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

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5. A more realistic mass function

The initial conditions for the mass function of the computation of model C (for collapsed cluster core) are chosen to be more realistic, in the sense that the mass function is flattened due to mass segregation in the previous evolution of the stellar system. The lack of detailed computations concerning the present-day mass function in the cores of globular clusters, justifies our choice to use a mass function similar to the one described by Verbunt & Meylan (1988). For the mass function of model C we consider three classes of objects: non-degenerate stars (main-sequence stars and giants), white dwarfs, and neutron stars.

The more massive stars have all evolved, and left inert remnants (white dwarfs or neutron stars). We assign a certain fraction of the total number of stars in the stellar system to each of these classes. All neutron stars (5% of the total number of stars) are assumed to have the same mass (of [FORMULA]). The mass distribution within the two other classes are described with power-laws with a slope of [FORMULA] for the main-sequence stars and the (sub)giants and a slope of [FORMULA] for the white dwarf progenitors. At the start of the dynamical modeling a total number fraction of main-sequence stars and giants of 70% is chosen, this number decreases as the stellar system evolves. The minimum initial mass of a main-sequence star is chosen to be 0.2 [FORMULA] instead of the 0.1 [FORMULA] for models S.

The numbers of stars in the different classes change as time evolves due to stellar evolution, encounters between stars, and due to the addition of a star, each time that the number of stars has decreased by one in a merger process.

Model C has a core radius of [FORMULA]  pc and a 1-dimensional velocity-dispersion for a 1  [FORMULA] star of 10 km/s. We switch-on the dynamics at [FORMULA] 10 Gyr and terminate the model at [FORMULA] 16 Gyr.

The number of stars used in the computation is higher than the calculated number of stars in the core for the parameters of model C ; as a result the Poissonian noise in our calculation is smaller than it would be in an actual core.

Fig. 3 shows for model computation C, the relative probabilities of encounters with various types of stars for a single [FORMULA] star, at an age of the cluster of 12 Gyr. At this age, products of previous encounters are already present in the cluster, and have a finite probability of undergoing another encounter. However, the most probable partner for an encounter with a [FORMULA] star is a white dwarf with a mass of about 0.7 [FORMULA].

[FIGURE] Fig. 3. Relative encounter probabilities in model calculation C, at time [FORMULA] Gyr, when the turnoff mass is [FORMULA], for a single star with [FORMULA] and [FORMULA] as a function of mass and radius of the other star involved in the encounter (similar to Fig. 3). The high encounter-rate and different mass function result in an enormous enrichment of collision products in the stellar system. Besides the small fraction of black holes (nominally with zero radius an with a mass larger than 2 [FORMULA]), there is also a rich population of blue stragglers (in the area with a mass larger than the turn off and a radius larger than about one [FORMULA]) and yellow stragglers (stars with a radius larger than that of the blue stragglers). Except for the neutron stars and black holes (nominally with zero radius an with a mass larger than 2 [FORMULA]) all stars with mass in excess of the turnoff mass are the products of previous encounters. The vertical bar in the upper left corner presents a scaling to the gray shades. The lowest square corresponds to an encounter rate of once every 21 Gyr decreasing with a factor of two for each subsequent square. The integrated encounter frequency of the [FORMULA] star is 1 encounter every 3.1 Gyr. Almost 15% of the encounters occur with a white dwarf with a mass of about [FORMULA] (black square in the middle and below).

[FIGURE] Fig. 4. Relative frequencies of various types of encounters (upper panel) and their outcomes (lower panel), for model computation C, integrated over the duration of the calculation (see also Fig. 1). Apart from the variation in the relative encounter frequencies between various types of stars, encounters between two giants become noticeable and Thorne [FORMULA] ytkow objects ([FORMULA] O) appear (in very small numbers) as the result of a collision.

The relative importance of the various types of encounters is very different in model C compared to model S, as illustrated in Fig. 5, and consequently the relative frequencies of merger outcomes are very different as well. The fraction of collisions that directly result in the formation of a blue straggler rises sharply as does the relative formation-rate of yellow stragglers and white dwarfs with a massive disc. Because the mass function in model C is flat, the region of the main sequence around the turn-off is well populated with massive main-sequence stars and consequently the total number of giants is much larger than in model S where a steep mass function is used.

[FIGURE] Fig. 5. Hertzsprung-Russell diagram of model C, at ca. [FORMULA]  Gyr. [FORMULA] stars (corresponding to about the total number of stars in the core) were selected randomly from all stars involved in the simulation.

5.1. An evolved H-R diagram

A Hertzsprung-Russell diagram of model C after about 12 Gyr is shown in Fig. 5. The dots (representing individual stars) that are positioned in the color magnitude diagram at a position that deviates from the isochrone of the stellar system are the result of a collision. Blue stragglers can be identified close to the zero-age main-sequence but are bluer and more luminous than the turn-off, whereas yellow stragglers are situated above the giant branch. Because the stars in our calculation evolve, the number of collision products present at any time in the core is not at all proportional to their formation rate. For example, blue stragglers (a main sequence star with mass [FORMULA]), formed by merging of two main-sequence stars, often evolve into giants before our calculation is stopped, because of the short main-sequence lifetime of more massive stars. Evolving blue stragglers turn into yellow stragglers, and in fact most of the yellow stragglers present in the cluster have evolved from blue stragglers. The yellow stragglers formed directly from collisions with giants evolve too fast to contribute as strongly to the presence of yellow stragglers. This is illustrated in Fig. 6, which also shows that the fraction of stars that are yellow stragglers is rather constant throughout the computation.

[FIGURE] Fig. 6. Fraction of stars in the computation of model C that are blue stragglers (upper solid line) and the fraction of stars on the main-sequence that were left behind as blue stragglers when primordial stars of equal mass evolved into giants (lower solid line), as a function of time. Due to the slow evolution on the main sequence, the lower line is less susceptible to Poissonian fluctuations. The dotted lines show the fraction of stars that are yellow stragglers, for all yellow stragglers (upper dotted line) and for those that evolved from blue stragglers (lower dotted line).

Merged main-sequence stars with a mass smaller than the turnoff mass upon formation are left behind as blue stragglers when the equally massive primordial stars leave the main sequence. As illustrated in Fig. 6 (lower solid line), the fraction of such blue stragglers is relatively small. On the other hand, the fraction of stars that are blue stragglers rises rapidly at first, but levels off when the evolution rate of blue stragglers into yellow stragglers and beyond becomes competitive with their formation rate. Thus, the fraction of stars that are blue stragglers does not rise much above 3% at any given time, even though 26% of the stars in the computation is directly turned into a blue straggler at some time or after a collision. The dotted line in Fig. 6 illustrates that the total number of yellow stragglers is roughly constant from the beginning of the dynamical simulation.

Giants which undergo a collision become more massive in our prescription, and thus evolve faster than their unperturbed counterparts. As a result, the number of giants in the model is smaller than it would have been in a cluster without collisions, as illustrated in Fig. 7. At the end of the computation the number of giants is depleted by roughly 70%. The fraction of stars on the horizontal branch is roughly 50% larger than expected from a non-dynamically evolving stellar system. This enhancement of the fraction of horizontal branch stars is the result of two effects: most collisions between a giant and another star result in aging of the giant which is then evolved closer towards the horizontal branch and the majority of the collisions between a main-sequence star and a white dwarf results in the formation of a star that is about to terminate its giant lifetime (e.g. close to or on the horizontal branch).

[FIGURE] Fig. 7. Total number of stars formed in model C during the computation with dynamical encounters divided by the number formed from a non-dynamical model as a function of time. The dotted line indicates the fraction of main-sequence stars, the dashed line the stars on the Hertzsprung gap and (sub)giant branch, and the solid line the relative fraction of horizontal-branch stars (averaged in 100 Myr intervals).Stars on the (sub)giant branch are more depleted than the main-sequence stars as time evolves. The fraction of horizontal-branch stars is roughly 50% larger in the stellar system where collisions are included. The Poissonian noise for the main-sequence stars is smallest, as expected, followed by that in the number of (sub)giants. The noise in the fraction of horizontal branch stars is largest. The [FORMULA] error bar (lower left) indicates the Poissonian error for the giants (the dashed line).
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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998