SECOND-GENERATION STELLAR DISKS IN DENSE STAR CLUSTERS AND CLUSTER ELLIPTICITIES

We run several N-body simulations of stellar disks embedded in an initially spherical cluster using ϕGPU (Berczik et al. 2011), a direct N-body code running on graphic processing units (GPUs). The code was developed based on ϕGRAPE, a previous code optimized to use GRAPE6 cards as hardware accelerators. ϕGPU code is fully parallelized using the MPI and CUDA C, the language developed to program NVidia GPUs.

We run our simulations both on the cluster Tamnun at the Technion and on the cluster Cytera in Cyprus. Tamnun is equipped with four nodes with one Nvidia Tesla M2090 each and other four nodes with three Tesla K20m. Cytera is an HPC cluster with 18 GPU nodes with dual NVidia M2070 GPUs. In order to smooth close encounters between particles, we used a softening length of 10⁻³ pc. As shown in Paper I, even a softening length as big as $5\times {10}^{-2}$ pc does not affect the results significantly. The relative energy variations at the end of the simulations are smaller than ${\rm{\Delta }}E/E=0.01$.

The initial spherical FG component of the cluster is a single-mass King model, similar to the best-fit model of the density profile of ω Cen (Meylan 1987), with ${W}_{0}=6$, core radius ${r}_{c}=4.4$ pc, and tidal radius ${r}_{t}=80$ pc. This system is sampled using N = 50,000 particles, whose mass is $20\;{M}_{\odot }$, accounting for a total mass of $M={10}^{6}\;{M}_{\odot }$.

We embed each of the SG disks we consider in the initially spherical FG stellar population. We assume that the cluster has already lost a large fraction of FG stars and we focus on its long-term evolution. The disk is allowed to relax inside the cluster until it reaches a quasi-stable state, which is then considered to be the initial configuration of the system. During the pre-initial stage, a fraction of the SG stars in the disk are expelled from the system and are removed from the N-body simulation, leaving behind a disk whose mass corresponds to 19%, 24%, 29%, 33%, and 50% of the total mass of the system (first- + second-generation populations) in the different models considered (see Table 1 for more details and for the simulations IDs used throughout the text). The profile of the FG component after the initial relaxation phase is still well described by the initial parameters. At the same time the disk inflates, achieving a flattening q = 0.4; 90% of its total mass is inside the core radius of the FG spherical cluster. The initial half-mass relaxation time, t_rh³ , of each simulated system is also given in Table 1. The FG spherical component needs to rotate to form a SG disk (Bekki 2010). However, in our case, the initial disks have a maximum rotational velocity of ∼7 km s⁻¹. Since this value corresponds to a peak rotational velocity smaller than 0.5 km s⁻¹ for the FG population (see Figure 3 in Bekki 2010), we neglect this initial effect. N-body simulation of GC systems are computationally expensive, and we therefore use a smaller number of particles to represent the FG cluster and SG disk stars to allow for faster computation. The mass of each N-body particle is therefore larger than the typical average stellar mass in GCs. Because this choice affects the dynamical timescales, we rescale the simulation time to the relaxation time of the corresponding real system using the procedure described in Paper I, assuming an average stellar mass of $\langle {m}_{\ast}\rangle =0.5\;{M}_{\odot }$ for the stars in a realistic GC. Each system is evolved for 12 Gyr of rescaled simulation time, comparable to the age of GCs.

The dynamics of stars born in a disk can give rise to different signatures observable in the cluster structure compared with the effects of SG stars born in a spherical subcluster. To compare these two different initial configurations, we run an additional simulation where the SG stars are distributed inside the core radius of the FG population with a King profile of the same parameters as used for the FG stars. The latter case is simulated only for a SG population comprising 50% of the total mass of the cluster (including both FG and SG stars).

The clusters are simulated in isolation; tidal effects due to the galaxy significantly affect only the most external regions of the cluster and are neglected here.⁴ We did not take into account stellar evolution. Generally, mass loss would slow down relaxation, and stronger kinematic signatures should be left. We did not consider a mass spectrum for the stars, which could also affect the relaxation time; however, massive stars are short-lived (and stellar black holes are rare) and most of the cluster evolution would be dominated by the dynamics of low-mass stars, for which the range of masses is modest.

Figure 1 shows the isodensity contours of each stellar disk at the beginning of the simulation and after 3, 6, 9, and 12 Gyr. As already found in Paper I and by Vesperini et al. (2013), the SG stars always remain confined within the central 10–20 pc without completely mixing with the FG population. The 19%, 24%, and 29% disks, at the end of the simulation, are almost spherical while the 33% and the 50% disks are still significantly flattened. We also note that the 50% disk is unstable and forms a central bulge that could lead to a faster angular momentum exchange with the FG population through collective effects rather than two-body relaxation (see also Paper I).

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Figure 2 shows the axial ratios of the whole system at the end of each simulation. The 19% disk is slightly prolate, with all axial ratios ∼1, while the other systems are oblate. The c/a axial ratio is smaller for larger disks, the 50% disk is almost perfectly oblate, while the clusters with lower-mass disks are more triaxial. The cluster is flattened at any radius, but the effect is larger in the central 10–20 pc, where most of the disk stars reside. In contrast, the spherical SG system (model S50s) and its host clusters system do not show any significant flattening, as expected.

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Figure 3 shows the final tangential and radial velocity dispersions of all the initially spherical components and their comparison to embedded disk cases. Both the disk and the cluster (FG) stars are centrally isotropic.

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In every case, the disk stars show a large radial anisotropy that increases with the distance from the center (see left panel of Figure 4). This is true also for the FG population that, nevertheless, shows a milder radial anisotropy. The velocity dispersion of the SG population remains lower than that of the FG population even after 12 Gyr, as also found for the low-mass disk simulated in Paper I. This difference is more apparent in the intermediate/external regions of the system. The level of radial anisotropy of the disk stars does not show a clear dependence on the initial disk mass. In contrast, as also seen in Figure 3, the FG population shows a distinct dependence of its radial anisotropy on the initial disk mass.

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The S50s case is highly radially anisotropic (see Figures 3 and 4). Both the first- and second-generation populations are more radially anisotropic than the systems hosting an initially flattened second stellar generation. This induced radial anisotropy was similarly found by Hénault-Brunet et al. (2015), who attributed it to the initially more centrally concentrated configuration of the SG stars. In addition, we find that the higher radial anisotropy is correlated with a higher-mass ratio between the SG and FG stellar populations. This correlation is due to the higher density of the disk and the shorter relaxation time in systems with a more massive SG in respect of the FG.

The right panel of Figure 4 shows the azimuthal velocity ${v}_{\theta }$ of the FG and SG populations as a function of the radius. In all cases, the FG populations show a mild central rotation with a velocity between 0.5 and 1 km s⁻¹. The SG populations, regardless of the mass of the disk, rotate with ${v}_{\theta }$ between 0.5 and 2 km s⁻¹ (within the central ∼20 pc). The rotation initially increases up to 3 pc and then decreases with the distance from the center of the cluster. As expected (see also Hénault-Brunet et al. 2015), the spherical case does not show any significant rotation.

To directly compare the morphological properties of the simulated systems with those of their observed counterparts, we evaluated the projected axial ratio q_p of each system as (Chakrabarty et al. 2008)

$$\begin{eqnarray}&&{q}_{p}^{2}=\displaystyle \frac{{q}_{1}^{2}{q}_{2}^{2}}{{q}_{2}^{2}{\mathrm{cos}}^{2}(i)+{q}_{1}^{2}{\mathrm{sin}}^{2}(i)}\end{eqnarray} \tag{ 1 }$$

where q₁ and q₂ are the three-dimensional (3D) major and minor axial ratios and i is the inclination of the line of sight in respect to the z axis. We evaluated q_p for all the systems after 12 Gyr of evolution (Figure 5). We adopted four different inclinations: 0°, 30°, 60°, and 90°. We find that the projected flattening grows with the disk mass. Note, however, that the observed flattening strongly depends on the inclination angle in respect to the line of sight; in particular, even very flattened clusters could appear spherical if seen face-on. Inclinations of 60° and 90° (edge-on) amplify the flattening compared with the other cases.

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As already found for the spatial axial ratios, the projected axial ratios of the cluster hosting a spherically distributed SG population are of order of unity and very similar, regardless of the adopted viewing angle.

Typical GCs are not spherical, but slightly ellipsoidal (see e.g., Geyer et al. 1983; White & Shawl 1987; Mackey & van den Bergh 2005; Chen & Chen 2010). Several origins for the observed ellipticity has been suggested, including primordial internal rotation, external tides, and/or pressure anisotropy (Fiestas et al. 2006; van den Bergh 2008; Varri & Bertin 2012; Bianchini et al. 2013b). Mergers of GCs, which have been proposed as a possible explanation for the presence of multiple stellar generations (van den Bergh 1996; Carretta et al. 2010; Amaro-Seoane et al. 2013), could also lead to significant flattening (Makino et al. 1991). However, the absence of binary Galactic GCs (see, e.g., Innanen et al. 1972; Surdin 1991; van den Bergh 2008) suggests that this process does not play an important role in the evolution of GCs in our Galaxy. Akiyama (1991) found that the external regions become more spherical with time as a consequence of gravothermal contraction and tidal stripping. At the same time, the tidal field can stretch the clusters, making them more elongate. Moreover, Schmidt (1996) found that moderate bright Galactic GCs are rounder than fainter and brighter clusters. The existence of a correlation between age and flattening (Makino et al. 1991, and reference therein) or between flattening and the presence of bluer horizontal branches in clusters are still debated (Norris 1983; van den Bergh 2008).

As also shown in Paper I in our simulated clusters, the flattening is caused by the angular momentum exchange between the second- and first-generation stars. When both the first- and second-generation are spherically symmetric, the system always keeps its initial morphology and does not rotate (see Sections 3.1 and 4.3). Note that since it does show significant radial anisotropy, the velocity anisotropies are not linked with the flattening (Sections 3.2 and 4.2).

Because the central density and the angular momentum increase with SG disk mass, both the total amount of the angular momentum transferred and the efficiency of the angular momentum exchange between the FG and SG populations are expected to be larger for more massive SG disks, as indeed was observed in our simulations.

The 3D flattening is always larger than the projected one (see Figures 2 and 5), thus observations can only provide a lower limit for this quantity. However, statistical analysis and future observations, e.g., potentially provided by the GAIA mission, could give more detailed information on the spatial structure of GCs, allowing a quantitative comparison between the theoretical expectations and the observations.

The projected axial ratios (see Equation (1)) of the simulated systems, averaged over the radial extent and evaluated at the half-mass radius, are shown in Figure 6 as a function of the SG mass fraction of the total mass of each system and for four different projection angles (0°, 30°, 60°, and 90°). If seen face-on (0°), all of the clusters look almost spherical. Different inclinations produce different values of the flattening and this could be connected to the large spread in q_p for the observed Galactic GCs (see Figure 6). The trend of ellipticity correlation with SG mass fraction can be clearly seen.

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Both the cluster hosting a spherical SG population and the clusters whose SG stellar populations have initially flattened structures show radially anisotropic profiles. Therefore, this feature cannot be used as a tracer for the initial SG configuration (see also Hénault-Brunet et al. 2015); however, it can be used as an indicator of the total mass of the SG population. The simulated clusters have similar relaxation times and their radial anisotropy grows with time. After 12 Gyr of evolution, the larger the SG mass (independent of the initial configuration), the larger is the radial anisotropy of the FG population. The SG population is more radially anisotropic than the FG one.

One difference between the spherical SG case and the flattened SG populations is that even when the mass of the spherical SG model is similar to that of the most massive simulated disk, its velocity dispersion is more radially biased than that found for the S50 case. This is possibly caused by the faster relaxation in the case of an initially flattened distribution for the SG population.⁵

A bulk average rotational velocity has been already observed in few Galactic GCs. Recently, Bianchini et al. (2013a) and Kacharov et al. (2014) found that the rotation plays an important role in the shaping the structure of ω Cen, M15, 47 Tuc, and NGC 4372 and it gives rise to the flattening of these clusters. Furthermore, Fabricius et al. (2014) found a tight correlation between rotation and flattening for the 11 Milky Way GCs in their sample.

The FG population in our simulation rotates with ${v}_{\theta }$ smaller than 1 km s⁻¹. The SG population has a maximum ${v}_{\theta }$ between 1 and 2 km s⁻¹, and all the systems reach the maximum velocity at ∼3 pc from the center of the cluster, regardless of the mass of the disk. The rotation is apparent even beyond 10 pc—the radius containing most of the SG stars. As a consequence of how the initial conditions are built, the spherical S50s system does not show any significant rotation in any of the components. Thus, in the AGB model or in any model where the SG stars form in a disk, the rotation can be potentially explained as the result of angular momentum exchange between an initially flattened and slowly rotating SG stellar population and an initially spherical FG population.

The strength of the signatures left by the SG disk depends both on the mass of the disk and on the relaxation time of the system, but all the signatures are evident even after 12 Gyr in any of the simulated cases. All of the systems studied have relaxation times shorter than a Hubble time and the presence of disks induce a more rapid core collapse of the cluster; much faster than that expected from non-centrally concentrated SG population. More massive disks lead to faster core collapse, and we therefore expect clusters with a larger SG fraction to be more dynamically evolved than their counterpart similar systems with smaller SG fractions.