| { | |
| "INITIAL SIZE DISTRIBUTION OF THE GALACTIC GLOBULAR CLUSTER SYSTEM ": [], | |
| "Jihye Shin1 , Sungsoo S. Kim1,2 , Suk-Jin Yoon3 , and Juhan Kim4 ": [], | |
| "Draft version March 5, 2022 ": [], | |
| "ABSTRACT ": [ | |
| "(<http://arxiv.org/abs/1212.6872v1>)arXiv:1212.6872v1 [astro-ph.GA] 31 Dec 2012 ", | |
| "Despite the importance of their size evolution in understanding the dynamical evolution of globular clusters (GCs) of the Milky Way, studies are rare that focus specifically on this issue. Based on the advanced, realistic Fokker–Planck (FP) approach, we predict theoretically the initial size distribution (SD) of the Galactic GCs along with their initial mass function and radial distribution. Over one thousand FP calculations in a wide parameter space have pinpointed the best-fit initial conditions for the SD, mass function, and radial distribution. Our best-fit model shows that the initial SD of the Galactic GCs is of larger dispersion than today’s SD, and that typical projected half-light radius of the initial GCs is ∼4.6 pc, which is 1.8 times larger than that of the present-day GCs (∼2.5 pc). Their large size signifies greater susceptibility to the Galactic tides: the total mass of destroyed GCs reaches 3–5×108 M⊙, several times larger than the previous estimates. Our result challenges a recent view that the Milky Way GCs were born compact on the sub-pc scale, and rather implies that (1) the initial GCs are generally larger than the typical size of the present-day GCs, (2) the initially large GCs mostly shrink and/or disrupt as a result of the galactic tides, and (3) the initially small GCs expand by two-body relaxation, and later shrink by the galactic tides. ", | |
| "Subject headings: Galaxy: evolution -Galaxy: formation -Galaxy: kinematics and dynamics -globular clusters: general -methods: numerical " | |
| ], | |
| "1. INTRODUCTION ": [ | |
| "Whereas the present-day mass functions (MFs) of globular cluster (GC) systems, which are nearly universal among galaxies ((<>)Brodie & Strader (<>)2006; (<>)Jorn´an et al. (<>)2007), are approximately log-normal with a peak mass Mp ≈2×105 M⊙, the MFs of the young massive star cluster (YMC) systems follow a simple power-law distribution ((<>)Whitmore & Schweizer (<>)1995; (<>)Zhang & Fall (<>)1999; (<>)de Grijs et al. (<>)2003, among others). Motivated by such a di erence between GCs and YMCs, numerous studies have examined the dynamical evolution of the GC MFs to determine whether the initial MFs of GC systems resemble those of YMC systems ((<>)Gnedin & Ostriker (<>)1997; (<>)Baumgardt (<>)1998; (<>)Vesperini (<>)1998; (<>)Fall & Zhang (<>)2001; (<>)Parmentier & Gilmore (<>)2007; (<>)Shin, Kim, & Takahashi (<>)2008, among others). In particular, (<>)Shin, Kim, & Takahashi ((<>)2008, Paper I hereafter) surveyed a wide range of parameter space for the initial conditions of the Milky Way GCs, and considered virtually all internal/external processes: two-body relaxation, stellar evolution, binary heating, galactic tidal field, eccentric orbits and disc/bulge shocks. They found that the initial GC MF that best fits the observed GC MF of the Milky Way is a log-normal function with a peak at 4×105 M⊙ and a dispersion of 0.33, which is quite di erent from the typical MFs of YMCs. ", | |
| "Using the outcome of N-body calculations, ", | |
| "(<>)Gieles & Baumgardt ((<>)2008) found that the aspect of mass loss in GCs varies with the tidal filling ratio ℜ≡rh/rJ , where rh is the half-mass radius and rJ is the Jacobi radius. More specifically, the mass loss of GCs in the ”isolated regime” (ℜ< 0.05) is driven mostly by the two-body relaxation, which induces the formation of binaries in the core and causes GCs to expand. On the other hand, the mass loss of GCs in the ”tidal regime” (ℜ> 0.05) is influenced by the galactic tides as well, which enables stars in the outer envelope to easily escape (evaporation). Thus, the cluster size (rh) is as important as the cluster mass (M) and the galactocentric radius (RG) in determining the dynamical evolution of GCs. ", | |
| "Can YMCs tell us something about the typical initial size of the Milky Way GC system? Observations show that the projected half-light radius Rh of YMCs (ages up to 100 Myr) in the local group ranges between ∼2 and ∼30 pc with a mean value of ∼8 pc ((<>)Portegies Zwart, McMillan, & Gieles (<>)2010), which is a few times larger than that of the present-day Milky Way GCs, Rh ∼2.5 pc. However, GCs could have formed in di erent environments and/or by di erent mechanisms from the YMCs. ", | |
| "Perhaps the best way to estimate the typical size of the GCs is to trace them back to their initial state by calculating their dynamical evolution. In this paper, we study the dynamical evolution of the Galactic GCs and identify the most probable initial conditions not only for the MF and radial distribution (RD), but also the size distribution (SD). Using the same numerical method and procedure as in Paper I, we perform Fokker-Planck (FP) calculations for 1152 di erent initial conditions (mass, half-mass radius, galactocentric radius and orbit eccentricity), and then search a wide-parameter space for the most probable initial distribution models that evolve into ", | |
| "Fig. 1.— Distribution of the Galactic ”native” (see the text for definition) globular clusters in the L–RG space (a), Rh–RG space (b), and Rh–L (c). Data are from the compilation by Harris (1996). the present-day Galactic GC distributions. ", | |
| "The paper is organized as follows. Section 2 describes the properties of the observed GCs, again which we compare our model results. Section 3 presents models and initial conditions for FP calculations, and Section 4 analyzes the aspects of the size evolution of GCs. We synthesize our FP results in Section 5 to construct the GC system, and examine common features of the best-fit MF, RD, and SD models in Section 6. We discuss characteristics of the final best-fit SD models of the Galactic GCs in Section 7. Finally, conclusions are presented in Section 8. " | |
| ], | |
| "2. PRESENT-DAY GC PROPERTIES ": [ | |
| "When comparing FP calculations to the present-day Galactic GCs, we consider the ”native” GCs only, i.e., ”old” halo and bulge/disc clusters, which are believed to be created when a protogalaxy collapses while ”young” halo clusters are thought to be formed in external satellite galaxies ((<>)Zinn (<>)1993; (<>)Parmentier et al. (<>)2000; (<>)Mackey & van den Bergh (<>)2005). Our native GC candidates do not include six objects that belong to the Sagittarius dwarf, seven objects whose origins remain unknown, two objects that have no size information, and fifteen objects that are thought to be the remnants of dwarf galaxies ((<>)Lee, Gim, & Casetti-Dinescu (<>)2007). The total number of our present-day Galactic native GCs is 93, and their observed properties, such as luminosity L, Rh, and RG, were obtained from the database compiled by (<>)Harris ((<>)1996). ", | |
| "Figure (<>)1 shows scatter plots between observed L, Rh, and RG values for the 93 Galactic native GCs. The L, Rh, and RG values range between 3.9×103–5.0×105 L⊙, 0.3–16 pc, and 0.6–38 kpc, where the mean values are located at 7.2×104 L⊙, 2.5 pc, and 4.1 kpc, respectively. The correlation between Rh and RG is tighter than the other two correlations (see Figure (<>)1b). This tight Rh– RG correlation could be just a result of the initially tight ", | |
| "Fig. 2.— Comparison of rh evolution between N-body simulations and our FP calculations for GCs with initial conditions of M = 104 M⊙, RG = 8.5 kpc, and rh = 1, 3, and 5 pc on circular orbits. N-body simulations were performed using Nbody4 code ((<>)Aarseth (<>)2003), and mass loss by stellar evolution was not considered in these test calculations (both N-body and FP). rh values of the two models agree well within ∼20% during the entire cluster lifetimes. ", | |
| "correlation between Rh and RG, or it could be due to the preferred disruption of large GCs near the Galactic center ((<>)Vesperini & Heggie (<>)1997; (<>)Baumgardt & Makino (<>)2003). Another possible cause is the expansion of initially small GCs up to rJ , which is roughly proportional to RG 2/3 for a given GC mass. One of the goals of this paper is to determine which of these possibilities is more feasible. ", | |
| "Previous studies on the evolution of the GC system assumed a certain constant mass-to-light (M/L) ratio, and converted the observed L to M when comparing their numerical values with observations. But the conversion of GC luminosity function (LF) to GC MF using a constant M/L ratio may lead to MFs in error because low-mass stars, which have higher M/L ratios than the high-mass stars, preferentially evaporate from the cluster and this causes the M/L ratio of the cluster to evolve with time ((<>)Kruijssen & Portegies Zwart (<>)2009). For the same reason, there is not a linear relationship between Rh and rh among di erent GCs. Thus, we transform M to L, instead of L to M, using the stellar mass–luminosity relation of the Padova model ((<>)Marigo et al. (<>)2008) with a metallicity of [Fe/H] = −1.16, which is the mean value for the Galactic native GCs. Our FP calculations, which will be described later, show that the present-day GCs can have M/L ratios ranging between 1.2 and 2.5 and rh/Rh ratios ranging between 1.0 and 2.5. ", | |
| "We use dynamical properties such as M and rh when constructing the initial distributions of the Galactic GC system and when calculating the dynamical evolution, while observed quantities, L and Rh, are used when comparing our FP results with the observations. " | |
| ], | |
| "3. MODELS AND INITIAL CONDITIONS ": [ | |
| "We adopt the anisotropic FP model used in Paper I, which was originally developed by (<>)Takahashi & Lee ((<>)2000, and references therein). The model integrates the orbit-averaged FP equation of two (energy-angular momentum) dimensions and considers multiple stellar mass components, three-body and tidal-capture binary heating, stellar evolution, tidal fields, disk/bulge shocks, dynamical friction, and realistic (eccentric) cluster orbit (see (<>)Kim & Lee ((<>)1999) for the tidal binary heating and Paper I for the detailed implementation of dynamical friction and realistic orbits). The model implements the Alternating Direction Implicit (ADI) method developed by (<>)Shin & Kim ((<>)2007) for integrating the two-dimensional FP equation with better numerical stability. ", | |
| "Parameters for our FP survey are the following four initial cluster conditions: M, rh, apocenter distance of the cluster orbit Ra, and cluster orbit eccentricity e. We choose eight M values from 103.5 to 107 M⊙, six rh values from 10−1 to 101.5 pc, and six Ra values from 100 to 101.67 kpc, all equally spaced on the logarithmic scale. For the eccentricity, we choose e = 0, 0.25, 0.5, and 0.75. We perform FP calculations for all possible combinations of these four parameters, thus the total number of cluster models considered in the present study amounts to 1152. ", | |
| "For the initial stellar mass function (IMF) within each cluster, we adopt the model developed by Kroupa (2001) with a mass range of 0.08–15 M⊙, which is realized by 15 discrete mass components in our FP model. Each mass component follows the stellar evolution recipe described by (<>)Schaller et al. ((<>)1992). The stellar density and velocity dispersion distributions within each cluster follow the King model ((<>)King (<>)1966) with a concentration parameter W0 = 7 and with neither initial velocity anisotropy nor initial mass segregation. We use only one value for W0, thus the tidal cut-o radius rt of the King profile is proportional to rh, while rJ varies depending on M and RG. Therefore, the Roche lobe filling ratio (rt/rJ ) and ℜof our FP models are functions of rh, M, and RG. ", | |
| "The aspects of mass and size evolution from our FP model are in a good agreement with those from N-body methods. A comparison of mass evolution between our FP calculations and the N-body simulations performed by (<>)Baumgardt & Makino ((<>)2003) for clusters on eccentric orbits with initial masses larger than 104 M⊙ shows good agreement of cluster lifetimes within ∼25%. For a comparison of size evolution, we run a set of N-body simulations using Nbody4 code ((<>)Aarseth (<>)2003) with M = 104 M⊙, RG = 8.5 kpc, and rh = 1, 3, and 5 pc (these correspond to ℜ= 0.04, 0.11, and 0.18), and find that the rh evolutions of the two models agree well within ∼20% during the entire cluster lifetimes (see Figure (<>)2). ", | |
| "Due to the expulsion of the remnant gas from star formation in the pre-gas-expulsion cluster, some of the low-mass pre-gas expulsion clusters can quickly disrupt, and even the surviving low-mass pre-gas-expulsion clusters will lose a significant fraction of their mass within the first several Myr and rapidly expand ((<>)Baumgardt & Kroupa (<>)2007; (<>)Parmentier & Gilmore (<>)2007). Since our FP model does not consider the e ect of gas expulsion, our initial GC models are to be regarded as models at several Myr after cluster formation. " | |
| ], | |
| "4. SIZE EVOLUTION OF INDIVIDUAL GLOBULAR CLUSTERS ": [ | |
| "The three main drivers of GC size evolution are the two-body relaxation, the mass loss by stellar evolution, and the galactic tides. In this section, we discuss the size evolution of individual GCs with a subset of our FP calculations. Figure (<>)3 shows the ratios between rh values at the present time (13 Gyr) and at the beginning from our FP calculations as a function of rh,0 and M0 for two di erent RG,0 values (subscripts 0 denote the initial value, hereafter). ", | |
| "Two-body relaxation causes GC core to collapse and the subsequent formation of dynamical binaries in the core makes the whole cluster expand. For GCs that have undergone core collapse in the early phase of evolution, the size of the post-core-collapse expansion follows a scaling relation rh ∝M0−1/3 t2/3 ((<>)Goodman (<>)1984; (<>)Kim, Lee, & Goodman (<>)1998; (<>)Baumgardt, Hut, & Heggie (<>)2002), and thus for a given initial mass and epoch, rh/rh,0 is simply proportional to rh,−10. Figure (<>)3 indeed shows that the size of the GCs with the same M0 tend to converge to a single value (rh/rh,0 ∝rh,−10), if the GCs have small trh,0 (log trh,0/yr . 9). ", | |
| "Mass loss by stellar evolution causes GCs to adiabatically expand to maintain virialization, and the GC sizes evolve following rh/rh,0 ∝M0/M when the stellar evolution is the main driver of the GC size evolution ((<>)Hills (<>)1980). The combination of Kroupa IMF and the stellar evolution recipe described by (<>)Schaller et al. ((<>)1992) yields a mass loss of ∼40% within 13 Gyr. Thus, GCs would expand by a factor of ∼1.67 as a result of the stellar evolution, if two-body relaxation or the galactic tides are relatively less important in driving the size evolution. Indeed, clusters with log trh,0/yr & 9 and ℜ< 0.05 have rh,13/rh,0 values between 1 and 2. ", | |
| "While stellar evolution and two-body relaxation cause clusters to expand, galactic tides make clusters shrink in general. A cluster extending farther than rJ (overfilling; rt > rJ ) loses stars outside rJ within a few dynamical timescales, and this naturally causes the mean size of the cluster to decrease. Since rJ ∝RG(M/MG)1/3 where MG is an enclosed mass of the Milky Way in a given RG, the size decrease caused by the galactic tides takes place mostly while the cluster approaches Rp. The cluster re-expands somewhat by two-body relaxation while approaching Ra ((<>)Baumgardt & Makino (<>)2003), but its size gradually decreases while repeating orbital motions. We find that clusters with 0.4 < rt/rJ < 1 can also shrink moderately as a result of the galactic tides even if it underfills, and clusters initially with rt/rJ < 0.4 (or ℜ< 0.05; i.e., ”isolated” GCs) can gradually move into the ”tidal” regime as they lose mass or expand by stellar evolution or two-body relaxation. Figure (<>)3 shows that GCs with larger ℜ0 are smaller at 13 Gyr for a given M0 and RG,0, as expected. ", | |
| "Among various initial GC parameters, rh,0 is the most important parameter in the size evolution caused by two-body relaxation and that resulting from galactic tides (ℜ0 ∝M0−1/3RG −2/3 rh,0 for a flat rotation curve). For this reason, initially small GCs gen-", | |
| "Fig. 3.— Ratios of rh values at 13 Gyr and at the beginning from some of our 1,152 Fokker–Planck calculations as a function of rh at the beginning for two di erent RG,0 values (4.6 kpc for the left panel and 46 kpc for the right panel) and four di erent initial mass (log M0/M⊙= 4, 5, 6, and 7). The approximate initial half-mass relaxation times and the initial tidal filling ratios are marked with di erent colors and symbol sizes, respectively. The blue line indicates the location of rh/rh,0 = 1.67, which is the expected expansion ratio mainly by the stellar evolution, and the red lines represents the relation rh/rh,0 ∝r h,−10, which is the expected result when the evolution is dominated by the two-body relaxation. ", | |
| "Fig. 4.— Comparison of mass functions (a), radial distributions (b), and size distributions (c) at 13 Gyr (solid lines) and at the beginning (dashed lines) from the best-fit initial parameter set for each SD model. ", | |
| "erally expand (by two-body relaxation), while initially large GCs generally shrink (by the galactic tides) as they evolve. The size evolution of intermediate GCs is determined by more than one dynamical e ect, and some GCs can even maintain their initial size over their whole lifetime. " | |
| ], | |
| "5. SYNTHESIS OF FOKKER–PLANCK CALCULATIONS ": [ | |
| "As discussed in Section 3, we performed a total of 1152 FP calculations with di erent initial cluster conditions in four-dimensional parameter space, M, rh, Ra, and e. The goal of the present study is to find the initial distribution of these variables that best describe the observed GCs. ", | |
| "For the initial MF model, we adopt a Schechter function, ", | |
| "(1) ", | |
| "and for the initial RD model, we use a softened power-law function, ", | |
| "(2) ", | |
| "We assume that the initial MF is independent of initial RG. For the sake of simplicity, we do not parameterize the distribution for e, and adopt the fixed isotropic distributions, i.e., dN(e) ∝e de. Unlike M, the RG of each FP model evolves by oscillating between Rp and Ra, and thus the model RD at 13 Gyr constructed from our population synthesis may su er from significant random ", | |
| "Fig. 5.— Comparison of size distributions at 13 Gyr (solid lines) and at the beginning (dashed lines) from SD Model 1. The upper panels are for initially small GCs (trh,0 < 0.5 Gyr or ℜ0 < 0.05), and the lower panels are for initially large GCs (trh,0 > 0.5 Gyr or ℜ0 > 0.05). ", | |
| "noise. To reduce this noise, we build a model RD by summing the probability distributions between Rp and Ra that are given by the orbital information at 13 Gyr, and we call this a phase-mixed RD. Hereafter, RDs in this paper refer to the phase-mixed RD. ", | |
| "For initial SDs, we use six distribution models (see Table (<>)1). Models 1, 2, and 3 represent a Gaussian distribution of rh, ˆh (mean density within rh), and ℜ, respectively, implying that the initial GCs have the preferred initial rh, ˆh, and ℜ, with dispersions. The initial rh of Model 1 does not correlate with the initial M or RG, while Models 2 and 3 have initial correlations of and rh ∝M1/3RG 2/3ℜ. In Models 4, 5, and 6, the initial rh is determined by powers of initial M and/or RG. Note that the power of Model 6 (rh ∝M0.615) corresponds to that of the mass– size relation derived from the Faber–Jackson relation for early-type galaxies ((<>)Faber et al. (<>)1989; (<>)Ha¸segan et. (<>)2005; (<>)Gieles et al. (<>)2010). ", | |
| "Once the calculations of the 1152 FP models are done, the aforementioned sets of initial MF, RD, and SD models are used to search for the best-fit parameters in five to seven dimensional space, depending on the SD models (Models 1–6). For this, we synthesize our 1152 FP calculations with appropriate weights to produce a given initial MF, RD, and SD, and find a set of parameters that best fit the present-day MF, RD, and SD for each of the six SD models. When finding the best set of parameters for each SD model, we minimize the sum of ˜2 values from all of the L, RG, and Rh histograms, which are constructed by using eight bins between 104 and 105.8 L⊙ for L, nine bins between 100 and 101.6 kpc for RG, and nine bins between 10−0.6 and 101.2 pc for Rh, all equally spaced on a logarithmic scale. Recall that we use dynamical (theoretical) properties M and rh for set-", | |
| "ting the initial distributions, while observable quantities such as L and Rh are used for comparing the models and observations. " | |
| ], | |
| "6. BEST-FIT INITIAL DISTRIBUTION OF THE GALACTIC GLOBULAR CLUSTER SYSTEM ": [ | |
| "The best-fit parameter sets that minimize the ˜2 values between observations and our calculations are presented in Table (<>)2 for the six SD models. We examine the characteristics of our best initial MFs, RDs and SDs in turn. " | |
| ], | |
| "6.1. Initial Mass Function ": [ | |
| "The best-fit values for all six SD models are quite low, ranging between 0.01 and 0.07. The best-fit log Ms/M⊙ values for all six SD models are similar to each other, having values between 5.8 and 5.9. Note that Schechter functions with such small values are similar to log-normal functions, while those of & 2 are closer to power-law functions. Thus, our small values suggest that log-normal functions better describe the initial MF of the Galactic GC system than power-law functions (see Figure (<>)4a), and this result is consistent with the result of Paper I. One way to explain the log-normal-like initial MF is expulsion of the remnant gas due to star formation in the pre-gas-expulsion cluster, which can quickly alter a power-law MF into a log-normal-like MF ((<>)Parmentier & Gilmore (<>)2007). Another possible mechanism resulting in a rapid change in the initial MF is the collisions of clusters with dense clouds or other clusters during the early phase of the galaxy ((<>)Elmegreen (<>)2010). " | |
| ], | |
| "6.2. Initial Radial Distribution ": [ | |
| "Initial RDs from the best-fit parameter sets for all six SD models have similar values (4.0–4.5) but a rather wide range of Rs values (0.3–3.6 kpc), and this is consistent with the result of Paper I ( = 4.2 and Rs = 2.9 kpc). ", | |
| "Figure (<>)4b shows that most of the GCs that disrupt before 13 Gyr are located in the bulge regime (RG,0 < 3 kpc), and most of the GCs formed in the bulge do not survive until now. We find that only 0.1–8.4 % of the total GC mass initially inside 3 kpc remains in GCs at 13 Gyr, and the total stellar mass that escaped from the GCs inside 3 kpc during the last 13 Gyr amounts to 5 ×107–3 ×108 M⊙, depending on the SD model. " | |
| ], | |
| "6.3. Initial Size Distribution ": [ | |
| "The initial SDs from our best-fit parameter sets are of larger dispersion than the present-day SDs for all six SD models (see Figure (<>)4c). The initial SDs evolve into the narrower present-day SDs by two main e ects: (1) expansion of GCs with small rh,0, which normally have small trh,0 and/or small ℜ0, due to two-body relaxation, and (2) shrinkage (evaporation) of large rh,0 GCs, which normally have large trh,0 and/or large ℜ0, due to the Galactic tides. Figure (<>)5 shows that the SDs of initially small GCs (upper panels) indeed shift to the larger rh region and those of initially large GCs (lower panels) shift to the smaller rh region after 13 Gyr. ", | |
| "Three p-values (significance levels) for ˜2 tests of LFs, RDs, and SDs are acceptably high, except for Model 3, which has relatively small ˜2 p-values for RDs and SDs (see Table (<>)2). However, the high p-values from the ˜2 ", | |
| "Fig. 6.— L (top left), R (top middle), and Rh (top right) histograms at 13 Gyr (thick solid lines) for SD Model 1 with the best-fit parameter set. Also shown together in the upper panels are the corresponding initial distributions (dashed lines) and the observed distributions (thin solid lines). The lower panels show the correlations between Rh and RG (bottom left), Rh and L (bottom middle), and L and RG (bottom right) relationships for the corresponding best-fit models in the upper panels (asterisks) and from the observations (open circles). ", | |
| "Fig. 7.— Same as Figure 5, but for SD Model 2. ", | |
| "Fig. 8.— Initial SDs of the GCs that survive until 13 Gyr in our best-fit SD models, Models 1 (thick solid line) and 2 (dashed line). Also shown is the currently observed SD (thin solid line with a shaded area). The overall size of the GCs were larger at birth than now even when only the surviving GCs are considered. ", | |
| "tests do not necessarily guarantee that the models with the best-fit parameters restore the observed correlation between L, RG, and Rh as well. Thus, we implement Student’s t-tests to see if our models with the best-fit parameters agree with the observed RG dependence of SDs (the Rh–RG correlation), the L dependence of SDs (the Rh–L correlation), and RG dependence of LFs (the L–RG correlation). For the Rh–RG correlation, we calculate ˜2 for the di erence of log Rh and ˙log Rh between the model and the observation as follows: ", | |
| "(3) ", | |
| "where subscripts o and m stand for the observation and the model, respectively, subscript j represents the equal number RG bins, and . . . denotes the averaged values. The same calculation is applied to Rh–L and L–RG correlation as well. We find that Models 3–6 have t-test p-values that are too small (. 1%) for at least one of the Rh–RG, Rh–L, and L–RG correlation. For this reason, we reject Models 3–6 as being a plausible initial SD candidate. Hereafter, we call SD models 1 and 2 “the final best-fit SD models”. ", | |
| "Figures (<>)6 and (<>)7 show our two remaining best-fit SD models, a Gaussian distribution of rh (model 1; rh,c = 6.4 pc, ˙rh = 2.7 pc) and a Gaussian distribution of ˆh (model 2; ˆh,c = 690 M⊙pc−3 , ˙ˆh = 4.6 M⊙pc−3). Note that rh,0 values are not correlated with the RG,0 in either model. This implies that the rh,0 of GCs probably does not depend on the strength of the galactic tides. Therefore, we interpret the observed, present-day Rh–RG correlation (see Figure (<>)1b) as an outcome of a preferential ", | |
| "disruption of the larger GCs at smaller RG due to the Galactic tides. " | |
| ], | |
| "7. DISCUSSION ": [ | |
| "The typical Rh,0 value from our final best-fit SD models (Models 1 and 2) is ∼4.6 pc (rh,0 ∼7 pc), and this is 1.8 times larger than that of the present-day GCs (∼2.5 pc). This result is rather di erent from a recent argument by (<>)Baumgardt et al. ((<>)2010) that most GCs were born compact with rh,0 < 1 pc. Our result implies that GCs initially have a rather wide SD, the typical value of which is similar to that of YMCs in parsec scale, and have evolved to have a narrower SD with a smaller mean value. ", | |
| "We also find that GCs formation favors a ”tidal” environment over an ”isolated” environment. The number of tidal GCs (ℜ0 > 0.05) at 0 Gyr from our final best-fit SD models is approximately five times larger than that of isolated GCs (ℜ0 < 0.05). The ratio of tidal to isolated GCs, however, drastically decreases as GCs evolve because tidal GCs are more easily disrupted, and this ratio becomes ∼0.2 at 13 Gyr. ", | |
| "Figure (<>)8 shows the initial SDs of the GCs that survive until 13 Gyr in Models 1 and 2. We find that these initial SDs are broader (˙(Rh) = 2.1 and 2.5 pc, respectively) and centered at higher values (Rh = 4.1 and 4.0 pc) than the currently observed SD (˙(Rh) = 1.2 pc, Rh = 2.5 pc). Thus, the overall size of the GCs were larger at birth than now by a factor of ∼2 even when only the surviving GCs are considered. ", | |
| "The initial total masses in GCs (Mt,0) of the final best-fit SD models are 2.8×108 M⊙ (Model 1) and 5.3×108 M⊙ (Model 2), and the masses that have left the GCs during the lifetime of the Galaxy (Mt) are 2.5×108 M⊙ (Model 1) and 5.0×108 M⊙ (Model 2). These give Mt/Mt,0 values of 0.89 and 0.94 for Models 1 and 2, respectively. Our Mt values are several times larger than previous estimates made by (<>)Baumgardt ((<>)1998, 4.0–9.5×107 M⊙), (<>)Vesperini ((<>)1998, 5.5×107 M⊙), and Paper I (1.5–1.8×108 M⊙). Our larger Mt values are due to the facts that (1) we consider virtually all disruption mechanisms in the calculations for the dynamical evolution of individual GCs, and (2) we use more a flexible initial rh distribution, which can have a relatively larger fraction of GCs with a large rh (larger GCs are more vulnerable to the galactic tide). Note that Mt will be larger if one considers the clusters that have been disrupted in the process of remnant gas expulsion. ", | |
| "We note that contrary to the finding in the present paper, detailed dynamical modeling of individual clusters shows that at least some of the clusters must have started with a very small size. For example, Monte Carlo calculations by (<>)Heggie & Giersz ((<>)2008) and (<>)Giersz & Heggie ((<>)2009, 2011) find 0.58 pc, 0.40 pc, and 1.9 pc as best-fit initial rh values for the observed current states of M4, NGC 6397, and 47 Tuc, respectively. These values are several times smaller than the typical initial rh found for the Galactic GC system from our calculations, ∼7 pc. However, we also note that the Monte Carlo models used for these three clusters all assume circular cluster orbits while M4 and NGC 6397 have moderate to high orbit eccentricities (0.82 and 0.34, respectively). We have performed several FP calculations for these two clusters and ", | |
| "find that consideration of appropriate eccentric orbits can increase the best-fit initial rh by a factor of 3–5. " | |
| ], | |
| "8. SUMMARY ": [ | |
| "We have calculated the dynamical evolution of Galactic GCs using the most advanced and realistic FP model, and searched a wide parameter space for the best-fitting initial SD, MF, and RD models that evolve into the present-day distribution. We found the initial MF of the Galactic GC system is similar to the log-normal function rather than the power-law function, and the RD of the GC system undergoes significant evolution inside RG = 3 kpc through the strong Galactic tides. We also found that the initial SD of the GC system evolves to narrower present-day SDs through two e ects: shrinkage of large GCs by the galactic tides and expansion of small GCs by two-body relaxation. The typical initial projected half-mass radius from the final best-fit model, ∼4.6 pc, is 1.8 times larger than that of the present-day value, ∼2.5 pc. The ratio of ”tidal” GCs to ”isolated” GCs is ∼5 at 0 Gyr and decreases down to ∼0.2 at 13 Gyr. ", | |
| "Since tidal GCs are found to be dominant in the beginning, one might expect the initial size of the GCs to be correlated with the Jacobi radius, i.e., to be a function of the galactocentric radius. However, our final best-fit SD ", | |
| "models (Models 1 and 2) do not seem connected to the galactocentric radius. This implies that the GC formation process favors a certain size and density, regardless of the tidal environment. Such a RG-independent initial SD evolves into a present-day SD, which shows a tight rh–RG correlation through evaporation and two-body relaxation. ", | |
| "We thank Holger Baumgardt and Mark Gieles for helpful discussion. This work was supported by Basic Science Research Program (No. 2011-0027247) through the National Research Foundation (NRF) grant funded by the Ministry of Education, Science and Technology (MEST) of Korea. This work was partially supported by WCU program through NRF funded by MEST of Korea (No. R31-10016). J.S. deeply appreciates Koji Takahashi for the help with his FP models. S.J.Y. acknowledges support by the NRF of Korea to the Center for Galaxy Evolution Research and by the Korea Astronomy and Space Science Institute Research Fund 2011 and 2012. S.J.Y. thanks Daniel Fabricant, Charles Alcock, Jay Strader, Nelson Caldwell, Dong-Woo Kim, and Jae-Sub Hong for their hospitality during his stay at Harvard-Smithsonian Center for Astrophysics as a Visiting Professor in 2011– 2012. " | |
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