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Astron. Astrophys. 328, 130-142 (1997)

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3. A static homogeneous environment with single stars

3.1. Initial conditions

In the present paper, we keep the dynamical environment as simple as possible, in order to focus on the stellar evolution recipes, that are introduced here and used in subsequent papers as well. The stellar distribution is take to be in thermal equilibrium, with a density that is constant in space and time. In addition, an additional simplification is obtained by excluding any primordial binaries, and ignoring binary formation channels. Within this setting, random encounters between single stars will lead to collisions resulting in the formation of merger products, the evolution of which can then be followed along with the evolution of the original single stars.

3.1.1. Initial mass function

While our main aim is to set-up and clarify our stellar evolution recipes, we present two calculations that could be interpreted as having a limited astrophysical interpretation, one for the core of an [FORMULA] -Centauri-like cluster (Sect. 4), and one for the core of an M-15-like cluster (Sect. 5). Our choice of constant density implies that we can only hope to model the history of a cluster core, not that of a cluster as a whole. To specify the mass distribution, we first take our standard choice: a Salpeter initial mass function (Sect.  4), which we will use to model a relatively unevolved core. Our second choice will be a much more flat distribution, which is more appropriate for a high-density post-collapse cluster core (Sect.  5).

3.1.2. Mass and number densities

If we specify the mass density [FORMULA] for the stars in our cluster case, we can use the mass function to determine the number density [FORMULA]. In a homogeneous medium the relation is linear, and for the simplest case of a powerlaw mass function [FORMULA], we find


where [FORMULA] and [FORMULA] are the lower and upper mass cut-offs, respectively. For the example of an initial Salpeter mass function, [FORMULA], we find


when we neglect the fact that the upper mass cut-off is finite. The inverse quantity [FORMULA] is the average stellar mass:


in our standard case where we take a lower cut-off mass of [FORMULA]. The median mass [FORMULA] for a Salpeter distribution is


which means that in our standard population most stars have a mass well below [FORMULA].

Even in our simple case of a homogeneous system, the linear relationship [FORMULA] involves a complicated time dependent factor [FORMULA]. Not only does the upper mass cut-off [FORMULA] (roughly the main sequence turn-off mass) depend on time, but what is worse, the distribution of remnants, in the form of black holes, neutron stars and white dwarfs, does not obey any simple power law, even if their progenitors did. In general, therefore, the coefficient [FORMULA] has to be determined numerically, as a function of time.

3.1.3. Velocity dispersions

In thermal equilibrium, equipartition of kinetic energy tells us how the velocity dispersions scale for stars with different masses. We only have to specify the three-dimensional velocity dispersion v for one particular mass, say [FORMULA], in order to determine the 3D velocity dispersion [FORMULA] for stars of general mass m:


3.1.4. Core radius and core mass

The three choices discussed so far, namely that of an initial mass function, a density, and a temperature, specify the intensive thermodynamic properties. This in turn enables us to calculate the local rate of collisions, per unit time, and per unit volume. In order to extract global information, we have to specify extensive quantities as well, such as the total volume or total mass of our system. This will allow us to determine a global collision rate per unit time, which we can then compare with that of an astrophysical system, such as the core of a globular cluster.

For an equal-mass cluster model that is close to thermodynamic equilibrium, the density drops by roughly a factor three, from the center to the edge of the core. This implies that the local density of collisions, which is proportional to the square of the density, drops by an order of magnitude. In the more realistic case of a mass spectrum the situation is even worse, since the density of the heavier stars drops off faster than that of the lightest stars. In the present paper we will not attempt to model these density dependent effects, and instead we will keep the density of all mass groups constant throughout the region of our simulation. It is clear, therefore, that our results are mainly for the purpose of illustration, and that any comparison with actual systems will have to be taken with many grains of salt.

The only question remaining is the definition of a core radius [FORMULA]. For an equal mass system, we have (Spitzer 1987)


In the presence of a mass spectrum, we have to modify this equation. Although the velocity dispersion is now quite different for different mass groups, the average kinetic energy per star [FORMULA] is independent of mass, with N the total number of stars in the core. Rewriting the above formula, we have:


where M is the core mass. For a general mass spectrum, we can substitute [FORMULA] which gives


In this expression, the right hand side contains only local quantities, and the global quantity [FORMULA] is given in terms of those.

Note that this is not the only possible generalization of the equal-mass expression, but it is a natural one, and it reverts to the original expression in cases where the mass of the core is dominated by stars in a relatively small mass range, as is the case, for example, in a post-collapse core of a globular cluster.

Other global quantities can be derived from [FORMULA], such as the core mass M:


where we have used the fact that in an isothermal sphere the average density in the core is roughly half the central density, a relationship, while not exact, is certainly good enough for our purpose of relating our results to astrophysical systems.

3.2. Recipes for stellar evolution

The stars in our computations are evolving using the stellar evolution model called SeBa (see Portegies Zwart & Verbunt 1996). To describe their evolution, we use the formulae fitted to the results of full stellar evolution calculations, by Eggleton et al. (1989). These formulae give the radius and luminosity (for population I stars) as a function of time, on the main sequence, in the Hertzsprung gap, the (sub)giant branch, on the horizontal branch, and on the asymptotic giant-branch. We use these population I recipes, because the more appropriate data for population II stars are not available in the same convenient form. In addition to the radius, we need the core mass for stars that have left the main sequence. We derive these from the luminosity, according to Eggleton et al. (1989), and core-mass luminosity relations, according to Boothroyd & Sackmann (1988), Paczynski (1970) and Iben & Truran (1978). The details of this procedure are described in Portegies Zwart & Verbunt (1996, Section 2.1).

3.3. Recipes for individual encounters

In this paper, we only treat single stars, and accordingly the only outcome allowed for a close encounter is a merged object. The merging between the two stars in an encounter in our calculation is generally assumed to conserve mass, which in fact may be a reasonable approximation (Benz and Hills 1987, 1989, 1992, Rasio & Shapiro 1991, 1992).

Only a limited number of simulations of encounters between stars has been performed, and these does not cover all possible combinations that may occur in a cluster. Also, different authors do not agree on the details of the outcomes for the same type of encounter. We therefore have chosen to use a set of simple prescriptions for the outcome of stellar collisions, often chosen without detailed justification. In the future these prescriptions can be refined, when more accurate calculations for collisions become available. Meanwhile, our results will help in determining which of all possible types of encounter are most frequent, and therefore deserve closer attention.

We describe our treatment of the possible outcomes of the encounters of two stars ordered by the evolutionary state of the more massive of the two, the primary. Table 1 summarizes this treatment.


Table 1. Simplified representation of possible merger outcomes. The four columns correspond to the four choices given for the type of massive star (primary), while the four rows indicate the type of less massive star (secondary): main-sequence star (ms), (sub)giant (sg), white dwarf (wd) and neutron star (ns). In this table we do not discriminate between stars in the Hertzsprung gap (Hg) or on the first and second ascent on the asymptotic-giant branch (AGB).

3.3.1. Main-sequence primary

If both stars involved in the encounter are main-sequence stars the less massive star is accreted conservatively onto the most massive star. The resulting star is a rejuvenated main-sequence star (see Lai et al. 1993, Lombardi et al. 1995). The details of this procedure are described in Appendix C4 of Portegies Zwart & Verbunt (1996).

If the less massive star in the encounter has a well developed core (giant or subgiant) this core is treated as the core of the merger product. The main-sequence star and the envelope of the giant are added together to form the new envelope of the merger. In general the mass of the core is relatively small compared to its envelope and the star is assumed to continue its evolution through the Hertzsprung gap. Note that this type of encounter can only occur when the main-sequence star is in itself a collision product (e.g. a blue straggler).

When a main-sequence star encounters a less massive white dwarf, we assume that the merger product is a giant, whose core and envelope have the masses of the white dwarf and the main-sequence star, respectively. We then determine the evolutionary state of the merger product, as follows. We calculate the total time [FORMULA] that a single, unperturbed star with a mass equal to that of the merged star spends on the asymptotic giant-branch, and the mass [FORMULA] of its core at the tip of the giant branch. The age of the merger product is then calculated by adding [FORMULA] to the age of an unperturbed star with the same mass at the bottom of the asymptotic giant branch. For example, a single, unperturbed 1.4 [FORMULA] star leaves the main-sequence after 2.52 Gyr, spends 60 Myr in the Hertzsprung gap, moves to the horizontal branch at [FORMULA] Gyr, and reaches the tip of the asymptotic giant branch after [FORMULA] Gyr, with a core of [FORMULA]. Thus, if a 0.6 [FORMULA] white dwarf mergers with an [FORMULA] main-sequence star, the merger product has an age of 2.87 Gyr, leaving it another 180 Myr before it reaches the tip of the asymptotic giant-branch.

If the less massive star is a neutron star a Thorne [FORMULA] ytkow object (Thorne & Z_ytkow 1977) is formed.

3.3.2. Evolved primary

When a (sub)giant or asymptotic branch giant encounters a less massive main-sequence star, the main-sequence star is added to the envelope of the giant, which stays in the same evolutionary state, i.e. remains a (sub)giant, c.q. asymptotic branch giant. Its age within that state is changed, however, according to the rejuvenation calculation described in Sect. C.3 of Portegies Zwart & Verbunt (1996). For example, an encounter of a giant of [FORMULA] and age [FORMULA] Gyr with a 0.45 [FORMULA] main-sequence star produces a giant of 1.4 [FORMULA] with an age of [FORMULA] Gyr.

When both stars are (sub)giants the two cores are added together and form the core of the merger product (see also the results of the smoothed particle hydrodynamics computations performed by Davies et al. 1991 and Rasio & Shapiro 1995). Half the envelope mass of the (less massive) encountering star is accreted onto the primary. The merger product continues its evolution starting at the next evolutionary state; thus a (sub)giant continues its evolution on the horizontal branch and a horizontal branch star becomes an asymptotic-giant branch star. The reasoning behind this assumption is that an increased core mass corresponds to a later evolutionary stage.

If the less massive star is a white dwarf then its mass is simply added to the core mass of the giant, and the envelope is retained. If the age of the giant before the encounter exceeds the total life time of a single unperturbed star with the mass of the merger, then the newly formed giant immediately sheds its envelope, and its core turns into a single white dwarf; if not then the merged giant is assumed to have the same age (in years) as the giant before the collision, and continues its evolution as a single unperturbed star.

If the encountering star is a less massive neutron-star a Thorne [FORMULA] ytkow object is formed.

3.3.3. White-dwarf primary

In an encounter between a white dwarf and a less massive main-sequence star, the latter is completely disrupted and forms a disk around the white dwarf (Ruffert & Müller 1990, Rasio & Shapiro 1991). The white dwarf accretes from this disk at a rate of one percent of the Eddington limit. If the mass in the disc exceeds 5% of the mass of the white dwarf, the excess mass is expelled from the disc at a rate equal to the Eddington limit.

If a white dwarf encounters a less massive (sub)giant, a new white dwarf is formed with a mass equal to the sum of the pre-encounter core of the (sub)giant and the white-dwarf. The newly formed white dwarf is surrounded by a disk formed from half the envelope of the (sub)giant before the encounter. If the mass of the white dwarf surpasses the Chandrasekhar limit, it is destroyed, without leaving a remnant (Nomoto & Kondo 1991 and Livio & Truran 1985).

Collisions between white dwarfs are ignored.

3.3.4. Neutron-star or black-hole primary

All encounters with a neutron star or black hole primary lead to the formation of a massive disk around the compact star. If the compact star had a disk prior to the collision, this disk is expelled. This disk accretes onto the compact star at a rate of 5% of the Eddington limit. An accreting neutron star turns into a millisecond radio-pulsar, or - when its mass exceeds [FORMULA] - into a black hole. Mutual encounters between neutron stars and black holes are ignored, as are collisions between these stars and white dwarfs.

3.4. Monte Carlo simulations of ensembles of encounters

Each star in our model can encounter any of the other stars. To reduce computational cost, we bin the stars in intervals of mass and radius, and compute the probability for encounters between bins, giving all stars in one bin the same mass and radius, and, through Eq. 2, choosing their velocities from the same distribution. The cross section [FORMULA] for a encounter with a distance of closest approach within d between a star from bin i and a star from bin j contains a geometrical and a gravitational focusing contribution:


where [FORMULA] is the relative velocity between the stars at infinity. For the minimum separation between the two stars that leads to a collision [FORMULA] is used. (The exact distance at which the transition between merger and binary formation occurs is not known - Kochanek 1992, Lai et al. 1993-, we choose the factor 2 arbitrarily.)

In the present paper, we model the stellar distributions as being spatially homogeneous. In order to make contact with astrophysical applications, we will consider our stars to be contained within in a fixed sphere with radius [FORMULA]. While we can consider this radius to stand for the notion of `core radius' in a post-collapse cluster, we want to point out that this interpretation is only an approximate one. In realistic star clusters, there is a significant drop in density across the core, from the center to the core radius. For most stars the density drops by roughly a factor of three, but for the heavier stars, such as neutron stars and especially black holes, this factor can be much larger.

The encounter rate [FORMULA] of stars from bin i with stars from bin j, anywhere in the volume of the sphere with radius [FORMULA] is given by two separate equations:


where [FORMULA] and [FORMULA] are the number densities of stars in bins i and j, respectively, and where [FORMULA] indicates averaging over the distribution of relative velocities [FORMULA] (Note that we should have written the last equation with an extra factor [FORMULA], if we would have summed over all combinations [FORMULA], in order to avoid double-counting of collisions).

Since the stars in bins [FORMULA] have Maxwellian velocity distributions with root-mean-square velocity [FORMULA] and [FORMULA], given by Eq.(2), the relative velocities [FORMULA] also have a Maxwellian distribution, with a root-mean-square velocity given by [FORMULA]. Hence


where we have defined


With this result, we write the encounter rates [FORMULA] in convenient units:


The total encounter-rate follows as


where N gives the total number of bins in mass and radius and [FORMULA] is the average time interval between two encounters.

The stellar population in our calculation changes both due to encounters between stars, and due to evolution of the stars. The shortest evolutionary time-scale of importance to us is the time scale on which the evolving stars expand; the fastest evolving star in the sample is used to set the evolution time scale


where R and [FORMULA] are the stellar radius and its time derivative, respectively.

At the beginning of each time step, we distribute the stars over the bins in radius and mass, calculate the number densities of stars in each bin, and the evolution and collision time scales [FORMULA] and [FORMULA]. The sum over all bins ij is less daunting as may appear at first sight, as many bins contain no stars. This is illustrated in Figs. 1 and 3. The time step to be taken is then calculated as


to ensure that changes in the stellar population are followed with sufficient resolution.

At this point, a rejection technique is used to keep track of collisions, as follows. We choose a random number between 0 and 1. If this number is larger than [FORMULA], we conclude that no collision has occurred. We evolve all stars over a time dt, and continue with the next step.

If the random number is smaller than [FORMULA], a collision has occurred. In calculating the sum (Eq.  15) over the bins, we keep track of the partial sum after addition of each bin combination ij. The first bin combination for which this growing partial sum exceeds the random number identifies the bins involved in the collision. We then assign a sequence number to each star in bin i, and select one of these numbers randomly; and repeat this for bin j. If i and j are identical, care is taken that the same star is not selected twice. From the prescriptions in the previous section, we decide the outcome of the collision between the two selected stars.

We then select another random number between 0 and 1, to see whether a second collision has occurred. If so, we determine its outcome. This procedure is repeated until a random number larger than [FORMULA] is found, which indicates that no further encounter has occurred in the time step under consideration.

After each time step dt a number of stars equal to the number of encounters that have taken place is lost from the stellar system; these stars have merged into single objects. For each lost star a new star is added to the computation, in order to guarantee a constant number of stars. The mass of this `halo guest' is determined by the present-day mass-function of the cluster.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998