Direct N-body simulations of globular clusters: (I) Palomar 14
We present the first ever direct -body computations of an old Milky Way globular cluster over its entire life time on a star-by-star basis. Using recent GPU hardware at Bonn University, we have performed a comprehensive set of -body calculations to model the distant outer halo globular cluster Palomar 14 (Pal 14). Pal 14 is unusual in that its mean density is about ten times smaller than that in the solar neighborhood. Its large radius as well as its low-mass make it possible to simulate Pal 14 on a star-by-star basis. By varying the initial conditions we aim at finding an initial -body model which reproduces the observational data best in terms of its basic parameters, i.e. half-light radius, mass and velocity dispersion. We furthermore focus on reproducing the stellar mass function slope of Pal 14 which was found to be significantly shallower than in most globular clusters.
While some of our models can reproduce Pal 14’s basic parameters reasonably well, we find that dynamical mass segregation alone cannot explain the mass function slope of Pal 14 when starting from the canonical Kroupa initial mass function (IMF). In order to seek for an explanation for this discrepancy, we compute additional initial models with varying degrees of primordial mass segregation as well as with a flattened IMF. The necessary degree of primordial mass segregation turns out to be very high, though, such that we prefer the latter hypothesis which we discuss in detail. This modelling has shown that the initial conditions of Pal 14 after gas expulsion must have been a half-mass radius of about 20 pc, a mass of about 50000 M, and possibly some mass segregation or an already established non-canonical IMF depleted in low-mass stars. Such conditions might be obtained by a violent early gas-expulsion phase from an embedded cluster born with mass segregation. Only at large Galactocentric radii are clusters likely to survive as bound entities the destructive gas-expulsion process we seem to have uncovered for Pal 14.
In addition we compute a model with a 5% primordial binary fraction to test if such a population has an effect on the cluster’s evolution. We see no significant effect, though, and moreover find that the binary fraction of Pal 14 stays almost the same and gives the final fraction over its entire life time due to the cluster’s extremely low density. Low-density, halo globular clusters might therefore be good targets to test primordial binary fractions of globular clusters.
keywords:globular clusters: Palomar 14 – initial mass function – methods: N-body simulations
The initial mass function (IMF) of stars is an important quantity for astrophysics. Since the properties and evolution of a star are closely related to its mass, the IMF is a link between stellar evolution and the evolution of stellar systems. It influences the chemical evolution of galaxies and the dynamical evolution of star clusters. The IMF is often expressed as a power-law function (). The traditional IMF that was proposed by Salpeter (1955) from analyzing stars in the solar neighborhood with masses between and , has , implying that high mass stars are rare compared to low mass stars. Later advances showed that the IMF slope flattens to for stars in the mass range while it remains Salpeter’s for massive stars (Kroupa, 2001).
Since stars in star clusters have similar metallicities, distances and ages, they are the best objects to study the stellar mass function. However, the mass function of stars in clusters evolves through stellar and dynamical evolution and one has to understand how these processes have affected the mass function of stars before one can extract the IMF from the observed mass function (Baumgardt & Makino 2003). For example, as a result of energy equipartition, the heavier stars tend to move toward the cluster centre while lighter stars tend to move further away from the cluster centre. Since observations of stars in globular clusters usually are radially limited, the effect of mass segregation has to be considered when deducing the global mass function from the observed local one. Moreover, as a result of preferential loss of low-mass stars, a cluster’s mass-to-light ratio evolves differently from that expected from pure stellar evolution, such that dynamical evolution leads to a decrease of the mass-to-light ratio (Baumgardt & Makino, 2003; Kruijssen & Mieske, 2009; Anders et al., 2009).
While the observed mass function shows local variations, e.g. as a consequence of dynamical mass segregation, the IMF is usually assumed to have no spatial variation throughout a cluster. However star formation simulations indicate that this assumption could be invalid (e.g., Tan et al 2006; Krumholz et al 2009). There is also some observational evidence for deviations from the standard initial mass function, at low and high mass ends, in many star clusters, which might be the result of dynamical evolution (e.g., see Elmegreen 2004 for review).
Some of the ideas that have been proposed to explain a shallowness of the slope at the high mass end include primordial mass segregation of stars in the cluster (e.g., Vesperini & Heggie 1997; Kroupa 2002; Mouri & Taniguchi 2002). But so far it is not clear if star clusters were primordially mass segregated from the star formation epoch or obtained mass segregation via dynamical evolution at a later time. Especially for some young star clusters it has been shown that the observed mass segregation cannot be explained dynamically from an initially unsegregated system (Bonnell & Davies 1998). The early mass loss due to stellar evolution in star clusters leads to shorter lifetimes, and faster expansion, and has a stronger impact on initially segregated clusters than on unsegregated clusters. (Vesperini et al. 2009b).
Measured mass functions of a sample of globular clusters surprisingly show that all high concentration clusters have steep mass functions (i.e., larger ), while low concentration ones have a smaller (De Marchi et al. 2007). This is puzzling, because concentrated clusters are expected to be dynamically more evolved and thus most of their low mass stars should have evaporated from the clusters. However, Marks et al. (2008) showed that for initially mass-segregated clusters, mostly low-mass stars are lost due to gas expulsion which implies a shallower slope in the low-mass range. Indeed this approach allows to deduce the evolution of the very early Milky Way in unprecedented time resolution as it allows us to deduce the changes in the cluster potential induced by gas-expulsion and the varying Galactic tidal field (Marks & Kroupa, 2010).
Several studies have been done to dynamically model the evolution of star clusters using -body simulations. Recently, Harfst et al. (2009) modelled the very young Arches cluster to find the best fitting initial model by comparing simulations with observational data. In this way they aimed at constraining the parameters for the initial conditions of the cluster such as IMF, size, and mass of the cluster by neglecting the Galactic potential and stellar evolution. Moreover, using -body simulations, a close correlation between the slope of the low-mass end of the stellar mass function and the fraction of the initial cluster mass loss has been found by several authors (Vesperini & Heggie 1997; Baumgardt & Makino 2003; Trenti et al. 2010).
In the present paper we report on simulations of the diffuse outer halo globular cluster Palomar 14 (Pal 14). Pal 14 with a mass of about M and a (three dimensional) half-light radius of 34 pc, is a particularly interesting object to study. It is one of the most extended Galactic globular clusters and, with a distance of kpc (Jordi et al. 2009) is also among the outermost clusters of the Milky Way. This large Galactocentric distance and its low stellar mass density make this cluster an ideal candidate to decide between Newtonian gravity and MOND (Baumgardt et al. 2005, Haghi et al. 2009). Its low density suggests that binary stars may be an important issue for interpreting its measured velocity dispersion (Küpper & Kroupa, 2010). Its low mass together with its large radius make it also possible to simulate Pal 14 on a star-by-star basis. However, simulations of the evolution of individual globular cluster’s long-term evolution have so far been done only with Monte Carlo or Fokker Planck simulations (Giersz & Heggie, 2009). We therefore performed a set of direct -body simulations to determine the most likely starting conditions of Pal 14 in terms of total mass, initial half-mass radius and stellar mass function. We also test if primordial mass segregation is necessary to reproduce the observed flattened mass function. Moreover, we check if primordial binaries get frequently disrupted within such a sparse cluster or whether Pal 14 has preserved its primordial binary content. It should be noted that all simulations performed in the present study start once the cluster is gas-free and has settled back into virial equilibrium.
This paper is organized as follows: in Section 2 we describe the observational data we use to compare our simulations to. Our methodology is described in Section 3. In Section 4 we explain the different N-body set-ups and compare the numerical results with the observed data. Conclusions are given in Section 5.
2 Observational data
We use observational data of Hilker (2006) and Jordi et al. (2009) who have presented a spectroscopic and photometric study of Pal 14. Jordi et al. (2009) have obtained the cluster’s mass function using HST/Wide Field Planetary Camera 2 (HST/WFPC2) data from the HST archive. Moreover, they determined Pal 14’s velocity dispersion by measuring the radial velocities of 17 giant stars using the Ultraviolet-Visual Echelle Spectograph (UVES; Dekker et al. 2000) at the Very Large Telescope (VLT) of the European Southern Observatory (ESO) in Chile and the High Resolution Echelle Spectrograph (HIRES; Vogt et al. 1994) on the Keck I telescope.
The distance of Pal 14 from the Sun was derived from the apparent magnitude of the horizontal branch in the colour-magnitude diagram and an isochrone fit to be about kpc (Jordi et al. 2009). This is closer to the Sun than previous estimates, e.g. Hilker (2006) and Dotter et al. (2008) derived larger distances of 74.7 kpc and 79 kpc, respectively. Pal 14 has a projected half-light radius of (Hilker 2006), which was derived from a curve-of-growth analysis of the integrated light of the member stars. This corresponds to a projected half-light radius of pc at the assumed distance of kpc and implies a 3D half-light radius of about pc.
Jordi et al. derived the age of the cluster to be 11.5 0.5 Gyr by adopting [Fe/H] = -1.50 for the metallicity (Harris 1996) from the best isochrone fitting of Dotter et al. (2007). Although previous work has shown that Pal 14 is 3-4 Gyr younger than typical halo GCs with the same metallicity, thus has an age of Gyr (Sarajedini 1997; Hilker 2006; Dotter et al. 2008), the recent estimated age by Jordi et al. reduces this difference.
The density profile of the cluster does not exhibit a significant ellipticity and the distribution of its stars is compatible with a homogeneous, circular distribution within the uncertainties (Hilker 2006). The core radius, and the central surface brightness of the cluster were determined from the best fitting King (King 1966) profiles. The concentration parameter which is defined as is about 0.85 (Hilker 2006), i.e. rather low compared to most Milky Way globular clusters.
The velocity dispersion of Pal 14 by measuring radial velocities of 17 red giant cluster members has been presented with and without including one probably binary or AGB star (hereafter star 15), whose velocity is more than 3 away from the mean of the other stars (Jordi et al. 2009). For the sample including star 15, the global line-of-sight velocity dispersion is 0.64 0.15 kms, and without star 15 it is 0.38 0.12 kms (Jordi et al. 2009). The quoted uncertainties reflect the 1- error.
The upper boundary of Pal 14’s main sequence corresponds to a magnitude of 23.4 mag at a wavelength of 555 nm. Based on photometric data, Jordi et al. (2009) used the masses given by the 11.5 Gyr isochrone by Dotter et al. (2007), and found that the observed stellar masses on the MS covered the mass range of 0.495 up to 0.825 . The mass function of Pal 14 was found to be flatter than a Kroupa or Salpeter IMF in the observed mass range, and the best fitting power-law mass function was obtained for a slope of .
The total mass of stars inside the half-light radius and within the mass range 0.525 to 0.825 has been estimated to be . With the measured slope for masses down to 0.5 and using a Kroupa-like mass function, for masses between 0.1 and 0.5 , Jordi et al. (2009) estimated the total mass of Pal 14 to be about without taking into account stellar remnants. Since this estimated total mass is a derived quantity and not directly measured, we use the number of bright main-sequence stars, to compare with our models, instead. For this aim, we used the star-count data with completeness factor greater than 0.50, which leads to the mass range from 0.525 M to 0.795 M and consequently to the number of Pal 14’s bright stars (Table 3 of Jordi et al. (2009)). This mass range corresponds to a magnitude range . is the HST filter, F555W, that is very similar to Johnson V (Fig. 7 of Jordi et al. (2009)). The stellar mass at the main-sequence turn-off point, , is about M (Dotter et al., 2007). Consequently the central and average density of the cluster are and stars/, respectively. The two-body relaxation time of Pal 14 is about 50 Gyr, i.e. several times larger than a Hubble time.
The total number of giant stars including the red giant branch (RGB) and horizontal branch (HB) stars is also another observable quantity. The photometry published by Hilker (2006) has not covered entirely the cluster (a segment in the West was missing) and was not corrected for foreground/background contamination. Therefore, the numbers in the tables of Hilker (2006) are not complete. We therefore redo a new selection and incompleteness calculation to count all RGB and HB stars. Figure 1 shows Pal 14’s colour-magnitude diagram (CMD). Note that some of the brighter RGB stars on the bluer side might actually be AGB stars. This cannot be differentiated from this photometry. From this CMD it also becomes clear that it makes no sense to count SGB stars. The photometric error is large at those magnitudes and the counts are highly incomplete. The spatial distribution of the observed stars is shown in Figure 2. We divided the regions into stripes/segments, called EE, E, W and WW from East to West. Everything outside the tidal radius is the background (B) which was used to count the contaminating objects. Obviously, the missing region WW should contain more or less the same number of RGB/HB stars as the region EE. But one can also make other sanity checks of number counts, for example whether there are the same amount of RGB/HB stars in the E and W stripes, or whether really the half-light radius divides the number of RGB/HB stars into half (assuming no mass segregation).
The total area within the tidal radius () is arcmin. There are different ways to derive the number of RGB and HB stars: 1) assuming that there are as many RGB stars in the WW segment as in the EE segment, 2) doubling the number of RGB stars in the East (E+EE). The first method gives N, the second N, agreeing very well with the first number. Both values give the background subtracted number counts, taking into account the ratio of the selected area vs. the background area, i.e. scaling the background counts to the selected area. This analysis shows that one can safely assume that the RGB counts in the West should be similar to those in the East. Another result is that there are slightly more RGB stars inside the half-light radius, N, than outside the half-light radius, N. The numbers are consistent within the errors, but might hint to mass segregation of RGB stars on the East side. By the same analysis for the HB stars in the tidal radius we find N for method 1) and N for method 2), both values being in very well agreement. The total number of giant stars (RGB+HB), N, is given in the tables to compare with those obtained from the modelled clusters.
In order to reproduce those observational values, e.g. the half-light radius, , the number of bright stars, , and the mass function slope, , we construct a set of -body models for Pal 14 in the next section.
3 -body Models
In total we computed a set of 66 models to find the initial conditions which reproduce the observations of Pal 14 best. The simulations were carried out with the collisional -body code NBODY6 (Aarseth 2003) on the GPU computers of the Stellar Population and Dynamics Group of the University of Bonn.
All clusters were set up using the publicly available code McLuster111www.astro.uni-bonn.de/~akuepper/mcluster/mcluster.html(Küpper et al. (2010), in preparation). We used a Plummer profile in virial equilibrium as initial mass distribution and computed all clusters for 11 Gyr. We chose the initial mass and size (i.e. half-mass radius) of the cluster as free parameters and kept the other parameters of the initial conditions fixed. The number of stars was varied in the range N such that the clusters have total masses in the range of M. Since the half-mass radius grows due to stellar-evolutionary mass loss and dynamical evolution over the cluster life time, we chose the initial 3D half-mass radii in the range of 18 pc to 22 pc to reach projected half-light radii of about 26 pc after 11 Gyr.
We have chosen two different initial mass functions for our models. First, we start with the canonical IMF (Kroupa 2001) using lower and upper mass limits of 0.08 M and 100 M, respectively. At the second step, the clusters were constructed using a flatter mass function with lower and upper mass limits of 0.08 M and 5 M, respectively, mimicking the initial conditions of an initially mass-segregated star cluster after gas expulsion at an age of about 100 Myr. Stellar evolution was modelled according to the routines by Hurley et al. (2000). Due to the low escape velocity of Pal 14, we assume a 0% retention fraction for neutron stars and black holes which form during the simulation. Note that the canonical (universal) two-part power-law IMF has for M and for M. Here we refer to as the index of the mass function in the stellar mass range 0.525 M to 0.795 M.
For simplicity, and since no information on the proper motion is available, the clusters move on a circular orbit through a logarithmic potential, ln, of the Milky Way at the assumed distance of Pal 14 and with a circular velocity = 220 km/sec.
All but one cluster do not have primordial binaries. We constructed one model with a primordial binary fraction of 5% to study the impact of binaries on cluster evolution and to estimate the fraction of binaries which dissolve in a Hubble time in a cluster as sparse as Pal 14. The binary population was set up using the Kroupa (1995) period distribution with random pairing of the binary companions. Random pairing of companion masses for primaries less massive than 1 M is the correct procedure to reproduce the pre-main sequence and main sequence binary-star data (Kroupa 2008).
The modelled clusters were assumed to be not initially mass segregated. In addition we computed a number of clusters with different degrees of primordial mass segregation to investigate the effect on the cluster’s evolution. An overview of the clusters is given in Tables 1, 2 and 3.
4 Finding the best fitting model
In order to find the most likely initial conditions, a grid of models with different initial half-mass radii and initial masses was calculated. The main observational values which were used to compare with each model, are the present day number of bright main-sequence stars, , the present day projected half-light radius, , and the slope of the mass function in the mass range 0.525 M to 0.795 M within the half-light radius, , with values
Note that, to match in the -body models we use only the giant stars since the projected half-light radius comes from Hilker (2006) and he could only see the giant stars in Pal 14. Moreover, for we count the main-sequence stars inside the half-light radius with masses between 0.525 and 0.795 . Finally, we extract the mass function for those main sequence stars within the half-light radius, with a mass in the range of , i.e. in the same way as Jordi et al. have done. We apply 10 bins between and . We reject the lowermost point in the observational data of Jordi et al. since it has a relatively high incompleteness correction, i.e. uncertainty. Since compact remnants are unobservable in Pal 14, as they are too faint, we ignored them in our measurement of the mass function.
To compare our models in an objective way and to measure the quality of the fit of the model to the observations, we define a goodness-of-fit parameter, . This parameter depends on the model parameters, and measures the closeness between the observations and the model as follow
where represents the corresponding model parameter. The goodness-of-fit parameter ranges from 0 to 1, where 1 describes an excellent fit. Since we have three main observables, , and , we also have three goodness-of-fit parameters, , and , respectively.
There is a significant statistical dispersion in the results if we repeat the calculations for one arbitrary model with the same initial parameters but with a different random seed. The required computational GPU time doesn’t allow us to do statistics for all parameters, though. Hence, in order to estimate the errors of the resulting parameters, we do several runs (10 times) for a particular model and discuss the resulting uncertainty. In the following investigation we will use these uncertainties for our discussion.
In the following sections we will investigate four sets of models. First, we investigate the initial conditions which we call ’regular’. We assume a Kroupa IMF without primordial mass segregation or primordial binaries. In the second step we assume that the IMF was flattened within the first 100 Myr through stellar evolution and gas expulsion processes. The third set of models is computed to test how much primordial mass segregation we have to add in order to explain the observed mass function of Pal 14. In the last section we test the influence of primordial binaries on the cluster evolution and on the evolution of the mass function.
4.1 Regular initial conditions
Figure 3 shows how the goodness-of-fit parameters of , , and vary for the different values of the model parameters, initial half-light radius and initial cluster mass. Moreover, the product of the goodness-of-fit parameters for and is shown in the lower right panel.
The darker areas show where the goodness-of-fit parameters are close to 1 in the parameter space, i.e. show the range in initial conditions that best fit to Pal 14. We find that many models (e.g. M54R22, M52R21, and M50R20) can reproduce Pal 14’s half-light radius and its number of bright main-sequence stars.
We look for models whose velocity dispersion lies within the uncertainties of the observational values. Therefore, we take only stars within a clusters tidal radius and with a mass higher than 0.8 M, and measure their velocity dispersion and compare it to the observed value of Jordi et al. (2009). According to Table 1, the line-of-sight velocity dispersion in most of our models is about km s.
Given the small sample size (i.e., radial velocity of 17 stars are used to determine the observed velocity dispersion), the observed velocity dispersion has a large error which could be much larger than the formal error (Gentile et al. 2010). In order to see how well a velocity dispersion of km s agrees with the observed one, we use the Kolmogorov-Smirnov (KS) test, which is the applicable statistical test for such a small sample size.
In Fig. 4 we show the cumulative distribution function (cdf) vs. radial velocity for both samples with and without star 15, separately. This cdf we compare to the cdf of one of the best fitting models (M50R20). With star 15, the maximum difference between the two cdfs is 0.13, which corresponds to P-value of 0.94, while in case without star 15 the maximum difference between the two cdfs and corresponding P-value are 0.14 and 0.91, respectively. This means that the models can be rejected with 6% (including star 15) and 9% (excluding star 15) confidence.
In addition, we find that the mean value of the velocity dispersion of all stars is similar to that of the red giant stars and main sequence stars with masses larger than 0.8 M. Therefore, the velocities of the giants are representative of all stars in the cluster. The number of giant stars, , is also in most of the modelled clusters in good agreement with the observations.
Figure 5 shows the mass function of one example model before and after the computation. The red and green points denote the number of stars inside and outside the half-light radius, respectively. We see that after 11 Gyr the mass function inside the half-light radius is significantly different from the one outside the half-light radius, which implies that mass segregation has taken place. The mass function of the stars inside the half-light radius is clearly flatter than those of the stars outside. It seems that even in a cluster as large as Pal 14, dynamical mass segregation plays an important role.
According to Table 1 and lower left panel of Figure 3, the slope of the mass function, , in all models is far from the observed value of Pal 14, though, even when accounting for observational errors. The upper limit of the observed value within the 1- error is , and only if we account for the statistical error for the modelled slope, which is 0.15, then two models (i.e., M48R19 and M50R18) are marginally compatible with the observed slope.
Our simulations have typically after 11 Gyr , i.e. only a mild decrease of the low-mass-star slope from the initial Salpeter value of due to mass segregation. In contrast, the observed slope is , which agrees with this value only at the 1.75 level. There is therefore some evidence that Pal 14 started with a mass segregated Kroupa IMF.
In order to see whether an extension of the initial parameter space would resolve the problem, we repeat the simulation for some extremely high and low initial values, i.e. we chose 25 pc and 15 pc for the half-light radius and 40000 M and 60000 M for the initial mass. The results are shown in Table 1. As can be seen, even by selecting such values for the initial conditions, the present day mass function of Pal 14 cannot be reproduced. Moreover, we find that the values for and get much worse in these directions of the parameter space, showing that we were searching in the right place initially.
4.2 Flattened mass function
As shown in Section 4.1, the initially unsegregated clusters with a canonical Kroupa IMF cannot reproduce the observed slope of the mass function well. The observed slope is but the slope that could be achieved from -body calculations is about 2. In this section we show that if we choose a flatter IMF for our model, it is possible to explain the observed slope of the mass function.
The evolution and disruption of star clusters due to gas expulsion and stellar evolution may play an important role in the evolution of the globular cluster mass function (GCMF). It has been shown that these early evolutionary processes, i.e. early cluster dissolution due to gas expulsion and mass loss from stellar evolution, can preferentially destroy low-mass clusters and significantly flatten the low-mass end of the power-law initial GCMF ( Goodwin 1997, Goodwin 1998, Kroupa & Boily 2002, Baumgardt et al. 2008b, Parmentier et al. 2008, Vesperini 2009c). That is, the birth process of a globular cluster can be very violent, depending on the exact initial conditions of its parent gas cloud (Baumgardt & Kroupa, 2007; Baumgardt, Kroupa, & Parmentier, 2008; Parmentier & Gilmore, 2007). Therefore, the first few million years can have a significant impact on a globular cluster’s evolution over a Hubble time. Marks, Kroupa, & Baumgardt (2008) find that, depending on the concentration of the initial cluster, gas expulsion can induce a flattening of the stellar mass function of the cluster. Unfortunately, this process cannot be easily computed in a direct way for a globular cluster of the pre-gas expulsive mass of Pal 14, because clusters can lose a large fraction of their birth stellar population as a result of gas expulsion and stellar evolution mass loss. We therefore have to use an approximation here.
Thus, following Marks, Kroupa, & Baumgardt (2008) and assuming that a certain flattening of the mass function slope has happened within the first 100 Myr of the cluster’s evolution, we change the initial mass function slope and start the simulations at a cluster age of 100 Myr. In this fashion we have performed a new series of -body simulations for models with various flattened initial slopes of the mass function instead of the canonical Kroupa IMF (see overview in Table 2). We have chosen three sets of slopes for the mass function: , , and , where the first number in each set is the slope of the mass function for stars more massive than 0.5 M and the second one is for low-mass stars (for comparison, the Kroupa IMF is ). The second column in Table 2 shows the name of the simulated models. For example, F0.6M50R20 represents a flattened model with the slope of , initial mass of 50000 M, and initial half-mass radius of 20 pc.
The maximum mass in the mass spectrum was set to 5 M, instead of 100 M. The reason can be attributed to the short lifetimes of the most massive stars, i.e., the stars with masses larger than 5 M will have died or turned into compact remnants within the first 100 Myrs. As stated earlier, it is reasonable to assume a retention fraction of 0% for compact remnants due to the low escape velocity from Pal 14.
In Figure 6 we plot the mass function evolution for one of the best fitting models, F0.5M50R18. We see that mass segregation has taken place, in addition to our artificial flattening of the mass function, by about the same amount as in the non-flattened case in Fig. 5.
The results are summarized in Table 2. We find that a number of models in Table 2 can reproduce the present day properties of Pal 14. From Fig. 7 we see that all models with a flattened IMF can reproduce the observed slope of the mass function much better. However, the half-light radius, due to the small value of the observational error, can constrain the calculated models. The number of bright stars is also well reproduced in some models. As can be seen in Fig. 7, models with a lower mass and slope pair , as well as models with a higher mass and slope pair are not compatible with the observed value of .
In order to calculate the uncertainties of our results we have simulated each model for several times (i.e., 2 to 6 runs for each model). This also gives us some insight to the uncertainties of the models in Section 4.1. The resulting errors are also given in Table 2. These are about the same as the errors of the models in Table 1.
Like in Section 4.1, the line-of-sight velocity dispersion in most of our models is about kms and is compatible with the observed velocity dispersion including or excluding star 15.
As can be seen in Table 2, due to the larger number of massive stars in the clusters with initially flattened mass-function, the average values of are larger than for clusters with regular initial condition. However, the number of giant stars in some of the best fitting models (e.g. F0.5M50R20, F0.6M45R18, F0.6M50R18, and F0.7M45R20) are still compatible with the observed value within the uncertainties.
4.3 Primordial mass segregation
Another possibility to explain the fact that a standard IMF gives a poor fit to the observed mass function of Pal 14 is the assumption that Pal 14 was primordially mass segregated. To test this hypothesis, we set up clusters with various degrees of mass segregation. The code McLuster allows to specify a degree of mass segregation parameter (hereafter, ). means no segregation and refers to full segregation, where the most massive star is in the lowest point of the cluster potential and the least massive star is in the highest point of the cluster potential (S̆ubr et al., 2008). McLuster uses the routine described in Baumgardt, De Marchi, & Kroupa (2008) to segregate the clusters. This routine allows to maintain the desired density profile when increasing the degree of mass segregation while also asserting the cluster to be in virial equilibrium.
For this part of the investigation we take one of our modelled clusters of Sec. 4.1 and add different degrees of primordial mass segregation. Thus, the initial 3D half-mass radius and initial cluster mass are fixed to and pc, respectively. The mass segregation parameter was changed in the range .
The results of the simulated models are shown in Table 3. In all cases the mass function flattens as the cluster loses stars. The amount of mass function flattening depends on the given amount of mass segregation, i.e. initially more segregated clusters have more flattened mass functions than unsegregated clusters. From Table 3 we see that low values of cannot reproduce the observed mass function slope, while highly segregated clusters () are able to fit the observed slope of the mass function.
We also find that primordial mass segregation in clusters leads to a stronger expansion than for unsegregated clusters. Figure 8 shows the evolution of 3D half-mass radius of clusters for different values of the mass segregation parameter. We see that by increasing primordial mass segregation, the final half-mass radius rises. For example, for a mass segregation degree varying between 0.5 to 0.95 and an initial half-mass radius pc, the final 3D half-mass radius changes from 39 pc to 67 pc. This expansion is due to both dynamical and stellar evolution. The stellar evolution in the first 100 Myr has a more important effect and leads to a jump in the cluster’s radius such that the cluster with a higher degree of segregation experiences a larger jump. Consequently, this increase of radius leads to a lower value of the mean velocity dispersion.
In order to reproduce the final compatible with the observed value, we therefore have to choose a value for the initial 3D half-mass radius as small as pc for a mass segregation parameter of . In this case, the final line-of-sight velocity dispersion in this model is about 0.6 kms, which is compatible with the observed value. Figure 9 shows the evolution of the mass function of this particular model. The mass function slope after 11 Gyr of evolution reproduces the observations reasonably well. This model can also reproduce all other observational parameters well, but such a high value for the mass segregation parameter implies a very tight correlation of a star’s mass and its specific energy within the cluster at birth. There is no conclusive evidence for such a high degree of primordial mass segregation in any observed star cluster. In addition, as can be seen in Fig. 9, the difference between the slope of the mass function inside and outside the half-light radius is larger than for other models. Thus, further analysis to determine the observed mass function outside the half-light radius would help to distinguish the mass segregation scenario from other scenarios.
Figure 10 compares the projected half-light radii, , and the projected half-mass radii, for the three different sets of modelled clusters from Sec. 4.1-4.3. As can be seen, in almost all models is smaller than . For initially segregated clusters this difference is more than 10% such that the assumption that mass traces light within a cluster is disputable. It must be mentioned that 10% difference in cluster radius leads to 30% difference in the number of stars, which is significant and should be taken into account in discussions.
4.4 Primordial binaries
Observations and theoretical work suggest that stars may be born with initially very large binary fractions and thus star clusters may contain a significant population of primordial binaries (Kroupa, 1995a, b). Low density stellar systems (i.e., less than Mpc) have a binary fraction of at least 50%, while in dense clusters (i.e., more than Mpc), the binary fraction is less than 10-20% (Hut et al., 1992).
The stellar density in Pal 14 is extremely low (0.1-0.2 Mpc), and thus it is expected that most of the primordial binaries survive during the evolution of the cluster. Thus, the present day binary population could be close to the primordial binary population which it had at birth (Kroupa, 1995a, b; Küpper & Kroupa, 2010) making Pal 14 a good target to determine the primordial binary fraction of a globular cluster. Here we discuss one model with primordial binaries.
Since tight binaries can significantly speed up the relaxation process, one might be able to explain the flattening of the observed mass function, since by adding binaries, one increases the mean mass of particles in the cluster (as a binary acts like one heavy particle), and by increasing the mean mass we decrease the relaxation time (Heggie & Hut, 2003).
Unfortunately, binaries slow down direct -body computations immensely because time steps have to be very small for their integration, such that most numerical investigations neglect binaries completely. It is only the peculiar configuration of Pal 14 with its relatively low mass and low density that enables us to perform such a simulation with a few thousand binaries over a time-span of 11 Gyr. Yet, such a computations takes about one month GPU time on a GeForce GTX 280 GPU.
We set up one of the modelled clusters of Sec. 4.1 with an initial mass of M=50000 M, initial half-mass radius of pc, a canonical Kroupa IMF without primordial mass segregation but with a non-zero primordial binary fraction. We add a few binaries to see if the present day binary distribution is a good match for the primordial one. A primordial binary fraction,
of 0.05 (i.e. about 10% of stars being in binaries because ), where is the number of binary systems in the cluster and is the number of single stars, is adopted, in order to see how many of them will be destroyed over the course of time and which effect on the cluster’s evolution they have. The initial period distribution (i.e. the post-gas-expulsion period distribution) used in this study is the birth period distribution of Kroupa (1995b, 2008). Note that the birth period distribution is not a Gauss function. But the period distribution function of bound binaries (i.e., the widest binary systems are excluded) that results from this birth distribution is approximately a Gauss distribution already at Gyr because the widest initial binaries are split due to crowding even in the extremely low density of our initial Pal 14 model.
Our simulation shows that such a small binary fraction does not increase the mass segregation enough, as 5% binaries do not increase the mean mass significantly. Hence, the obtained slope of the mass function, which is about is still not compatible with the observed value, within a 1- level of confidence.
After 11 Gyr evolution of the cluster, the binary fraction reaches , which means that during the evolution most of the binaries (85%) survived. In other word, the present day binary fraction that we see in Pal 14 is almost equal to the primordial binary fraction at its birth. Figure 11 shows the period distribution of binary systems at the beginning and after 11 Gyr of evolution. The parameters of the best-fitting Gaussian distributions to the initial and final period distributions are presented in Fig. 11. Since the mean period does not change with time (within the error), most of the missing binaries might have escaped the cluster rather than gotten disrupted.
Moreover, adding the binaries increases the mean velocity dispersion such that it reaches km/sec if we consider all giant stars and main sequence stars with mass higher than M. We measure km/sec, if we add a cut-off in velocities leaving out the stars that are km/sec away from the mean and hence would not be considered as cluster members in a cluster like Pal 14. It should be noted that the km/sec limit is only applied for this test to compare the velocity dispersion of our model clusters with the observations since observers would not count stars with such discrepant velocities as members. In all other cases stars were considered as members only based on their distance to the cluster center. An overview of the results for this cluster can be found in Table 3. In order to see how well these velocity dispersions are compatible with the observed sample including star 15, we use a KS test again. Here, the -value is for both cases with and without adding the cut-off for high-velocity stars. This means that the data presented in Jordi et al. (2009) is compatible with our modelled cluster with primordial binaries with 68% confidence. This is compatible with recent results by Küpper & Kroupa (2010).
We performed a series of -body computations in order to create a realistic model for the outer halo globular cluster Pal 14 and show that it is possible nowadays to directly calculate the evolution of small Milky Way globular clusters over a Hubble time by direct N-body simulations. For the sake of comparison, and as a rough estimate, the modelling of a typical halo globular cluster (i.e. with the initial mass of about M and half-light radius of about 5 pc) over 12 Gyr will take about times longer than Pal 14, given the same resources. Moreover, about half of the Galactic halo globular clusters are located at kpc, hence the tidal field and cluster evaporation would be stronger, thereby implying a higher initial cluster mass and longer time of simulation.
Pal 14’s large radius together with its low-mass, thus low-density allow us to simulate it on a star-by-star basis. In this way we are not only able to learn more about the dynamical history of such kind of globular clusters, but are also able to study the process of cluster formation as this period leaves its imprint on its ensuing evolution. Moreover, we show that there is valuable synergy between simulations and observations, as simulations can make predictions for observations and, on the other hand, can test observational parameters on their consistency.
Therefore, we generated 66 models divided into four categories: clusters with a canonical Kroupa IMF (Sec. 4.1), clusters with a flattened IMF (Sec. 4.2), clusters with a Kroupa IMF but with primordial mass segregation (Sec. 4.3), and finally one cluster with primordial binaries (Sec. 4.4). By varying the initial half-mass radius and the initial mass of the model clusters, we searched through parameter space to obtain the model that matches the available observations of Pal 14 best.
We used the most recent observational data to constrain the modelled clusters. Thus, for all simulated clusters we compared the half-light radius, the total number of bright stars inside the half-light radius, and the present day mass function slope inside the half-light radius in the mass range M with the observed values. Furthermore, we checked the line-of-sight velocity dispersion and the number of giant stars for their consistency with the numerical results.
From the computations we find that all clusters segregate significantly over the course of time, although the relaxation time of Pal 14 is longer than a Hubble time. We furthermore find that this mass segregation leads to an increasing flattening of the mass function inside half-light radius, which is what we were looking for since the observational value for the mass function slope of Pal 14 in the respective regime is only (Jordi et al., 2009) compared to a Kroupa IMF value of 2.3. The initially unsegregated clusters with a canonical Kroupa IMF do not produce enough depletion in the slope of the mass function in order to be in good agreement with the observations, though. Only a few models of this set marginally agree within the uncertainties with this value for . Besides that, the other observational constraints are matched reasonably well by most of the models in this set (Table 1 and Fig. 3).
The models with the flattened initial mass functions can make up for the discrepancy in , though some of them fail in the other parameters (Table 2 and Fig. 7). Primordial mass segregation also increases the flattening of the mass function but the necessary degree of primordial mass segregation turns out to be very high (Table 4.3).
The initially flattened mass function can be explained by loss of low-mass stars from an initially mass-segregated cluster across the tidal radius during the gas-expulsion phase (Marks & Kroupa, 2010). Therefore, the flattened initial mass function of Pal 14 we uncover here strengthens the idea that this cluster formed in a stronger tidal field rather than in isolation. This may have been the case if either the Galactic tidal field has since then evolved, or if the cluster was born closer to the Galactic centre (eccentric orbit), or, more speculatively, the cluster birth site was a now detached/disrupted dwarf galaxy. However, if Pal 14 has always been on the same circular orbit, it is hard to imagine that the tidal field would have played a role during the cluster response to gas expulsion (Parmentier & Kroupa 2010). Primordial binaries () do not help to reproduce the observational value of the mass function.
Thus, we are able to find some scenarios that can reproduce all observations reasonably well. We find that the initial mass of Pal 14 was about 50000 M and that it started with a half-mass radius of about 20 pc. The cluster must have been significantly mass-segregated with stellar mass function depleted in low-mass stars. These are the signatures of a post-gas expulsion cluster (Marks, Kroupa, & Baumgardt, 2008). The inferred post-gas expulsion ”initial” half-mass radius is larger than the half-mass radii of most globular clusters which is of the order of 3 pc. Moreover, we show that the different scenarios show different mass functions outside the half-light radius, such that finding the observational value for this region would help to discriminate between the different scenarios.
The effect of unresolved binaries is an important issue that has to be considered in the observational values of the velocity dispersion and the mass function. If Pal 14 has a significant binary population, which we expect due to the low density of this cluster, the true (binary-corrected) mass function should be steeper, because of the existence of unresolved components (Kroupa, Gilmore, & Tout, 1991). Therefore, the slope of the mass function is the most uncertain parameter of the observations of Pal 14. Even without taking binaries into account Jordi et al. (2009) estimate the uncertainty to be 0.4. Furthermore, as Küpper & Kroupa (2010) have shown, if the binary fraction was normal (i.e. more than 50%), the binary-corrected velocity dispersion of the cluster would also be smaller than the value reported by Jordi et al. (2009).
From the set of computations we furthermore find that primordial mass segregation leads to a larger impact of stellar evolution on cluster expansion. This is, segregated clusters are more affected by mass loss through stellar evolution (Fig. 8). We furthermore quantify the effect of mass segregation on the ratio of half-mass radius to half-light radius and find the latter to be up to 10% smaller (Fig. 10), leading to significant observational uncertainties for the number of stars within this radius in such cases.
Moreover, we measured the line-of-sight velocity dispersions of our models taking into account all stars, and compared it with the observed values. We additionally measured the velocity dispersion considering only giant stars, and find that both values do not show significant difference between each other, showing that giant stars are good tracers of a cluster’s global velocity dispersion. Most clusters show a velocity dispersion of km/sec, suggesting that star15 of the sample of Jordi et al. (2009) might be a normal cluster member.
Finally we find from the model with a primordial binary fraction of 5% that, due to the low stellar density of Pal 14 the initial binary distribution stays almost unchanged by the dynamical evolution of the cluster, and only about 15% of binaries escape or get disrupted in the course of time (Fig. 11). Low-density clusters like Pal 14 are therefore good objects to study the primordial binary distribution of globular clusters.
A.H.Z and H.H thank the Stellar Population and Dynamics Group of the Argelander Institute for Astronomy and the Institute for Advanced Studies in Basic Sciences (IASBS) for providing financial support for this research.
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