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 ). The mass distribution within the two other classes are described with power-laws with a slope of for the main-sequence stars and the (sub)giants and a slope of 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 instead of the 0.1 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 pc and a 1-dimensional velocity-dispersion for a 1 star of 10 km/s. We switch-on the dynamics at 10 Gyr and terminate the model at 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 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 star is a white dwarf with a mass of about 0.7 .
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.
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 ), 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.
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).
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998