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Cluster

Shashikant Joint Program Student Department Of Physics Indian Institute Of Science,Bangalore. December 10, 2003

1 Contents

1 Introduction 4

2 Color Diagram Of 4 2.1 ...... 4 2.2 Branch ...... 6 2.3 ...... 7 2.4 and Beyond ...... 8 2.5 Blue Stragglers ...... 8

3 Globular Cluster Ages 9

4 Chemical Properties 10

5 binaries and stellar remnants 10

6 radii of globular clusters 11 6.1 core radius ...... 11 6.2 median radius ...... 12 6.3 tidal radius ...... 12

7 time scales in the evolution of globular clusters 13 7.1 orbital time scale ...... 13 7.2 relaxation time ...... 13

8 Fokker Planck description and evaporation time of cluster 14 8.1 Fokker Planck description ...... 14 8.2 evaporation of globular cluster ...... 16

9 binary in globular clusters 17

10 disk shocking 18

11 formation of globular clusters 19

2 Abstract

In this article i am introducing some features of star clusters.I have discussed mostly globular clusters as open Custer’s are open question today.Globular clusters are the oldest in , their chemical properties are almost similar. Finally i have tried to discus the theoretical models to describe the main properties of globular clusters.

3 1 Introduction

We know that stars are formed in gaseous clouds. When stars formed in a cloud it seems reasonable to expect that there will be a tendency for a large number of them to be close together and hence they form star clusters. Clusters are divided into two broad categories known as and globular cluster. Open clusters are made up of population I stars. These clusters are found in galactic disk. They are relatively young objects. Their median age is around 108 years. Some observations suggests that formation of open clusters is an ongoing process. These clusters are small in size . They contain 102 to 103 stars in a region of size 1 to 10 pc. There are 105 open clusters in milky way. Since these clusters are found in galactic disk, it is hard to identify them.This is the reason that we do not have much more knowledge about open clusters. On the other hand globular clusters are quite old.They are made up of population II stars. A typical globular cluster contain 104 106 stars → within a median radius of 10 pc. There age is around 1010 years. Glob- ular clusters are spherical in shape but these spheres are inhomogeneous and concentrated toward center.

2 Color Magnitude Diagram Of Globular Clus- ter

Figure 1 shows the C.M. diagram of a globular cluster. It illustrates a number of basic features of globular cluster C.M. diagram. These include the main sequence, The giant branch, and the horizontal branch, each of which is discussed in the following subsections.

2.1 Main Sequence One of the key features of globular clusters is the well-defined Main Sequence extending from the turn-off to fainter magnitudes and redder colors. Globu- lar cluster stars on the main sequence derive their energy from the conversion of hydrogen to helium in the . the low end of the main sequence is determined by the magnitude limit of the observations. (there is

4 Figure 1: this figure shows the color magnitude diagram of M5 globular cluster also a theoretical lower limit to the main sequence corresponding to a stel- lar mass around 0.08M , Below which hydrogen burning no longer takes place in the stellar core.) A characteristic feature of the color-magnitude di- agrams of galactic globular clusters is that the turn-off of the main sequence occurs at fainter than for most star clusters in the solar neigh- borhood. This was first established by sandage arp, and others [1]. It was soon realized that the fainter luminosity of the main-sequence turn-off indi- cated that these globular clusters are old [2, 21] In the cores of more massive stars,Hydrogen is exhausted more rapidly,so that older stellar populations have main-sequence turn-offs at lower stellar masses and thus luminosities. As discussed in more detail later studies of the location of the main-sequence turn off in well studied galactic globular clusters give turn off masses of about 0.8M and corresponding ages of roughly 15 gyr. The sharp main sequence turn off of M5 is typical of globular cluster color-magnitude diagrams, Indicating that the stars within an individual globular cluster all formed at roughly the same time. another well studied cluster is M92 . The form of the main sequence turn off in M92 limits the age

5 spread between the constituent stars to about2.4% of the age of this cluster, or around 0.4 Gyr . Figure 1 illustrates another characteristic of globular cluster main sequences- they are very narrow. This narrowness indicates that all the stars in the globular cluster have a very similar chemical composition. It also constrains the fraction of binary stars within globular clusters. Unresolved binaries are expected to produce a population just above the main sequence, since the combined luminosity of the two stars exceeds that of a single star at the same color. (however, if the primary and secondary have significantly differ- ent masses, this effect is difficult to detect.) There are globular clusters where such a population of binaries has been detected, Such as NGC288 , where the inferred binary fraction is around 10% [5].Future observations,particularly with are likely to result in better constraints on the binary fraction in more globular clusters [6].

2.2 Red Giant Branch - The red giant branch (RGB) in globular clusters extends from the which connect it to the main sequence to brighter magnitudes and redder colors until the tip of the RGB is reached. Observational properties and characteristics of the RGB , As well as a survey of earlier literature, are given by stetson (1993) . While the detailed evolution of stars on the RGB is a complicated topic, The salient feature is that such stars possess hydrogen- burning shells, which advance outward as they ascend the giant branch. the ascent is terminated by the ignition of the degenerate helium core which forms in the center of a star during this period of its evolution. Details of theoretical studies of this process are reviewed by Iben (1974) and Renzini (1977) . Like the main sequence, the RGB of most individual globular clusters is narrow and well defined, placing limits on chemical in homogeneities, and on variation in stellar physics among stars on the RGB. A comparison of the RGBs of different clusters reveals that globular clusters of higher exhibit giant branches that are shallower and redder than low metallicity clusters [7]. Like stars on the main sequence, higher metallicity giant stars have increased opacity, due to electrons from metals, which allows stars to maintain equilibrium at a lower temperature. the astrophysics of stars on the RGB is complex, and the exact location of the RGB is dependent on mass

6 loss rates and the details of the convective processes within such stars, often treated in terms of“mixing length theory” [8].

2.3 Horizontal Branch The horizontal branch is composed of stars with helium burning cores which have evolved off the RGB. The horizontal branch is identified on color mag- nitude diagrams as a strip of stars. bluer than the RGB and brighter than the main sequence, which have a range of colors but similar luminosities(thus horizontal on the usual color magnitude diagram). The presence of RR Lyrae variables within the or RR Lyre gap on the horizontal branch is a defining feature of population II stars. A color magnitude diagram of the population I stars of the galactic disk is free of stars in this region. The horizontal branch of globular clusters has a particular importance on understanding these systems,As well as shedding light on stellar and galactic evolution. a simple method of describing the morphology of the horizontal branch is through a measure of the relative numbers of stars blueward and redward of the RR Lyrae gap. One way of quantifying this is through the parameter: C = (B R)/(B + V + R) (1) − Where B is the number of stars on the horizontal branch on the blue side of the RR Lyrae gap, R the number of stars on the red side, and V is the number of variables on the horizontal branch . In order to explain the detailed location of stars on the horizontal branch, it has been known for some time that mass loss during the earlier RGB phase must be invoked [9]. Observationally, the horizontal branch morphology (i.e., the color distri- bution of horizontal branch stars) of milky way globulars exhibits a broad range, and is determined by a number of physical effects. in general, the most important parameter is the metallicity of the cluster. Horizontal branch stars of higher metallicity are redder than those of lower metallicity as a re- sult of higher opacity on their envelopes. Metallicity is therefore the ”first parameter” in determining horizontal branch morphology. There are other parameters like age, helium abundance etc.

7 2.4 Asymptotic Giant Branch and Beyond When the helium in the core of a horizontal branch star is exhausted, The core contracts and helium begins burning in a shell around the core. Beyond this helium shell is the hydrogen burning shell. The increase in energy gen- eration means that the star ascends the giant branch for the second time. This second ascent is known as the asymptotic giant branch (AGB) phase of . Cohen (1976) studied AGB stars in globular clusters and discovered that many exhibited significant mass loss in the form of stellar winds. Other peculiarities include helium shell flashes, believed to be a con- sequence of the helium shell being spatially thin, which can cause the star to migrate briefly to the instability strip of the color magnitude diagram. Like the RGB phase, the details of the astrophysical processes of stars on the AGB are complex and beyond the scope of this article. Calculations of the post AGB phase of stellar evolution are made more difficult by the occurrence of loss. However,a low mass star eventually exhausts its hydrogen and helium shells and loses its extended envelope, ending up as a .

2.5 Blue Stragglers One oddity of the color magnitude diagram of globular clusters is the pres- ence of” blue stragglers”. The blue stragglers give the appearance of being an extension of the main sequence beyond the turn-off point. Until relatively recently, only a handful of globulars were known to contain such stars, al- though open clusters in the milky way frequently exhibit the phenomenon. However, subsequent observations have revealed that blue stragglers are a common feature of globular clusters as well. Various formation mechanisms for blue stragglers have been proposed, and at present it seems likely that more than one may play a significant role [10, 19]. Perhaps the simplest explanation of blue stragglers is that they are stars that formed later than the bulk of the stars in a globular cluster [12]. Such objects would still sit on the main sequence, beyond the turn off point of the majority of stars. The primary problem with this picture is the large implied age difference between the blue stragglers and the other stars in an environment which contains little, if any, gas. There does not seem to be any material out of which such late forming stars could have been produced.

8 A more promising scenario involves stellar mass transfer between roughly equal mass components on binary systems [13, 14, 15]. The more massive of the two stars will eventually evolve along the branch, during which time it expands and overflows its Roche lobe. mass is then transferred from this subgiant to its companion, which is still on the main sequence.(angular momentum loss in the form of a magnetic wind can accelerate the process by bringing the stars together more rapidly, in which case the primary need not have evolved significantly.) while the subsequent detailed evolution of the system has different routes, one possible end point is a common enve- lope binary in which the components eventually merge to form a single star. Clearly the product of such a merger will have a mass greater than the main sequence turn off mass, but not more than twice this mass. It has addition- ally been proposed that, under certain (probably rare) conditions, helium enriched material can end up in the envelope of the resulting , leading to a luminosity somewhat in excess of that expected for the star’s mass [16]. Another popular picture invokes stellar collisions to produce blue strag- glers [17]. This has the attraction that the high stellar densities in globular clusters make collisions relatively likely. (A simple calculation shows that the typical time for stellar encounters in globular cluster cores is significantly less than the age of globular clusters.) Even in low density systems, the collision rate may be sufficiently high thanks to close encounters between binary-single and binary-binary systems [18].

3 Globular Cluster Ages

The ages of globular clusters has been a topic of great interest for many years. The primary reason for this interest is cosmological: the universe must be older than the objects within it. Milky Way globulars contain the oldest stars for which reliable age estimates are available, and thus provide a lower limit to the age of the universe. Derived globular cluster ages have frequently been comparable to cosmological age determinations, such as those obtained through measurements of the Hubble constant. The fundamental method of establishing the absolute age of a globular cluster is to determine the mass of stars at the main sequence turn off. The observational parameters that provide the input to this process are the color

9 and of the turn off, along with the distance to the clus- ter and the reddening along the line of sight. This allows a calculation of the Mv of the main sequence turn off. models are then used to determine the total energy produced in the core of the star and the effective temperature of the stellar . Additional input into these models includes metallicity and helium mass fraction. The bolometric magnitude and effective temperature are compared with the re- sults of stellar interiors models which relate these quantities to stellar mass and age as follows

L t lg T O [0.019 (lg Z)2 + 0.065 lg Z + 0.41 Y 1.179] lg L ! ≈ − Gyr ! + 1.246 0.028 (lg Z)2 0.272 lg Z 1.073 Y (2) − − − where t is the age in Gyr.

4 Chemical Properties color magnitude diagrams are sensitive to stellar helium abundance, since an increase in the helium fraction leads to bluer stellar colors. similar argu- ments can be applied to the giant branch of globular cluster color magnitude diagrams. here, elements such as Fe,Si and Mg have a significant effect on the opacity of the outer regions of the stars, so that star to star variations in these elements produce a dispersion in the color of the giant branch. with the observation data we find very small dispersion. which indicates that metallic- ity homogeneity within globular cluster is remarkable an interesting aspect of observational studies is that carbon depletion is weaker in metal rich clusters, which implies that deep mixing is more efficient in low metallicity stars.

5 binaries and stellar remnants the great age of milky way globulars means that a significant fraction of their original stars has evolved off the main sequence. the final fate of these stars includes the formation of white dwarfs, neutron stars, and possibly black holes. the high central densities of globular clusters are also relevant, since they increase the probability that binary systems of compact objects form.

10 with the help of HST the study of white dwarfs has advanced. the primary conclusions are that white dwarfs exist in globular clusters, searches for accreting white dwarfs have also been undertaken. x-ray ob- servations are a useful way to search for these objects. observations made at much higher resolution with the ROSAT high resolution images have con- firmed the existence of low luminosity x-ray sources in globular clusters, and allowed a more reliable identification of sources in globular clusters [19] bright x-ray sources in globular clusters are generally accepted to be pro- duced by accreting neutron stars. when a captures a companion, the accretion of material produces x-ray emission and the object is observed as a low mass x-ray binary. while such observations provided the first evi- dence for neutron stars in globulars, the finding has been confirmed by the presence of millisecond in globulars. [4] at present, there is no compelling evidence for the presence of black holes in globular clusters. surface brightness profiles in some post core collapse clusters may indicate the presence of a central black hole, but this is currently a matter of debate. if black holes do exist in globular, they are, in principle, detectable through their x-ray signatures.

6 radii of globular clusters the spatial distribution of stars in a globular cluster is fairly nonuniform and is centrally concentrated. thus we define three different radii that are of use.

6.1 core radius core radius is observationally determined as the radius at which the projected surface brightness of the globular cluster falls to half its central value. there are several models are to explain stellar evolution. out of these king model is found to be a reasonable description of the globular cluster. the distribution function for the king model is given by

0 if E > E f(E) = 0 K[e−β(E−E0) 1] otherwise ( −

where K, β and E0 are constants and E is the energy per unit mass.

11 for such a model, the core radius is defined as

9 rc = (3) s4πGρ0β !

where ρ0 is the central density.

6.2 median radius it is the radius of the sphere that contains half of all the light emitted by the cluster.

6.3 tidal radius this determines the outer edge of the cluster beyond which a member of the cluster will be influenced more by the external gravitational field of other matter in the rather than by the self of the cluster. let Φ(X) denote the gravitational potential of the smooth distribution of matter in the galaxy, where the origin is taken to be the center of the galaxy . let φ(x) denote the gravitational potential of the stars in the globular cluster itself, with x = 0 denoting the cluster center. tor an order of magnitude estimate, we can take

g 2Φ x(4πGρ ) bg ≈ 5 ≈ bg and

g x 2φ x(4πGρ ) self ≈ 5 ≈ cluster the tidal radius can be estimated from the distance at which

gbg = gself (4) it is clear that at tidal radius

ρcluster = ρbg (5)

12 7 time scales in the evolution of globular clus- ters

7.1 orbital time scale the first time scale that characterizes a gravitational many body system is the orbital time scale. it is defined as

t R/v (6) cross ≈ where R is the size of the system and is the typical velocity. if the system is in steady state, then from virial theorem we have GM v2 (7) ≈ R so that GM tcross (8) ≈ s R3 this time scale also decides the duration over which the initial phase of violent relaxation operates in a self gravitating system.

7.2 relaxation time this time scale takes into account due to gravitational collisions. this is the time scale over which the granularity of the system can make the deviate significantly from the paths that stars would have followed under the action of the gravitational field generated by the smooth density of matter. mathematically relaxation time is as N trelax α (9) ≈ lnN  where α os a numerical coefficient , usually α 0.1 for a globular cluster, ≈ (N/lnN) 8.7 103 ≈ ∗ and hence t 109yr relax ≈

13 thus we see that

trelax < tage this means that relaxation effects would have played a major role in deter- mining the present structure. there are three key effects that arise because of trelax tage for the globular cluster. the first one is the obvious feature that the distributionh of stars will tend toward a state of maximum entropy over this time scale and it leads to inhomogeneity for a self gravitating system. second the system will tend toward a state with equipartition of kinetic energy in a time scale t trelax . stars with larger kinetic energy will, on the average, lose energy to stars≈ with less kinetic energy. if the initial distribution function is determined by violent relaxation, thin the initial velocity distri- bution of stars will be independent of stellar mass and hence more massive stars will have larger kinetic energy. over a time of the order of trelax , more massive stars will lose energy to less massive stars and will sink toward the center of the system. this will lead to mass segregation in the system. third effect is , the escape of high velocity stars in the tail of the Maxwellian distribution.(the velocity distribution of stars in a globular cluster is de- scribed by Maxwellian distribution)

8 Fokker Planck description and evaporation time of cluster

8.1 Fokker Planck description globular clusters contain a large number of stars, so we can choose a distri- bution function f(x, v, t) such that dN = f(d3xd3v) denotes the number of stars at time t in the range (x, x + d3x) and (v, v + d3v) . we can choose f as the relative probability of finding a star at some location in phase space . in general, the distribution function will evolve by the equation df = c (10) dt where c denotes the collision terms. for systems dominated by a long range force like gravity, even distant encounters can play an important role in

14 determining the form of bf c . the effect of distant encounters can be treated as a diffusion in the velocity space. the form of the collision term well then be ∂jα c = (11) −∂pα where jα is a current in the momentum space and pα = mvα is the momtum. the particle current jα is given as

2 1 ∂(σαβf) jα(p) aα(p)f(p) (12) ≡ − 2 ∂pβ therefore the collision term becomes 2 ∂(aαf) 1 ∂(σαβ f) C = α + (13) − ∂p 2 ∂pα∂pβ the functions aα(p) and σαβ(p) can be thought of as diffusion coefficients in the momentum space. the above equation is similar to the Fokker Planck equation in statistics. changing the description from momentum space to velocity space the collision term C becomes as ∂ 1 ∂2 C[f] = [f(w)D( vi)] + [f(w)D( vi vj)] (14) −∂vi 4 2 ∂vi∂vj 4 4 these diffusion coefficients may be written as

2 ∂ D( vi) = 4πG ma(m + ma) ln Λ h(v) (15) 4 ∂vi and 2 2 2 ∂ D( vi vj) = 4πG ma ln Λ g(v) (16) 4 4 ∂vi∂vj now we consider an specific example, in which the distribution function is Maxwellian with a velocity dispersion σ2 such that

ρ 2 2 f(v) = e−v /2σ (17) m(2πσ2)3/2 the diffusion coeff. becomes now in terms of error function 8πG2ρm ln Λ D( v ) = G(X) (18) 4 k − σ2

15 and 4√2πG2ρm ln Λ erf(X) G(X) D( v 2) = [ − ] (19) 4 ⊥ σ X where X = (v/√2σ) and

1 2X 2 G(X) = [erf(X) e−X ] 2X2 − √π given the diffusion coefficients for the velocity, it is possible to calculate the corresponding coefficient for any other function of velocity.

8.2 evaporation of globular cluster one possible definition of relaxation time could be

v2 t (20) relax ≡ D( v2 ) 4 parallel because vrms = √3σ hence X = 1.225 for v = vrms thus we get the relaxation time σ3 t = 0.34 relax G2mρ ln Λ or 10 3 3 −3 1.8 10 yr σ M 10 M pc trelax = ∗ [ ][ ] (21) ln Lambda "10mk/s# m ρ because the density and σ can vary across a system, trelax has no deep signifi- cance for an inhomogeneous system. now we replace ρ with the density inside the system’s median radius r , 3σ2 with v2 and Λ with r v2 /Gm 0.4N h h i hh i ≈ this yields 8 1/2 6.5 10 yr M M rh 3/2 trh = ∗ [ 5 ] [ ] (22) (0.4N) 10 M m 1pc the subscript is changed from relax to rh as a reminder of these substitu- tions. this revised and more precise estimate of relaxation time can be used to provide a simple model for the evaporation of stars from a cluster. the sim- plest model for a cluster is based on the assumption that the cluster evolves self similarly and shrinks as the stars escape from it. because the stars that escape are predominantly due to weak encounter, they have negligible energy,

16 and we might assume that the total energy of the cluster remains constant as it shrinks. taking the energy of the cluster to be E (GM 2/R) ,we get M(t) 2 ∝ − the scaling that R(t) = R0[ M0 ] , where M0 and R0 are the initial values. assuming that a fixed fraction λM of all mass is lost because of evaporation in a time scale t t , the mass loss rate is governed by the equation ≈ rh dM = λM/t (23) dt − rh from equation 22 we have t2 MR3 rh ∝ so that 0 3 3 1/2 trh = trh (MR /M0R0 ) (24) combining these results, we get 7/2 dM λM0 = 0 5/2 (25) dt trh M this is a differential equation having the solution 7λt 2/7 M(t) = M0(1 0 ) (26) − 2trh this suggests that the cluster evaporates completely in a time scale tev = 0 (2trh /7λ) if λ 0.007 than ≈ 2 0 tev = 10 trh (27) observations suggest that there are no globular cluters with t < 3 107 yr, rh ∗ which is 1% of the currently estimated age of the galaxy. this is consistent with the ∼hypothesis that if there were many globular clusters initially, those with shorter relaxation time would have evaporated by now.

9 binary stars in globular clusters earlier in the observation part we have seen that some binary stars are found in globular clusters. we can classified these binaries into two main types depending upon their binding energy.i.e. Gm m 1 1 2 < m σ2 (28) 2a 2 a

17 where m1 and m2 are masses of the stars and a is the semi major axis. σ is the relative velocity of the binary and the field star. if the magnitude of binding energy of the is small compared with the random kinetic energy of the field stars then the binary is called hard. i.e. Gm m 1 1 2 > m σ2 (29) 2a 2 a let us see what happens in a collision of a field star and a binary. if we use energy equipartition we can easily see that the soft binaries become softer by the collision. any encounter between a field star and a soft binary can be thought of as a perturbation to the binary of a particle with reduced mass. because the binary is very soft, the kinetic energy of the effective (reduced mass) particle is much less than the kinetic energy of the field star. as the system moves toward energy equipartition, encounters will increase the energy of the particle with reduced mass, that is the internal energy of the binary. hence soft binaries not play any vital role in the evolution of the cluster. on the other hand, the evolution, is significantly affected by the encounter between field stars and hard binaries. the relative velocity of the binary and the field star is σ , which is much smaller than the orbital speeds in a hard binary. in an encoun∼ ter, if the magnitude of the internal energy E of the hard binary is increased (so that the binding becomes harder), | | then the field star gains energy and escapes back to infinity. however, if E decreases, the field star is likely to become bound to the binary, thereby |forming| a temporary triplet. eventually one of the stars will be ejected from the triple system with an escape speed typically of the same order as that of the orbital speed. it follows that the energy of the ejected star is higher than that of the incoming single star and hence, by energy conservation, the binding energy E must decrease, making E increase. hence we conclude that in an encounter of a field star and a hard| | binary, hard binary becomes harder.

10 disk shocking the discussion so far has treated clusters as isolated physical entities. but this is not true always. let us see what happens when globular cluster passes through galactic disk? the stellar distribution in solar neighborhood can be

18 −3 parameterized by a mean density ρ 0.18M pc and a vertical scale height of z 350pc . this scale height is m≈uch larger than the typical tidal radius 0 ≈ of a cluster, and hence a cluster will be totally immersed in the disk while it passes through it. the mean density of stars within the cluster decreases rapidly from the center to the edge of the cluster, whereas the mean density of the material is reasonably uniform. it follows that stars sufficiently away from the center of the cluster will be influenced strongly by the gravitational force of the background material in the disk rather than by the self gravity of the cluster. this effect is called disk shocking. the time scale for disk shocking is roughly

−2 −3 12 vc Mc R R R r1/2 tsh 5 10 ( −1 )( 5 )( ) exp − yr. ≈ ∗ 220kms 10 M 1kpc  − hR  1pc ! (30) this equation assumes that the cluster follows a circular orbit with a rotation velocity vc . the basic interpretation of this time scale is that this represent the characteristic elapsed time before a globular cluster is destroyed. recent theoretical work has refined the analysis of disk shocking and indicates that the destructive effects of this process may be considerably more important than indicated by above equation. [20]

11 formation of globular clusters at present there is no widely accepted theory of globular cluster formation. large number of models are proposed, but non of them is completely sat- isfactory . we divide these models into main three classes, primary model, secondary model and tertiary models. primary formation models are those in which globular clusters form before . secondary models have galaxy and globular cluster formation occurring roughly contemporaneously. while tertiary models are those in which globular clusters form after their host galaxies. its an open question and I leave it for future.

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