (Thoul et al., in preparation) :
the s-process element enrichment occurs only in the external layers.
With time, some of those stars can be
ejected from the globular clusters through various
dynamical processes, such as evaporation or disruption, and populate the
galactic halo. These stars account for our PopIIb. Theoretically foreseen
for a long time (see for example Applegate, 1985; Johnstone, 1992;
Meylan and Heggie, 1997),
these dynamical processes dislodging stars from
globular clusters begin to rely upon observations.
De Marchi et al. (1999) have observed the
globular cluster NGC 6712 with the ESO-VLT and derived its mass
function. Contrary to other globular clusters, NGC 6712 mass function
shows a noticeable deficit in stars with masses below
0.75 .
Since this object, in its galactic orbit, has recently penetrated very
deeply into the Galactic bulge, it has suffered tremendous gravitational
shocks. This is an evidence that tidal forces can
strip a cluster of a substantial portion of its lower mass stars,
easier to eject than the heavier ones.
Our scenario requires that a significant fraction of the field stars now
observed in the halo have been evaporated from globular clusters at an
earlier epoch.
Indeed, Johnston et al. (1999) claim
that GC destruction processes are rather efficient : a significant
fraction of the GC system could be destroyed within the next Hubble time.
However, McLaughlin (1999) and Harris and Pudritz (1994) argue that
these various destructive mechanisms are important only for low
mass clusters. Therefore, these clusters cannot have contributed
much to the total field star population because of their small
size. As one can see, the origin of the field halo stars
is still a much debated question.
In relation with the different steps proposed above to explain the
observations, the scenario was labeled EASE which stands for
Evaporation/Accretion/Self-Enrichment
(Fig. 2).
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3. Model description
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3.1. The proto-globular cluster clouds
|
According to Fall and Rees (1985), PGCCs are cold
(Tc 104K)
and dense primordial gas clouds in pressure equilibrium with a hot
(Th 2 × 106K) and
diffused protogalactic medium.
As already mentioned in section 1, the PGCCs can be maintained at
a temperature of 104K only if some external heat sources were
present at the protogalactic epoch. Fall and Rees (1988) and Murray and
Lin (1993) have proposed that the UV flux
resulting from the hot protogalactic background could be sufficient to
offset the cooling below 104K in a gas with a metallicity less
than [Fe/H] -2.
As the PGCC is assumed to be made up of primordial gas
(we deal with a self-enrichment model and not a
pre-enrichment one), we will suppose it is indeed the case.
Within the context of this preliminary model,
the following assumptions are made :
(r) scales as r-2 :
, T and
Ph
are respectively the mean molecular weight
( = 1.2), the
temperature of the PGCC and the pressure of the
hot protogalactic medium confining the PGCC. Equations (1) and (2)
lead to a relation between the radius and the cubic root of the
mass of the PGCC :
.
Upon the assumption of a temperature of 104K for the cold
clouds, 
is defined by 
,
our determination is based on protogalactic conditions.
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3.2. The formation of the first generation
|
It is well known that star formation can only occur in the coolest and
densest regions of the ISM.
We assume that the UV external heating provided by the hot
protogalactic background is shielded by the bulk of the PGCC gas.
Therefore, H2 formation and thermal cooling are assumed
to occur only near the center
of the PGCC : the formation of the star first generation takes
place in the PGCC central area, the densest and the coolest regions
of the cloud. For the value of the SFE, i.e. the
ratio between the mass of gas converted into stars and the total mass of
gas, we refer to Lin and Murray (1992). Their computation shows that the
early star formation in protogalaxies was highly inefficient, leading to a
SFE not higher than one percent. The mass spectrum is described with the
following parameters :
,
)
have very thin shells of heavy elements when they explode
and contribute only very slightly to the chemical enrichment even if they
are numerous. However, their dynamical impact on PGCC must be taken into
account (stars with masses between ml2 and mu
end their live as supernovae,
but only the ones with masses between ml3 and mu
participate to the chemical enrichment).
|
3.3. Supershell propagation
|
The model of Castor et al. (1975) primarily describes the evolution of
a circumstellar shell driven by the wind of an early-type star.
Their study can be extented to multiple supernova shells if the supernova
progenitors (the first generation massive stars) are closely associated.
In this case, all supernova shells will merge into a single supershell
propagating
from the center to the edge of the PGCC. Following the remarks of the
previous section, we assume that this is indeed the
case.
The blast waves associated with the first supernova explosions sweep the
PGCC material in a thin, cold and dense shell of radius
Rs and velocity 
.
This shell surrounds a hot and low-density region, the bubble, whose
pressure acts as a piston driving the shell expansion through the
unperturbed ISM. The following equations settle the expansion law
Rs(t) of the shell during its propagation in a given PGCC
(Castor et al., 1975; Brown et al., 1995) :
and the dominant energy loss of the bubble comes from
work against the dense shell, hence the total energy of the bubble
Eb is the solution from 
supplied by the supernova explosions is constant in
time, then an analytical solution of the type
Rs(t) = v × t describes
the shell motion. In this expression, v is the constant speed of
the shell during its progression through the PGCC. For an isothermal cloud
of temperature T and mean molecular weight
, relations (7-10)
show that v
depends on the PGCC mass and radius and on the supernova rate :
is set to 1.2,
the expansion speed v10 is in units of
10km.s-1 (the sound speed value in a medium of this
temperature), M6 is the mass of the cloud in
units of millions solar masses, N is the total number of SNeII,
E51 is the energy provided by each Type II supernova explosion
in units of 1051 ergs and
t6 is the
supernova phase duration expressed in millions years. Equation (13)
shows that the velocity of the shell depends on the number of SNeII
and is independent on the hot protogalactic background pressure.
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4. Level of self-enrichment
|
We have assumed that stars form in the central regions of
a given PGCC. This first generation of stars is spontaneous, i.e. not
triggered by any external event, a shock wave for instance. This results
in a low SFE. After a few millions years,
the massive stars
(9 < M ;< 60 )
end their lives as SNeII. The consequences are :
|
4.1. Dynamical constraint
|
As the energy released by one SNII is typically 1051 ergs
(E51 = 1), the energy of a few supernova explosions
and the gravitational binding energy of a PGCC are of the same order of
magnitude. This is the major argument used against self-enrichment.
Actually, it is often argued that successive supernovae will disrupt the
proto-globular cluster cloud.
To test whether this idea is right or not, we can compare the gravitational
energy of the cloud to the kinetic energy of the expanding shell, supplied
by the supernova explosions, when it reaches the edge of the cloud.
Indeed, taking the following criterion for disruption :
and te is the time at which
the shell reaches the edge of the cloud, i.e. the time at which the whole
cloud has been swept-up in the shell, we have
t6 = 30).
Therefore, a PGCC, here described by a
pressure-truncated isothermal
(T 104K)
sphere of gas in
hydrostatic equilibrium, can sustain {\it numerous} SNII explosions.
|
4.2. Chemical constraint
|
The first generation of stars is not triggered by any event (shock wave
for instance) and therefore the SFE is likely to be be very low.
The typical halo metallicity of [Fe/H] ~ -1.6 must however be
reached despite this low first
generation SFE. We now show that this is indeed the case.
We compute the relation between the mass of primordial gas
and the number of first generation supernovae for two different
cases. In the first case, we assume a given SFE
(plain curve in Fig. 3),
while in the second one, we impose the final metallicity (dashed curves in
Fig. 3).
In what follows, all the masses are in units of one solar mass.
The mass distribution of the first generation of stars obeys the
following power-law IMF,
is the
power-law index and k is a coefficient depending on the total mass
of the first generation. 
is the solar heavy element mass
abundance. = 2.35 (Salpeter 1955),
ml1 = 3(),
ml2 = 9(),
ml3 = 12(),
mu = 60(),
M* / M = 0.01 and three different values for
the final metallicity : [Fe/H] = -2, -1.5 and -1.
We will justify these values for the IMF parameters in the
next section.
The point represents the dynamical constraint (17), namely
the maximum number of SNeII a PGCC can sustain without disruption.
The PGCC mass is set by the value of Ph (Eq. 6).
The points located left/right
correspond respectively to stable/unstable PGCCs. The last ones
can't give rise to GCs as, following Eq. (14), they are disrupted.
Similarly, the clouds with masses and number of SNeII above/under the
plain curve (SFE=0.01) correspond to clouds with SFEs smaller/greater
than one percent. The three dashed curves represent relations between
the primordial gas mass and the number of SNeII at a given metallicity.
From top to bottom, these metallicities are [Fe/H] = -1, -1.5,
and -2. We deduce from Fig. 3 that we can
achieve a metallicity [Fe/H] ~ -1.5 with an SFE of one percent.
Hence, the mean metallicity of the
galactic halo can be explained by the activity of a spontaneous first
generation of stars, even with a low SFE. Furthermore, the graphical
comparison of the dynamical constraint with the set of curves corresponding
to the three different metallicities shows that the dynamical constraint
is also compatible with the metallicities of some halo globular clusters
. For instance, under a hot background pressure of
8 × 10-11
dyne.cm-2, the mass of a PGCC is
1.2 × 106
and the metallicity that can be achieved through self-enrichment is
-1.75. Table 1 illustrates the sensitivity
of the model to the choice of Ph.|
5. Discussion
|
|
5.1. The IMF of the first generation
|
The first generation of stars forms out of a gas very poor, or even free, in heavy elements. We now examine the consequences for our model.
|
5.1.1 The shape of the IMF in a
metal deficient medium
|
As already mentioned in section 3.2,
we assume that the gas
temperature can decrease significantly below 104K in the
central regions of the PGCC only.
We now focus on what might happen in this central region where
star formation is expected.
According to Larson (1998), the Jeans scale can be identified
with an intrinsic scale in the star formation process.
It is defined as the minimum mass that a gas clump must have in order for
gravity to dominate over thermal pressure (although the thermal
Jeans mass is not universally accepted as relevant to the present-day
formation of stars in turbulent and magnetized molecular clouds).
It scales as
and decline into the brown dwarf regime.
The uncertainty about the behaviour at the lower mass end comes from the
fact that the IMF below 0.1
remains very poorly known. Whatever the exact solution, a single
power-law Salpeter IMF can be ruled out at masses lower than
1 , the IMF slope
becoming clearly much smaller than the Salpeter value.
The Larson IMFs (21) and (22) are in agreement with the observations since
they reproduce the Salpeter's behaviour for large
masses while they allow a large range of possibilities
at the lower mass end depending on the value adopted for the mass scale
m1. The essential result is to alter the relative
number of low-mass stars compared with the number of more massive ones.
If the mass scale was higher at early times, relatively fewer low-mass
stars would have been formed at these times.
star. In other words, instead of any flattening or decline of the
IMF, there is in this case a sharp cut off at the low mass end.
|
5.1.2 The star formation efficiency
|
The star formation efficiency is an important parameter, since the final mass fraction of heavy elements released in the primordial medium depends linearly on it. How stars are formed out of a gaseous cloud is still poorly known and it is not easy to estimate the value of the SFE. One of the most crucial step in the star formation process is the creation of molecular hydrogen (Lin and Murray 1992). H2 cooling results in a rapid burst of star formation which continues until massive stars have formed in sufficient number to reheat the surrounding gas. The massive stars produce a UV background flux which destroys the molecular hydrogen by photodissociation and shuts down further star formation. Applying this principle of self-regulated star formation by negative feed back to a protogalactic cloud, Lin and Murray (1992) have calculated the UV flux necessary to destroy the molecular hydrogen and the required number of massive stars to produce this UV flux. Finally, the mass and the SFE of this first generation of stars in the protogalaxy are estimated. They find a value of the order of one percent. Under the asumed IMF, an SFE of one per cent corresponds to a metallicity of -1.5 (see Fig. 3).
|
5.1.3.The upper limit of the mass spectrum
|
In the model we have adopted in section 4,
the mass of heavy elements released
by a star is proportional to its total mass (see Eq. (20)).
However, above a critical mass mBH, a star can form a black hole
without ejecting the heavy elements it has processed. This critical mass
is given by
mBH =50 ± 10
(Tsujimoto et al., 1995). Moreover, Woosley and
Weaver (1995) have shown that zero initial metallicity stars have a final
structure markedly different from solar metallicity stars of the
same mass. The former ones are more compact and larger amounts of
matter fall back after ejection of the envelope in the SN explosion.
In this case, more heavy elements are locked in the remnant left by the
most massive stars. In order to evaluate the consequences for our model,
we now define two different upper mass limits for the IMF. The mass of
the most massive supernova contributing to the
enrichment of the ISM is chosen to be
mu1 = 40 .
But the more
massive stars will have a dynamical impact on the ISM and
contribute to the trigger and the early expansion of the supershell,
even though they will not contribute to the self-enrichment.
All stars with masses between ml2 and mu2 end their
lives as supernovae, but only the ones with masses between ml3
and mu1 contribute to the PGCC self-enrichment.
We adopt
mu2 = 60
as the mass of the most massive supernova
progenitor. This value is the same as in
section 4.2 and therefore
the plain curve in Fig. 3 (given SFE) is
not modified. Keeping the same SFE, if we decrease mu1 from
60
to 40 ,
there is a reduction of ~ 0.2 dex in the final metallicity
(Fig. 5).
An IMF with a Salpeter slope doesn't favour at all the highest mass stars
and these stars are quite rare compared to the less massive supernovae.
This explains why the final metallicity is not strongly dependent on
the value of mu1.
|
5.1.4. The lower limit of the mass spectrum
|
We have adopted the point of view of Nakamura and Umemura (1999) who
assume that there is a sharp cutoff at the lower mass end of the IMF
(ml1 = 3 ).
If we consider a pure Salpeter IMF, this
parameter is very critical. Indeed, if we take
ml1 = 0.1
instead of 3, while keeping the SFE unchanged, the mass of heavy elements
ejected by SNeII is decreased by a factor of four, leading to
a decrease in metallicity of 0.6 dex.
The Larson's modified IMFs provide less sensitive results.
|
5.1.5. The slope of the IMF
|
Following Larson (1998), we have used the Salpeter value. Changing the slope of the IMF will have the same consequences as changing the value of the lower mass cut off. If we decrease the slope, we will increase the ratio of high mass stars over low mass stars. The same result can be obtained by increasing the lower mass cut off of the mass spectrum.
|
5.2. Observational constraint:
the RGB narrowness
|
With the exception of 
Cen,
and perhaps M22, galactic globular clusters share the
common property of a narrow red giant branch, indicative of chemical
homogeneity within all stars of a given globular cluster.
This observational property is also
often used as an argument against self-enrichment.
We now show that self-enrichment and a narrow RGB are in fact compatible.
|
5.2.1. The mass of the first
generation stars
|
If we adopt the lower mass limit for the mass spectrum of initially
metal-free stars proposed by Nakamura and Umemura (1999), we get rid of one
of the major arguments against self-enrichment scenarios in globular
clusters :
the existence of two distinct generations of stars with clearly different
metallicities. Indeed, if the first generation of stars is biased towards
high mass stars as previously suggested, these stars are no more
observed today and the current width of the RGB is not affected.
However, following Larson (1998), we could also allow low mass star
formation
during the first phase but under a different mass-scale than today.
Using the IMF given by equation (26), we have computed the ratio between
the currently observed numbers of low-mass
(0.1 < M < 0.8 )
stars, which are produced in both generations. According to
Brown et al. (1995), the second generation of stars will form a bound
globular cluster if its SFE is at least 0.1. We therefore assume a value
of 10 for the ratio SFE(2nd generation)/SFE(1st
generation). The second generation mass scale m1
(see Eq. (26)) has been fixed to
0.34
to match the solar neighborhood potential peak located at
m = 0.25
(see section 5.1.1).
The mass peak of the first generation stars is left as a parameter
and we allow it to vary between 1 and
3 .
The ratio R of the number of second generation low mass stars to
the number of first generation low mass stars is shown in
Fig. 6.
For one metal-free star,
the number of second generation stars lies between 100 and
4000 depending on the first generation mass peak.
Thus, even if low mass stars were formed in the metal-free
PGCC, their relative number observed today is so small that the existence
of the first generation is not in contradiction with the RGB narrowness
observed on globular cluster CMDs.
|
5.2.2. The time of formation of
the second generation
|
Another controversial point about self-enrichment concerns the ability of
the shell to mix homogeneously the heavy elements with the primordial gas.
If the mixing is not efficient,
inhomogeneities will be imprinted in the second
generation stars formed in the shell and will show up as a broader red
giant branch in CMDs, contrary to observations.
Brown et al. (1991) have established
that the accretion time by the blast wave propagating ahead of the
shell is one to two orders of magnitude larger than the mixing time due to
post-shock turbulence in the shell. In other words, the material swept by
the shell is more quickly mixed
than accreted and the post-shock turbulence insures supershell homogeneity.
But even if the supershell is chemically homogeneous
at a given time, the chemical composition varies with time :
metallicity is increasing as more and more supernova explosions occur
at the center of the bubble. So, one can argue that the second generation
will not be homogeneous : stars which are born early will be more
metal-poor than stars which are born later, when self-enrichment has gone
on. However, shell fragmentation into molecular clouds,
in which second generation stars
will form, cannot take place too early, at least not before the death of
all first generation O stars. Indeed, these ones
are the most important UV flux emitters and, as such, prevent the formation
of molecular hydrogen.
It is very interesting to plot the increase of metallicity versus time
when the shell has emerged in the hot protogalactic medium (all the cloud
material has been swept in the shell whose mass is now a constant).
For simplicity, we have assumed that the ejecta of one supernova mix
with all the PGCC gas before the next supernova explosion.
In Fig. 7,
the parameter values are the same as in section 4
(same IMF, N = 201,
Ph = 8 × 10-11 dyne.cm-2)
and the relation
between the mass of a SNII progenitor and its lifetime on the Main Sequence
is given by 
(Mc Cray, 1987).
In this case, the shell reaches the edge of the cloud 2.3 millions years
(explosion of the 35 supernovae)
after the first explosion. We see in
Fig. 7 that after a rapid increase in
metallicity, as expected for
a metal-free medium, the increase in metallicity slows down and saturates.
After 9 millions years, when all stars more massive than
19
have exploded, the further metallicity increase is less than 0.1 dex,
the upper limit of the RGB metallicity spread.
Therefore, there is no conflict between a
self-enrichment scenario and the RGB narrowness if the second generation
of stars is born after this time . Even if supernova
explosions still occur, the self-enrichment phase has ended. This point
was already underlined by Brown et al. (1991) from a dynamical point
of view.
|
6. Conclusions
|
We have investigated the possibility that globular clusters have undergone
self-enrichment during their evolution.
In our scenario, massive stars contribute actively
to the chemical enrichment and to the gas dynamics in the early Universe.
When a stellar system is formed, supernovae enrich the remaining gas
in such a way that the next generation of stars is more metal-rich than the
first one. In this paper, we assume the birth of a first generation of
stars in the
central areas of PGCCs. When the massive stars end their lives, the
corresponding SNeII explosions trigger the expansion of a
spherical shell of gas, where the PGCC primordial gas and the heavy
elements ejected by supernovae get mixed. Because of the dynamical impact
of supernova shock waves on the ISM, the gas is compressed into a dense
shell and this high density favours the birth of a second generation of
stars with a higher SFE.
This scenario of triggered star formation is now confirmed by observational
examples in the disk of our Galaxy and in irregular galaxies.
The second generation stars formed in these compressed layers of gas are
the ones
we observed today in GCs. Others authors have proposed scenarios where
these stars are also formed in triggered events, namely in gas layers
compressed by shock waves, but the origin of the trigger is different.
For instance, following Vietri and Pesce (1995) and Dinge (1997), the
propagation of shock waves in the cloud could be respectively promoted by
thermal instabilities inside the cloud or cloud-cloud collisions.
Thus, in these
scenarios, there is no first generation massive stars as shock wave
sources: this is the major difference between our scenario and theirs.
It has long been thought that PGCCs were
not able to sustain SNeII explosions because of the associated
important energetic effects on the surrounding ISM.
In this paper, we have shown that this idea may not be true.
For this purpose, the criterion for disruption proposed by Dopita and Smith
(1986) was used. Nevertheless, we have extended it to more general
conditions. Owing to the shell motion description proposed by Castor et
al. (1975), the spacing in time of the supernovae explosions was taken
into account. Also, we have not considered a tidal-truncated cloud as
Dopita and Smith did but a pressure-confined one, which is certainly more
suitable to protogalactic conditions. With this model, we have computed
the speed of propagation of the shell through the PGCC for a given
supernova rate and a given external pressure.
We have demonstrated that a PGCC can sustain many supernova explosions.
Moreover, the dynamical upper limit on the number of SNeII
is compatible with an enrichment of the primordial gas clouds to
typical halo globular cluster metallicities.
This conclusion is quite robust to changes in
IMF parameters. Our result depends on the hot protogalactic pressure
confining the PGCC and implies therefore a relationship between the
metallicity and the radial location in the protogalaxy. We have also
pointed out that a scenario which involves
two distinct generations of stars is not in contradiction with the
RGB narrowness noticed in CMDs of nearly all galactic globular clusters
providing that the birth of the second generation of stars is not
triggered before the 19
supernova explosions have occurred.
In a forthcoming paper, the correlations expected from this
self-enrichment model will be deduced and compared to the observational
data of the galactic halo GCs.
|
7. Figures
|
 Figure 1 |
 Figure 2 |
 Figure 3 |
 Figure 4 |
 Figure 5 |
 Figure 6 |
 Figure 7 |
||
|
Acknowledgements
|
We are very grateful to Dean McLaughlin, whose suggestions as referee of this paper have resulted in several improvements over its original version. This work has been supported by contracts ARC 94/99-178 "Action de Recherche Concertée de la Communauté Française de Belgique" and Pôle d'Attraction Interuniversitaire P4/05 (SSTC, Belgium).
|
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