Globular Star Clusters Author: Staš Jevševar (28030543) prof. Mentor: Tomaž Zwitter Abstract In this seminar globular star clusters are represented as very important objects, that have been studied for over 100 years. How they formed is still a matter of debate but with Hubble Space Telescope a lot of progress has been made due to increase of empirical data. I will present what their special characteristics and general dynamics are. Several globular clusters around a galaxy form a globular cluster system, which I will show has several useful and interesting properties.
Contents 1 Introduction 2 2 Formation 2 2.1 Pre-galactic.................................. 3 2.2 Proto-galactic................................. 3 2.3 Post-galactic.................................. 4 3 Composition 4 3.1 Structure................................... 4 3.2 Dynamics................................... 5 3.3 Hertzsprung-Russel diagram of GCs..................... 6 3.4 Blue Stragglers................................ 6 3.5 RR Lyrae stars................................ 8 4 Globular cluster systems 8 4.1 Luminosity functions............................. 10 4.2 Metallicity................................... 11 4.3 Black holes and GC systems......................... 11 5 Epilog 12 1
1 Introduction A group of stars, that is gravitationally bound and smaller than 35 pc is called a star cluster. Anything larger would be in the region of dwarf galaxies. There are two basic types of star clusters, open clusters and globular clusters. This seminar is going to be about globular star clusters, GCs. Open cluster is a group of a few thousand stars weakly gravitationally bound, which makes them easily disturbed by other clusters and giant molecular clouds as they move through the galaxy. When this occurs stars belonging to the cluster still move in the same direction, even though they aren t gravitationally bound any more and are then referred to as a stellar association. On the other hand globular clusters are heavily gravitationally bound. Their name is derived from the Latin word globulus, meaning a small sphere, denoting their almost perfect spherical shape. They are found on a halo around the galaxy and contain considerably more stars than open clusters and are much older. Up to a million stars can belong to a single cluster while the age can reach 10 10 years and more. The Milky Way galaxy contains around 150 globular clusters, with several still to be found. Globular clusters have been studied for over 50 years, and have shown that all stars from a single cluster belong to single age, single metallicity. This strongly suggests the stars in a cluster formed at the same time. Also given their compact nature, all of these stars lie on roughly the same distance from the Sun. Both these traits make them ideal candidates for determination of age and distance [1]. Distances to globular clusters can be measured simply from the parallax of a few member stars or through RR Lyrae variable stars. All these characteristics make them experimental test beds to be used by astronomers in their studies of stellar evolution. 2 Formation Early models of globular cluster formations were based on empirical data that all globular clusters were old and massive. Consequently, these models tended to exploit physical conditions unique to the early universe that might give rise to bound clusters of stars with masses around 10 5 M. Over the last decade or so, observations of extragalactic globular cluster systems and the discovery of young globular clusters have dramatically expanded the empirical basis of globular cluster formation theories. All stars, in any globular cluster, belong to Population II stars, which are metalpoor and are found on a halo up to 10 kilo-parsecs away from the center of the Galaxy. These clusters, like everything else in the Galaxy, are in orbit around the galactic center. Even though they require many million years to complete one orbit, investigations in the current velocities and locations show they exhibit elongated orbits, randomly oriented. In contrast to halo objects, younger clusters appear to be in orbits that are more circular in shape and oriented near the galactic plane. Also called thick-disk population, they are found closer to galactic plane with distances from 1 to 2 kpc. In comparison, Population I stars are lying within 0.4 kpc of galactic plane. Stars in these clusters also belong to intermediate Population II stars, but with higher metallic content. It is convenient to divide models for the formation of GC systems into three broad categories: pre-, proto- and post-galactic. While these seem distinct, they rely on similar 2
Figure 1: ω Centauri, the biggest GC of our galaxy. With naked eye, it is barely visible in the southern hemisphere [2]. processes to explain the formation. All clusters begin with a mass distribution function similar to that for interstellar clouds, which is approximately N(M)dM M 2 dm, where N(M) is the number of stars that has mass M. 2.1 Pre-galactic In the picture of pre-galactic model the formation of clusters pre-dates that of their host galaxy. This was suggested by Dicke & Peebles (1968), who found that the Jeans mass in cosmological epoch, at which charged electrons and protons first became bound to form electrically neutral hydrogen atoms called time of recombination, was the same as the mean mass of Galactic GCs. The lack of a dominant mass of dark matter inside globular clusters (Moore 1996; Heggie & Hut 1996) makes it unlikely that they formed through direct cosmological collapse, and more likely that they resulted from fragmentation during the process of galaxy formation. 2.2 Proto-galactic During the proto-galactic stage GCs could have formed within dense cores of supergiant molecular clouds, suggested by Harris & Pudritz (1994). These SGMC were supported against gravitational collapse by magnetic field pressure and magneto-hydrodynamic turbulence. Cosmological N-body simulations by Weil & Pudritz (2001) confirmed that gravitationally bound objects having masses and sizes similar to these super giant molecular 3
clouds do indeed form and have roughly power-law mass spectrum, N(M)dM = M α dm, with α 1.7. Although this agreement lends credibility to the results, it is important to recall that the inferred mass spectrum is much steeper than measured luminosity function of galaxies in the local universe. Thus, the formation of GC systems seems inevitably linked to the so-called missing satellite problem (Klypin et al. 1999; Moore et al. 1999). 2.3 Post-galactic One of characteristics od globular cluster systems of galaxies is specific frequency S N, which is number of clusters per unit galaxy luminosity. In numerical terms it is defined simply as S N = N 10 0.4(M V +15) for a cluster population N and galaxy luminosity M V. Interestingly there is a factor of 2 difference between S N of normal elliptical and spiral galaxies. This might be explained if normal elliptical galaxies formed as a result of major mergers of disk galaxies and the merger process led to the formation of new, presumably metal-rich, globular clusters. An N-body simulation by Bekki et al. (2002) confirmed that the end-products of such mergers do indeed resemble elliptical galaxies, with distinct metal-poor and metal-rich globular cluster sub-populations. But simulation also showed that the mean metallicity of the metal-rich clusters far exceeded those measured for metal-rich clusters in real elliptical galaxies [3]. 3 Composition Globular clusters are generally composed of hundreds of thousands of low-metal, old stars. The type of stars found in a globular cluster are similar to those in the bulge of a spiral galaxy but confined to a volume of only a few million cubic parsecs. They are free of gas and dust and it is presumed that all of the gas and dust was long ago turned into stars. The average star density in a Globular Cluster is about 0.4 stars per cubic parsec. In the dense center of the cluster, the star density can increase from 100 to 1000 per cubic parsec. GCs contain a variety of exciting objects by themselves worth a continuous investigation, for instance strong and weak x-ray sources, neutron stars and millisecond pulsars, white dwarfs, cataclysmic variables, binaries, blue stragglers, planetary nebulae, etc. Moreover, they contain one of the most popular intrinsic variable stars, the so-called RR Lyrae stars [2]. 3.1 Structure In contrast to open clusters, most globular clusters remain gravitationally bound for time periods comparable to the life spans of the majority of their stars. However, a possible exception is when strong tidal interactions with other large masses result in the dispersal of the stars. Several distances describe a globular cluster: core radius r c, half-light radius r h and tidal radius r t. The core radius is defined to be the radius at which the surface brightness has dropped to half the central value. The half-light radius is the radius that contains 4
half of the light of the cluster and the tidal radius is defined as the radius beyond which the external gravitational field of the galaxy can separate individual stars from the GC. Typically values of these radii are from 1.5 pc for core radius to 50 pc for tidal radius [4]. 3.2 Dynamics Globular clusters can be described as a spherical N-body system, or more simply an ideal gas, whose elements are stars that interact through gravity. Three characteristic time scales come up when dealing with GC dynamics, which are crossing time, relaxation time and evaporation time. Crossing time is time required for a star, with typical cluster velocity, to travel the half-mass radius. It s in order of 10 6 years. The relaxation time tells us how long it takes a star to meaningfully change its velocity as a result of travelling through a GC. t relax 0.1 N ln N t cross (1) Evaporation time is a typical time of dissolving of globular cluster through loss of stars that gain sufficient velocity through encounters to escape its gravitational potential. The in this time γ fraction of all stars leaves the cluster. dn dt = γ N t relax = N t evap (2) The value of γ can be determined from escape velocity, which is related to gravitational potential Φ by equation ve 2 = 2Φ. Mean square escape velocity from a GC with a density ρ( r) is: v 2 e = ρ( r)v 2 e dv ρ( r)dv = 2 ρ( r)φ( r)dv ρ( r)dv = 4W p M where W p is the total potential energy of the cluster and M is its total mass. We can assume thermodynamic equilibrium and use the virial theorem W p = 2W k = M v 2 e. Combining the equations we get v 2 e = 4 v 2 (4) By assuming speed is distributed with Maxwell distribution we get that γ > 7.38 10 3 fraction of stars has v > v e. By doing the calculation, with N = 10 5, r h 10 pc and v 10 km s, the outcome of typical evaporation time is t evap = 10 10 years, which is comparable to the observed age of globular clusters. Evaporation is accelerated by tidal shocks, which implant additional kinetic energy to the stars, like a passage through the galactic disk or through a bulge of the galaxy. In old globular clusters core collapse can occur. From the virial theorem and total energy E = W k + W p we get E = W k, which, from ideal gas is E = 3 2 Nk bt. We then calculate specific heat: C V = de dt = 3 2 Nk B (5) In such a system energy flows from inner core into envelope, which raises cores temperature due to its negative specific heat, and expands the envelope. The center continues to contract and get hotter, sending out heat to the outer parts [5]. At one point this 5 (3)
Figure 2: Plot of logarithm of the central density against time, in units of initial half-mass relaxation time [6]. contraction is stopped due to influx of energy supplied by the formation and evolution of binary stars which provides a huge reservoir of energy; the binding energy of a hard binary can store the total energy of the whole cluster, which reverses the outward flow of energy. This effect governs a series of collapses and expansions of the cluster core, called gravothermal oscillations [6]. 3.3 Hertzsprung-Russel diagram of GCs Globular clusters are ideal for HR diagrams that show the properties of assemblies of stars with the same age and chemical composition. Most of the stars belong to main sequence, where they spend 90 percent of their lives. As the mass, upwards the main sequence line, increases, we come to the so called turn-off point. All main sequence stars, on a isochrone, have, after this point, consumed all their hydrogen in their cores and go into next, the sub-giant phase, and afterwards to red giant phase. 3.4 Blue Stragglers Blue stragglers are stars that seem no more than one or two billion years old, as they are massive and should have died compared to other cluster stars. It is thus surprising they are located in centers of 13 billion year old globular clusters. Origin and evolution of these stars are puzzling. The main question is: how can ancient star clusters, devoid of star-forming gas, construct new stars billions of years after star formation in the cluster ceased? Main theories of formation of these blue stars come down to 2 different types of origin, from a single star or from a binary stars or a collision of 2 stars. Single star formation has been ruled out with best indicator that BSS have very little lithium in comparison to normal main sequence stars. Absence of lithium is a sensitive 6
Figure 3: Schematic color magnitude of GC HR diagram [2]. Figure 4: Color magnitude diagram of M67. Blue stagglers are large dots in the box on the left [2]. 7
indicator of stars surface material temperature history. The reason for this is that lithium has a fragile nucleus, destroyed by proton capture at temperatures exceeding 2 million degrees and so large-scale mixing destroys surface lithium. This leaves us 2 types of origins where 2 stars are involved in the formation: Stellar collisions Binary star evolution Binary star evolution consistently dominates blue straggler production, but in some globulars blue straggler formation must be enhanced by dynamical encounters either via direct collisions or by stimulating mass transfer in binary systems [7]. 3.5 RR Lyrae stars RR Lyrae stars are a type of variable stars that are often found in globular clusters, and have been initially called cluster-type variables, although they are also found elsewhere. The amplitudes of their light curves range between 0.2 and 1.8 magnitudes in B. Their periods range between 0.25 and 1.2 days. Since their mean absolute magnitude is constant and fairly independent of metallicity (to within 0.3 mag), the RR Lyrae variables and the GCs, in turn, are ideal standard candles to measure distances [2]. Globular cluster HR diagrams generally show horizontal branches that have a prominent gap in the horizontal branch. This suggests there are no stars in that area of HR diagram, but are in fact just hidden as data on them is usually not adequate to be plotted. The importance of RR Lyrae variables for a large number of astronomical problems stems from their unique place in the HR diagram on the horizontal branch of globular cluster colormagnitude diagrams, showing that they have only a small spread in absolute luminosity as a group. They are also used to calibrate extragalactic distances and the expansion rate of the universe. 4 Globular cluster systems Almost all galaxies possess a family of globular clusters. Study of these system has boomed in 1990s with the launch of Hubble space telescope, that has made possible to study color magnitude and spatial distributions of galaxies of up to 100 Mpc away. The most basic property of a GC system is the number of GCs associated with a galaxy. Dwarf galaxies have the smallest number of GCs meanwhile giant elliptical galaxies contain over 10000 globular clusters. This number is roughly correlated to luminosity of host galaxy N L 2. As was already mentioned, specific frequency was defined to facilitate comparison between galaxies. It also shows how efficient a galaxy was at formation of globular clusters compared to stars elsewhere. S N = N10 0.4(M V +15) (6) Cluster ellipticals have S N ellipticals around 15 [2]. around 5 and their field counterparts about 3, while giant 8
Figure 5: The light curve characteristics of the three Bailey types of RR Lyrae stars of classes a, b and c [2]. Figure 6: Hertzsprung-Russel diagram of several galactic clusters with RR Lyrae stars in their gap [2]. 9
Figure 7: Two subsets of Milky Way cluster system, both showing typical peak at around M V = 7.4 ± 0.2 [8]. 4.1 Luminosity functions In a given galaxy, the number of globular clusters per unit magnitude interval M, is the luminosity function or GCLF of the cluster system. The GCLF is the visible manifestation of the cluster mass spectrum, and must result from both the initial mass function of cluster formation and its subsequent long-term erosion by various dynamical processes. The GCLF data now available for several galaxies show that it can be simply and accurately described by a Gaussian distribution, N(M) = Ae (M M V ) 2 /2σ 2 (7) where A is the simple normalization factor representing the total population N, M V is the mean of the distribution, and σ is the dispersion. This log-normal function is introduced purely as an empirical match to the real GCLF and as a convenient way to compare different galaxies. It is, however, an extremely successful match to the actual data. No significant departures from a strictly Gaussian form have been found even in samples containing 2000 clusters. What is remarkable is that the peak at M V = 7.4 ± 0.2 is nearly universal, as all well studied systems have shown. If this is correct globular cluster systems can be used to determine distance, although no well established theory exists that this is really true. 10
Galaxy < M V > σ(m V ) < [F e/h] > Fornax de -7.06 ± 0.42 0.95-1.85 ± 0.14 N147-5.99 ± 0.92 1.8: -2.05 ± 0.4 N185-6.49 ± 0.71 1.42-1.65 ± 0.25 N205-7.27 ± 0.27 0.72-1.45 ± 0.10 LMC -7.51 ± 0.32 0.96-1.66 ± 0.1 M33-7.01 ± 0.17 1.17-1.40 ± 0.2 Milky Way -7.36 ± 0.17 1.28-1.35 ± 0.05 M31-7.4 ± 0.5 1.2-1.21 ± 0.05 N1399-7.3 ± 0.3 1.4-0.85 ± 0.1 N3031-1.46 ± 0.31 N3377-7.05 ± 0.6 1.2 N3379-7.59 ± 0.4 1.2 N4365-6.95 ± 0.18 1.58 N4406-6.9 ± 1 1 N4472-7.40 ± 0.11 1.50-0.8 ± 0.3 N4486-7.32 ± 0.13 1.70-1.0 ± 0.2 N4649-7.45 ± 0.14 1.29-1.1 ± 0.2 N5128 1.35-0.84 ± 0.1 Table 1: Table of characteristic of several GS systems, with M V clearly similar everywhere [9]. 4.2 Metallicity Metallicity is defined as [Fe/H] = log 10 ( NFe N H ) star ( ) NFe log 10 N H sun with metallicity of the sun approximately 1.8 percent by mass and N Fe and N H are the number of iron and hydrogen atoms per unit of volume respectively. Out of all metals in a star iron is chosen because it is most easily detected with spectrographic observations. In recent years Hubble space telescope has made it clear that massive elliptical galaxies have two distinct sub-populations of globular clusters. The typical separation between the relatively metal-poor and metal-rich is around 1 dex, which is the unit often used for metallicity, a contraction of decimal exponent. Metal rich cluster thus have a similar metallicity to the host galaxy stars. The existence of multi-modal color distributions indicates that GCs formed in distinct star formation episodes from different metallicity gas, perhaps at different times.[2] 4.3 Black holes and GC systems It is worth mentioning that recent study on the relation between GCs and super massive black holes in the center of galaxies shows strong correlation: (8) M = m / N 1.08±0.04 GC (9) 11
Figure 8: Metallicity histogram for the giant elliptical galaxy NGC 5846 showing clear bimodal distribution [2] Where M is mass of the black hole and m / average GC mass in a galaxy. Thus, to a good approximation the super massive black hole s mass is the same as the total mass of the globular clusters in a specific galaxy. The study sampled from only 13 galaxies but to a reasonably good approximation we can say that a relationship exists [10]. 5 Epilog Globular clusters as well as GC systems represent a powerful tool to obtain a deep insight into a large variety of astrophysical and cosmological problems. Their study still represents a benchmark and a major field of interest in the international astronomical community. Perhaps one of the most remarkable impacts of GC research on other fields of astronomy is provided by the estimate of the ages of the Milky Ways globulars. GCs are, in fact, among the few objects in the Galaxy for which relatively precise ages can be derived. On the other hand, their age distribution and how ages vary with varying metallicity, spatial location in the Galaxy and kinematic properties make these systems direct tracers of the chronology of the first epoch of star formation in the Galactic halo and may help in understanding the whole process of galaxy formation. 12
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