relaxation-limited evaporation of globular clusters

New Astronomy 17 (2012) 411–423

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New Astronomy

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Relaxation-limited evaporation of globular clusters

Maurice H.P.M. van PuttenKorea Institute for Advanced Study, Dongdaemun-Gu, Seoul 130-722, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 September 2011Received in revised form 2 November 2011Accepted 3 November 2011Available online 13 November 2011

Communicated by E.P.J. van den Heuvel

Keywords:Globular clustersEvaporationRelaxation

1384-1076/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.newast.2011.11.003

E-mail address: [email protected]

Evaporative evolution of stellar clusters is shown to be relaxation limited when the number of starssatisfies N� Nc, where Nc ’ 1600. For a Maxwell velocity distribution that extends beyond the escapevelocity, this process is bright in that the Kelvin–Helmholtz time scale, f�1

H trelax , is shorter than the Ambar-tsumian–Spitzer time scale, f�1

N trelax , where fH > fN denote the fractional changes in total energy and num-ber of stars per relaxation time, trelax. The resulting evaporative lifetime tev ’ 20.5trelax for isolated clustersis consistent with Fokker–Planck and N-body simulations, where trelax is expressed in terms of the half-mass radius. We calculate the gray body factor by averaging over the anisotropic perturbation of thepotential barrier across the tidal sphere, and derive the tidal sensitivity dlntev/dy ’ �1.9 to �0.7 as a func-tion of the ratio y of the virial-to-tidal radius. Relaxation limited evaporation applies to the majority ofglobular clusters of the Milky Way with N = 104–106 that are in a pre-collapse phase. It drives streamsof stars into the tidal field with a mean kinetic energy of 0.71 relative to temperature of the cluster. TheirS shape morphology leads in sub-orbital and a trails in super-orbital streams separated by 3.4 r/X in theradial direction of the orbit, where X denotes the orbital angular velocity and r the stellar velocity dis-persion in the cluster. These correlations may be tested by advanced wide field photometry andspectroscopy.

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1. Introduction

Globular clusters represent some of the oldest stellar systems asindicated by their stellar ages of at least 12 Gyr (Peebles and Dicke,1968; VandenBerg et al., 1996; Hansen et al., 2002) consistent withthe ages of white dwarfs in the thin disk of the Milky Way (Nohand Scalo, 1992; Salaris et al., 2000; DeGennaro et al., 2008; Isernet al., 2008; Renedo et al., 2010), the oldest halo field stars such asHE 2523-0901 of 13.2 Gyr ± 2.7 Gyr (13.4 ± 1.8 Gyr based on U/rchronometry alone, Frebel et al. (2007)) and the age of the Universeof 13.75 ± 0.11 Gyr (Jarosik et al., 2010). The population of clustersof the Milky Way as observed today is but a shadow of a primordialpopulation (Chernoff and Weinberg, 1990; McLaughlin and Fall,2008; Prieto and Gnedin, 2008), that may have formed in the eraof reionization (VandenBerg et al., 1996). Their ages hereby exceedtheir relaxation times by a factor of a few to an order of magnitude.They are essentially virialized close to thermal equilibrium (Mey-lan and Heggie, 1997) and some 80% satisfy the King model (KM,King (1966) and Chernoff and Weinberg (1990)), which bears outwell also in numerical simulations (Chatterjee et al., 2010). Theexception is a minor fraction of clusters in the direction of thegalactic halo with a centrally peaked core, commonly designatedas post core-collapse (PCC) (Chernoff and Djorgovski, 1989; Trageret al., 1993; Johnson et al., 2008).

ll rights reserved.

Prior to core collapse, the leading-order evolution of stellar clus-ters is well-described by two-body interactions in the small anglescattering approximation (Spurzem and Aarseth, 1996; Aarsethand Heggie, 1998), that naturally gives rise to evaporation of thecluster (Ambartsumian, 1938; Spitzer, 1940). This process is amena-ble to detailed modeling by Fokker–Planck and N-body simulations(Hénon, 1961; Spurzem and Aarseth, 1996), there results of whichcan be interpreted statistically (reviewed in Meylan and Heggie(1997)) in terms of evaporation, tidal effects, stellar mass loss, masssegregation and heating and crossings of the galactic disk (Gnedinand Ostriker, 1997; Watters et al., 2000; Joshi et al., 2001; Fregeauet al., 2002; Heggie, 2001; Chatterjee et al., 2010; Heggie, 2011).

There is an increasing interest in these various processes to de-velop efficient semi-analytic approaches to the birth and evolutionof large numbers of stellar clusters as tracers of cosmological evo-lution of galaxies (Kruijssen et al., 2011). Evaporation stands out asa common process in many-body systems that includes for in-stance atmospheric science (e.g. Chamberlain (1963), Hays andLiu (1965), Chamberlain and Campbell (1967), Shizgal and Black-more (1986), Johnson et al. (2008) and Johnson (2010)), where itextracts the lightest elements from an otherwise nearly thermalstate (e.g. Chavanis et al. (1996, 2002)).

Evaporation is a radiation process that defines a finite Kelvin–Helmholtz time scale due to energy extraction by the escapingstars. Key to their evolution, therefore, is an accurate determina-tion of the energy loss carried off by stars escaping the cluster a

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412 M.H.P.M. van Putten / New Astronomy 17 (2012) 411–423

determination of the time scale of evaporation, in particularwhether it follows the relaxation time scale or the crossing timescale (Heggie, 2001) or possibly a combination thereof (Baumgardt,2001). The energy extraction by escaping stars is potentially usefulin classifying post core-collapse clusters (Küpper et al., 2008b).

Evaporation of stellar clusters has received considerable atten-tion (reviewed by Johnstone (1993)) with an emphasis on massloss typically more so than energy loss (but see Wielen (1967)).Yet, the enthalpy distribution function of all stars, those remainingin the globular cluster and those expelled by evaporation, continu-ously spreads towards increasingly negative and positive valuesassociated with contraction and heating. The kinetic energies ofevaporated stars in the halo hereby carry a thermal imprint ofthe history of the clusters. Since the enthalpy distribution functionis unbounded in the point particle limit, there is the potential forextended lifetimes already at small binary mass fractions. How-ever, observational evidence for a dynamically significant popula-tion of compact binaries has not been demonstrated, nor in thecorrelation between the observed population of luminous X-raybinaries and the binary star-star interaction rate (Pooley et al.,2003) that points to relatively small binary populations that donot necessarily affect the lifetime of the clusters.

In this paper, we explore the parameter range for relaxation-limited evaporation and some observational consequences for thepresent state of globular clusters in the Milky Way. We discussthe thermodynamics of evaporating globular clusters with tidalperturbation in the large-N limit of equal mass stars as a functionof a tidal parameter

y ¼ Rrt

ð1Þ

representing the ratio between the virial radius R (close to the half-mass radius, to be discussed further in Section 2 below) and the ti-dal radius rt of the cluster. For old clusters such as of the Milky Way,rt is much larger than the radius of the cluster. The tidal sphere hasthe effect of lowering the potential barrier for stars to escape alongorbits into the halo of the galaxy and, generally, introduces stronganisotropy in the directions of the escaping stars defined by the La-grange points L1 and L2 (Hayli, 1967). Existing approaches to incor-porate tidal effects commonly use the minimum energy for escapeat L1 (Spitzer, 1987; Lee and Ostriker, 1987; Chernoff and Weinberg,1990; Takahashi et al., 1997; Ross et al., 1997), effectively applyinga spherically symmetric extension thereof to the tidal sphere. How-ever, since last scatterings arise typically from within the half-massradius (Baumgardt et al., 2002), the escape rate is effectively deter-mined by the probability of escape derived from spherical averagingover the anisotropic tidal field for randomly oriented ejections. Wethus derive the evaporative lifetime up to the moment of core-col-lapse to be

tev ’ 20:53 1� 1:90yþ 2:78y2 � 2:40y3 þ 0:84y4 þ Oðy5Þ� �

trelax;

ð2Þ

where the relaxation time trelax expressed in terms of the half-mass radius Rh (the coefficient is 13.85 when trelax is expressedin R). While our approach is a largely explorative extension toAmbartsumian (1938) focused on the sensitivity of the evaporativelifetime to energy loss, some confidence can be gained by comparing(2) with known results reviewed in Spitzer (1987), Takahashi et al.(1997), Meylan and Heggie (1997) and Heggie and Hut (2003).

Early in their evolution over the first few Gyr, globular clustersmay have experienced considerable mass loss in stellar winds(Aarseth and Heggie, 1998; Baumgardt and Makino, 2003) due tostellar evolution (e.g. Applegate (1986)), that created a crucial fac-tor in the likelihood of survival of a cluster after its formation(Chernoff and Weinberg, 1990). A common approximation used

in Fokker–Planck and N-body simulations is an exponential evolu-tion on a time scale tm (Aarseth, 1985; Chernoff and Weinberg,1990; Meylan and Heggie, 1997). The same approach is readilyincorporated in a thermodynamic model as a regular perturbationprovided that tm� trelax. Evaporation has also been found to nar-row the mass spectrum due to a relatively enhanced rate of escapeof the lightest stars, by which older clusters tend evolves similarlyto a system of equal mass stars (Chernoff and Weinberg, 1990). Theaforementioned analogy with atmospheric science is useful here,noting that evaporation leads to an early escape of light moleculessuch as hydrogen and helium from the terrestrial atmosphere leav-ing only the relatively heavy molecules of nitrogen and oxygen tosurvive. Early evolution of star clusters therefore deviates from theevaporation driven evolution of single mass clusters. However, forthe old clusters of the Milky Way at the present epoch, these con-siderations are less critical, and will not be considered here further.

As an application, we focus on high velocity stars in and aroundglobular clusters that may provide ‘‘real-time’’ snapshot of lastscattering events (Hénon, 1961). These last scattering events fromwithin the half-mass radius (in the limit of isolated clusters, Baum-gardt et al. (2002)) and most frequently derive from the core,where the density is highest and the local relaxation time is theshortest (Aarseth, 1974; Johnstone, 1993). This includes tidal tailsproduced by ejections that, indirectly, serves as a diagnostic forthe cluster temperature as, e.g., in NGC 6254 (Leon et al., 2000)and 47 Tuc (Lane et al., 2010a,), where the velocity dispersion risesto about 60% at the tidal radius relative to the velocity dispersioninside the cluster. Observational data on tidal tails hereby providesa signature that may be used to calibrate numerical models whenapplied to individual clusters, e.g., M15 (Grillmair et al., 1995; Dru-kier et al., 1998), Palomar 5 (Odenkirchen et al., 2001; Meylanet al., 2001; Dehnen et al., 2004), NGC 5466 (Belokurov et al.,2006) and Palomar 14 (Sollima et al., 2011) and their tails by N-body simulations (e.g. Heggie (2001), Lee et al. (2006)). This mayinclude further processes such as additional heating by past diskcrossings for clusters near the center of the galaxy, but possiblyless so for the outer clusters (Gnedin and Ostriker, 1997; Leonet al., 2000). Some of these tidal tails display an S–shaped morphol-ogy (Heggie, 2001; Belokurov et al., 2006; Kroupa, 2008) along theLagrange points L1 and L2 on short scales, e.g., in NGC 5904 (Leonet al., 2000; Jordi and Grebel, 2010), and along the orbit of the clus-ter on long scales, e.g., NGC 5644 (Fellhauer et al., 2007), where theleading (trailing) tail is inside (outside) the cluster orbit around thegalactic center. The temperature of the clusters varies broadly,from about 1 km/s in Pal 12 (Leon et al., 2000) to about 15 km/sin x-Centauri (van der Ven et al., 2006), with evidently differentprospects for the manifestation of these tidal tails.

The planned Large Synoptic Survey Telescope (LSST, 2009), theBaryon Oscillation Spectroscopic Survey (BigBOSS, a multi-objectspectrograph, Schlegel et al. (2009)) the Japan Astrometry SatelliteMission for INfrared Exploration (JASMINE Gouda et al. (2010)),and GAIA (Lindegren, 1995) seem to be ideally suited for studyingthese tidal tails by large field of view photometry and spectros-copy. This development offers novel opportunities for testing thephysics of evaporation by measuring correlations between thekinematics of stars in tails to the thermal state of the clusters.

In Section 2, we define notation and the basic quantitiesdescribing the thermodynamic state of virialized star clusters, theircrossing and relaxation time. In Section 3, we introduce the generalset-up for the evolution equations. in Section 4, the parameterrange for relaxation-limited evaporation is identified. In Section 5,we calculate the evaporation coefficients for the number of starsand energy emitted per unit of relaxation time. In Section 6,we extend our model by regular perturbation analysis toinclude tidal interactions. In Section 7, we discuss the remainingrelaxation-limited evaporative lifetime of the globular clusters in

M.H.P.M. van Putten / New Astronomy 17 (2012) 411–423 413

the Milky Way. We give model predictions for observational teston tidal tails in Section 8 and summarize our results in Section 9.

2. Virialized star clusters

By the large number N = 104 � 6 of stars, the globular clusters ofthe Milky Way are essentially virialized with constant mass-to-light ratio in their cores (Meylan and Heggie, 1997). For globularclusters of the Milky Way, corrections to the Virial Theorem dueto tidal forces are generally small (Lee and Rood, 1969). In lightof the aforementioned KM morphology, we begin with a homoge-neous and isolated cluster of single stars of the same mass, m, andno binaries. The system is then described by a total number ofstars, N and virial radius, R. The Newtonian binding energiesUij = �Gm2/jri � rjj between stars i and j (i – j) give rise to a totalenergy H representing the sum of total potential energy U betweenthe stars and the total kinetic energy Ek of their individual motions,satisfying

H ¼ U þ Ek ¼14

Ri–jUij ’ �GN2m2

4R; ð3Þ

where we used the Virial Theorem, 2Ek þ U ¼ 0, on the mean values

U ¼ 12N

Ri–jUij ’ �GmM

2R; Ek ¼ N�1Ek; ð4Þ

and where (3) is a defining relationship for the virial radius R (cf.Meylan and Heggie (1997)). The ratio M/R is a hereby a measurefor the mean kinetic energy in the system.

The relaxation time for a homogeneous population of starsarises out of elastic 2-body encounters, produce random deflectionangles D/ ’ ±Uij/Ek, where Uij = �Gm2/b denotes the binding en-ergy associated with the impact parameter b for an interaction be-tween stars i and j. High velocity encounters hereby experiencerelatively small angular deflections, giving rise to asymptotic free-dom generic to scattering problems. In a crossing time

tc ¼Rv ¼

R3=2ffiffiffiffiffiffiffiffiffiffiffiGmNp ; ð5Þ

up to a factor of unity, the numerous encounters of a star wanderingthrough a cluster give a non-zero expectation value of the varianceof these deflection angles. (The crossing time corresponds to thefree-fall time scale within a factor of two.) The result can be sum-marized in a relaxation time for an expectation of unit variance(Spitzer, 1940)

trelax ¼N

6 ln Ntc ¼ 2:4� N1=2R3=2

1

ln NMyr; ð6Þ

where R = R1 pc. The approximations involved allow for slight vari-ations, while preserving the main dependencies on the overall char-acteristics of the stellar system. The details of these differences shallnot concern us here.

The virial radius R is closely related to the half-mass radius rh

are closely related, e.g., based on the Plummer model, we have(Aarseth and Fall, 1980; Meylan and Heggie, 1997)

GM2c

2R¼ �U ¼ 3pGM2

c

32Nðrh=1:30Þ ¼ 0:38GM2

c

rh; ð7Þ

whereby R = 1.30rh. Consequently, trh ’ 0.67trelax, where trh denotesthe relaxation time estimated on the basis of rh.

3. Equations for evaporative loss of mass and energy

Evaporation is a radiation process in which stars with sufficiententhalpy escape, out to infinity for isolated clusters or out into thetidal field of the host galaxy. This process can be described by frac-

tional changes fN and fH in the number of stars, N, and, respectively,the energy of the cluster per unit of a characteristic time, that isgenerally close to the relaxation time. It represents the statisticaloutcome of a relatively large phase space of possible stellar trajec-tories, that emerges from, most frequently, soft two-bodyinteractions.

When the relaxation time is intermediate between a dynamicalcrossing time and the evaporative lifetime of the cluster, the es-cape probabilities can be calculated in a thermodynamic limit sub-ject to the Virial Theorem and radiation boundary conditions. Inthe absence of thermodynamic equilibrium, the ratio fH/fN is any-where from zero, e.g., when all stars are trapped or escape withat most the escape velocity, to arbitrarily large, e.g., in the unlikelyevent of a star escaping with a kinetic energy on the order of thetotal gravitational binding energy of the cluster.

Ambartsumian (1938) described evaporation of isolated clus-ters by a singular perturbation away from thermal equilibrium gi-ven by the tail of a Boltzmann velocity distribution representing aflux of stars with positive energy to escape to infinity on the relax-ation time scale trelax, that is,

dNdt¼ �fNNt�1

relax; ð8Þ

where fN ’ 1/135. (Here, fN is sometimes referred to as ne, e.g., Spit-zer (1987)). Thus, (8) defines the Ambartsumian–Spitzer time scale

tAS ¼ f�1N trelax: ð9Þ

The true evaporative lifetime of cluster is shorter than (9), however,by continuous heating of the cluster in a fully nonlinear evolution,upon including the energy radiated away by the escaping stars. Inaddition to trelax, the characteristic time scales for a bound cluster,therefore, may further include the crossing times tcr,i = {R/vt,Rh/vt,rt/vt}, where vt represents the mean thermal velocity of the starsin the cluster. These considerations generalize (8) to

dNdt¼ �fNNFðtrelax; tcr;iÞ; ð10Þ

with accompanying loss of total energy

dHdt¼ fHHFðtrelax; tcr;iÞ: ð11Þ

Early attempts to include energy loss use order of magnitudescaling arguments (Spitzer, 1940) or by explicit calculation of en-ergy fractions in the tail of the velocity distribution (von Hoerner,1958; King, 1958a; Ossipkov, 2008). This approach seems justifiedin the KM morphology, but requires careful estimation of thenumerically small fractions involved, that represent a singular per-turbation away from thermal equilibrium. The early estimates ofKing (1958a) and von Hoerner (1958) give lifetimes shorter than135trelax, but still much larger than what is found in numerical sim-ulations by a few hundred percent due to an underestimate fH < fN.Present numerical estimates of the time to core-collapse rangefrom tev = 10 � 20trh in terms of the relaxation time associated withthe half-mass radius, rh, that are considerably smaller than theAmbartsumian (1938) evaporation time by Fokker Planck and N-body simulations (Spitzer and Shapiro, 1972; Cohn, 1980; Inagaki,1985; Lee and Ostriker, 1987; Spurzem and Aarseth, 1996; Drukkeret al., 1999; Heggie and Hut, 2003). Here, relatively longer timesare found in King models with medium and low central concentra-tions (Quinlan, 1996) and intermediate times in a Plummer model(Chernoff and Weinberg, 1990), while short times are obtained inmulti-mass models with broad mass distributions (Chernoff andWeinberg, 1990; Johnstone, 1993) with minor dependency onvelocity anisotropies (Takahashi, 1995).

In this paper, we shall formulate the evaporative lifetime interms of the coefficients ðfN; f HÞ, derive (2) and determine its


sensitivity to energy loss and tidal interactions, going beyond thatpredicted by (9) alone. To pursue this, we first identify a criticalnumber of stars, for evaporation to be limited by relaxation.

4. Relaxation-limited evaporation when the number of stars islarge

At a given location in the cluster, the Boltzmann distributionn(E) = N0be�bE for the kinetic energy E (equal to the total energyas measured by a local observer) is the limit for detailed balancein a thermal state, where N0 ¼

R10 nðEÞdE denotes the total number

of particles in a local neighborhood b = 1/kBT is the inverse temper-ature and kB denotes Boltzmann’s constant. Relaxation of smallperturbations away from the Boltzmann takes place on the relaxa-tion time scale trelax associated with small angle scattering in ourpresent approximation. The normalized distribution �n ¼ ebEnevolves to a uniform distribution in E. The process of relaxationfor �n is therefore equivalent to a diffusion process in a mediumwith uniform affinity (a vanishing chemical potential throughout),whereby @t �n ¼ t�1

relax@2E �n, i.e.,

@tnðE; tÞ ¼ t�1relax @

2En� b@En

� �: ð12Þ

Evaporation is a process in which particles of energy E > Ec es-cape from the system, where Ec depends on the location in thecluster, on the time scale of the crossing time, tcr, set by the sizeof the system, that provides an additional relaxation time for radi-ation leaking out to infinity. With the scaling x = E/Ec, a = Ec/kBT,

kBT ¼R Ec

0 EnðEÞdER Ec

0 nðEÞdE; ð13Þ

we are led to consider the diffusion equation

@tn ¼ t�1relax @

2x n� a@xn

� �� t�1

cr Uðx� 1Þn; ð14Þ

where U denotes the Heaviside function. About a thermal distribu-tion, Generally, the evolution of n = n(x, t) depends on both timescales(trelax, tcr). The case of trelax� tcr represents slow relaxation, relative towhich evaporation on the time scale tcr is essentially immediate. Itwill be appreciated that (14) is similar in form to an inhomogeneousKompaneets equation. The bound for this asymptotic result to holdallows by balancing the second and third terms in (14), i.e.,

trelax

te’ a2; ð15Þ

with a corresponding critical value N = Nc satisfying Nc ’ 218logNc,i.e.,

Nc ’ 1600 ð16Þ

by (5), (6) and the Ambartsumian (1938) value a ’ 6. Thus, evapo-ration of clusters with N� Nc is limited by relaxation, which justi-fies the starting point (8) and (9) and hence (10) and (11) with

F�1 ¼ trelax ð17Þ

and the corresponding Kelvin–Helmholtz time scale

tKH ¼ f�1H trelax: ð18Þ

The parameter range for relaxation-limited evaporation (17) is am-ply satisfied by the globular clusters of the Milky Way withN = 104�6. Young open star clusters in the disk of the Milky Waycan be different with N as low as a few hundred. For these systems,a full equation of the type (14) must be used.

A gradual departure away from the linear scaling of evaporativelifetimes / trelax has been observed in N-body simulationsby Baumgardt (2001) in the range N = 102–104, which we hereidentify with the low-N regime of (14) defined by (16), where

evaporative lifetimes depend on trelax and tcross. Baumgardt (2001)models the low-N scaling with

F�1 ¼ t1�brelaxtb

cr ¼6 ln N

N

� �b

trelax; ð19Þ

where b ’ 0.25. However, Eq. (12) in Baumgardt (2001) from which(19) derives, does not recover the Boltzmann distribution in the ab-sence of evaporation (defined by tcr ?1). Hence, it fails to predict acorresponding critical value in N that explains this transition. By(16), we are able to ascertain that the discussions of Baumgardt(2001) (and Lamers (2005a,b), Gieles and Baumgardt (2008)) per-tains strictly to low-N asymptotic behavior in N < Nc with no bear-ing on high-N asymptotic behavior relevant to the present-dayglobular clusters with KM morphology in the Milky Way (with mosthaving relatively small tidal parameters (1)). A similar conclusion isgiven by McLaughlin and Fall (2008).

The existence of a transition to relaxation-limited evaporationin the large N limited has been anticipated but not specified inBaumgardt (2001) and McLaughlin and Fall (2008). The abovedetermines the transition to be around (16) for N. The precise valueof Nc is not critical, as the transition is rather smooth, as may beseen from numerical results in Fig. 4 of Baumgardt (2001).

In what follows, we shall continue in the parameter rangeN � Nc , where the evaporation process is governed essentially bytrelax.

5. Evaporation coefficients fN and fH

For an isolated cluster, an individual star is bound to escapewhenever its enthalpy is positive, i.e., ek + u > 0, where ek denotesthe kinetic energy and u the gravitational binding energy to allother stars in the cluster. For star i, we have u ¼ RN

j¼1Uij (sum notincluding j = i). By (4) and the Virial Theorem, the kinetic energyof the escaping stars hereby satisfies (Ambartsumian, 1938)

�ek > ��u ¼ �2U ¼ 4Ek: ð20Þ

Correspondingly, the escape velocity is about twice the mean veloc-ity of the stars (Ambartsumian, 1938; Spitzer, 1940). For a thermaldistribution, it represents a fraction

fN ¼R1

2 e�32s2 ð4ps2ÞdsR1

0 e�32s2 ð4ps2Þds

¼ 1134:44

ð21Þ

described by a Boltzmann distribution, where s2 ¼ Ek=Ek; Ek ¼ 32 kBT ,

where T denotes the temperature kBT ¼R1

0 EnðEÞdE=R1

0 nðEÞdE.Alternatively, we may identify the cluster with the collection ofbound stars whose velocities are restricted to the truncated Boltz-mann distribution. In this event,

f �N ¼R1

2 e�js2 ð4ps2ÞdsR 20 e�js2 ð4ps2Þds

¼ 1112:43

; ð22Þ

where j = 1.452165 represents the modified relation Ek ¼ jkBT ob-tained from (13), i.e.,R 2

0 s2e�js2 ð4ps2ÞdsR 20 e�js2 ð4ps2Þds

¼ 1: ð23Þ

The difference in (21) and (22) may be considered representative forthe uncertainty in the present thermodynamic approach, and wewill proceed with both to keep track of uncertainties in any of thederived quantities.

The relaxation time (6) is the instantaneous time scale for therelaxation of a stellar system, given its current state. It definesthe rate at which the high velocity tail in the Boltzmanndistribution of escaping stars is replenished by upscattering of


low velocity stars, giving rise to the evaporation rate (10) and theinstantaneous evaporation time (9), or 21% less based on (22). Theestimate (9) represents an extrapolation based on the initial condi-tions of the system, which otherwise evolves nonlinearly over multi-ple relaxation times. Because stellar systems are self-gravitating,their temperatures increase as they loose stars by evaporation.Therefore, (9) gives an upper bound for complete evaporation, notthe true time to complete evaporation in a fully nonlinear evolution.

In the process of evaporation, the system evolves also in re-sponse to energy loss due to positive energy of the escaping stars.A complete system of equations, therefore, includes energy bal-ance. Analogous to previous arguments, the total rate of energyloss (‘luminosity’) changes the total energy H of the cluster at a rate(11) with

fH ¼R1

2 s2e�32s2 ð4ps2ÞdsR1

0 s2e�32s2 ð4ps2Þds

¼ 128:75

: ð24Þ

Here, we identify H with the total energy of the stars below thethreshold (all the stars that make up the cluster) and fH with the frac-tion of total energy carried off by the evaporating stars, that replenishthe tail above the threshold in each relaxation time period.

Note that fH is calculated by the relative fraction in kinetic energyin the tail of the velocity distribution, representing a flow in momen-tum space from local interactions (in coordinate space) between‘‘low’’ velocity stars (below the threshold, in the cluster), where ki-netic energy and potential energy are virialized. Thus, replenishingthe tail above the threshold with kinetic energy is tightly correlatedto H defined by the stars below the threshold and vice versa. Follow-ing (22), a more precise definition is, therefore,

f �H ¼R1

2 s2e�js2 ð4ps2ÞdsR 20 s2e�js2 ð4ps2Þds

¼ 123:74

; ð25Þ

which differs from (24) by 17%.The system (10) and (11) is closed by (3). Integration of the two

ordinary differential equations gives

TT0¼ N

N0

� ��c

;RR0¼ N

N0

� �a

; ð26Þ

−0.2 0 0.2

0.51

1.52

2.53

3.54

f N/f N

(0)

Δ

fffN(0)=0.74%

−0.2

0.51

1.52

2.53

3.54

f* N/f* N

(0) f*N(0)=0.88%

−0.2 0 0.2

0.51

1.52

2.53

3.54

f H/f H

(0)

Δ

fH(0)=3.5%

−0.2

0.51

1.52

2.53

3.54

f* H/f* H

(0)

f*H(0)=4%

Fig. 1. (First and second column.) Shown are the evaporation fractions (fN, fH) and ðf �N ; f �HÞover the range D ¼ � 1

4 ;14

, including quadratic approximations thereto (continuous lin

essentially the same for both pairs of fractions (continuous and thick lines, respectively). Thfaster than exponential decay in the tail of the Boltzmann velocity distribution.

where

c ¼ fN þ fH

fN¼ 5:71; a ¼ cþ 1 ¼ 6:71 ð27Þ

based on c = 5.712 from (21) and (24). As a ratio, the estimate (27) isstable in the first two digits in view of c = 5.736 as follows from (22)and (25). The correlations (26) and (27) are independent of thechoice of evolution time scale F.

As illustrated by (21), (22), (24) and (25), evaporation is a singu-lar perturbation away from thermal equilibrium which introduceshigh sensitivity (of the coefficients (fN, fH)) to the critical escapevelocity, even though some dimensionless results are relativelystable such as (27). For example, a variation of +1% in the thresholddefining the tail gives rise to a variation of about +10% in tev (34).For small changes D, we have

fNðDÞ ¼R1

2�D e�32s2 ð4ps2ÞdsR1

0 e�32s2 ð4ps2Þds

¼ 1þ 5:57Dþ 13:92D2 þ OðD3Þ135:44

ð28Þ

fHðDÞ ¼R1

2�D s2e�32s2 ð4ps2ÞdsR1

0 s2e�32s2 ð4ps2Þds

¼ 1þ 4:73Dþ 9:45D2 þ OðD3Þ28:75

: ð29Þ

Simular results hold for the fractions ðf �N ; f �HÞ, i.e., f �N=fNð0Þ’1þ5:3781Dþ12:9308D2; f �H=f �Hð0Þ’1þ4:5419Dþ8:6492D2. Fig. 1shows both (fN, fH) and ðf �N; f �HÞ as a function of D. Corresponding to(28) and (29), the evaporation time satisfies

sevðDÞ ¼ 12:82 1þ 0:05ln N0

� �1� 5:01Dþ 14:13D2 þ OðD3Þ� �

ð30Þ

and, similarly, sevðDÞ¼10:60 1þ 0:05lnN0

� �1�4:82Dþ13:15D2þOðD3Þ� �

in the approximation defined by ðf �N;f �HÞ. Consequently, we have

fH

fN¼ 4:71 1� 0:8434Dþ 0:23094D2 þ OðD3Þ

� �ð31Þ

and similarly f �Hf �N¼ 4:74 1� 0:8460Dþ 0:2150D2 þ OðD2Þ

� �, which

evidently is essentially the same for (fN, fH) and ðf �N ; f �HÞ. We concludethat the impact of evaporation on energy is about five times largerthan that on mass.

0 0.2Δ

0 0.2Δ

−0.2 0 0.2

101

102

Δnorm

aliz

ed e

vapo

ratio

n tim

e

−0.2 0 0.24

4.55

5.56

6.57

Δ

γ

as a function of the relative departure D from the nominal escape velocity (circles)es), which differ by about 20%. (Third column.) The resulting c ¼ 1þ fH

fN(27) are

e normalized evaporation time 1qfN

in (34) shows a steep dependency on D due to the


5.1. The evaporative lifetime of an isolated cluster

For an isolated cluster, integration of (10) and (11) with (6), (26)and F ¼ t�1

relax gives

lq�1 dlds¼ �fN 1þ ln l

ln N0

� �; ð32Þ

with s ¼ t=t0relax, where t0

relax ¼N

120R

320

6 ln N0denotes the initial relaxation

time defined by N0 and the initial radius R0; l ¼ NN0

the normalizedparticle number and q ¼ 2þ 3

2 c ¼ 10:57. A finite-time singularitysolution obtains by ignoring the logarithmic term in (32),

N ¼ N0 1� ssev

� �1q

; ð33Þ

where sev ’ q�1f�1N ¼ 12:82. A next order approximation for sev ob-

tains by including a time average of the logarithmic term following(33). With lnl ’ �sfN for most of the time (away from the singular-ity), the time to complete evaporation extends to

sev ’1

qfN1þ 1

2q ln N0

� �’ 12:82 1þ 0:05

ln N0

� �: ð34Þ

For all practical purposes, the evaporation time is about 13 timesthe instantaneous relaxation time, apart from a weak positivedependence on the initial particle number N0 by no more than 5%.

Solutions of the type (33) are canonical for a broad range ofradiation processes with generally different coefficients, thatwould include evaporation of relatively low N star clusters accord-ing to Lamers (2005b) on the basis of (19).

The evaporative lifetime (34) serves as a lower bound to the core-collapse time of isolated clusters. Quite typically, scattering of starsoccasionally has an impact parameter much smaller than bmin = Gm/v2, in which case the interaction can be inelastic by tidal interactionsthat may lead to tidal capture and binaries formation (e.g. (Pooley

40 60 80 100 120−100

−50

0

50

100

v0=0.943

0 20 40 60 80 100 1200

20

40

60

80

100

120

Ω t=10

−

−

Fig. 2. (Top.) Formation of S-shaped streams shown at two times Xt = 10 and Xt = 40uniformly in all directions from the cluster with mass M = 10�5MG with orbital angularcenter at r0 = 0.1rt, where rt is the tidal radius rt = 1.5 (red; 2rt in green), relative to a distanThe results are shown for an initial velocity v0 which is 0.943 times the escape velocTransformed to the rest frame of the galactic center, the streams form extended tails as ththe galactic disk on the orbital time scale of the cluster (not shown). (For interpretationversion of this article.)

et al., 2003)). This dissipative effect is ignored in idealized point-par-ticle N-body simulations. While numerical simulations suggest thatbinary fractions of a 0.1% can already produce an order of magnitudeincrease in the lifetime of the cluster (Joshi et al., 2001), a detailedstudy identifying the threshold for a population of compact binariesto affect the lifetime of a cluster appears to be lacking.

6. Stimulated evaporation by tidal fields

In clusters bound to a host galaxy, the escape velocity will be per-turbed by the tidal gravitational field of the galaxy, which generallyfacilitates stars to spill over across the Lagrange points L1 and L2, i.e.,ek + u >�d following (20), where d > 0. Ab initio, d depends on thedirection relative to the line passing through L1 and L2 with an addi-tional deflection due to the angular momentum of the globular clus-ter, as can be seen in direct N-body simulations (Fig. 7.4 in Meylan andHeggie (1997)). The distance to L1 and L2 is given by the tidal radius(Spitzer, 1987; Chernoff and Weinberg, 1990; Takahashi et al., 1997)

rt ¼ aMc

3MG

� �13

; ð35Þ

where MG denotes the mass of the host galaxy and a the separationof the cluster-galaxy system. The resulting formation of tidalstreams is illustrated in Fig. 2.

Existing approaches to incorporate tidal effects commonly usethe minimum energy for escape at L1 (Spitzer, 1987; Lee and Ostr-iker, 1987; Chernoff and Weinberg, 1990; Takahashi et al., 1997;Ross et al., 1997), effectively applying a spherically symmetricextension thereof to the tidal sphere. However, the probabilityfor escape depends on the orientation relative to the anisotropic ti-dal field, and hence requires averaging over the directions of lastscattering events, here illustrated by Figs. 3 and 4 below. Thepossibility of long lived delays in escape (Baumgardt and Makino,

−200 −100 0 100 200150

100

−50

0

50

100

150

0 20 40 60 80 100 1200

20

40

60

80

100

120

Ω t=40

in the corotating frame of the cluster by the trajectories (blue) of 20 stars ejectedvelocity X around the host galaxy of mass MG. The trajectories emanate from thece of 100 from the center of mass of the cluster-galaxy system located at the origin.

ity v0 of the same cluster when viewed in isolation with no tidal field. (Bottom.)ey would aggregate into diffuse rings in the absence of destruction by crossings withof the references to colour in this figure legend, the reader is referred to the web

Fig. 3. Shown are the trajectories (blue) of 30 stars ejected uniformly in all directions of a globular cluster similar to Fig. 2 with initial velocities v0 increasing by 0.5% in eachwindow, normalized to the critical velocity

ffiffiffiffiffi2Mr0

qof the same cluster in isolation with no tidal field. The tidal sphere forms an effective threshold for stars to escape. The results

serve to illustrate the critical onset of anisotropic evaporation of high velocity stars in the tail of an extended velocity distribution in the presence of a tidal field. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Shown are the trajectories as Fig. 3, now with a choice of initial velocities v0 increasing by 2.2% in each window. The results serve to illustrate escape across the tidalsphere with an increasingly isotropic outflow of stars with increasing energies.


2003) conceivably modifies the escape probability, but this effectwill be small in the limit N� Nc.

As the relaxation time is smallest in the high density centralregion of a cluster, most of the escaping stars derive from high


velocity jumps produced in the center (Johnstone, 1993). The tidalperturbation of the evaporation process can hereby be effectivelymodeled by studying the escape of stars that are emitted from a re-gion about the center that is small relative to the tidal radius of thecluster (relative to the distance between the Lagrange points L1 andL2) in essentially randomly oriented directions; see also Baumgardtet al. (2002), Baumgardt and Makino (2003).

The trajectory of a single star in the potential outside of the coreof a globular cluster of mass Mc = NM is the solution of a restrictedthree-body solution described by an effective potential in a Carte-sian frame of reference (x,y,z) = (ac + rcosh,rsinhcos/,rsinhsin/)with angular velocity X2 ¼ MGþMc

a3 about the z-axis, given by

U ¼ �Mc

rc� d; d ¼ Mg

rgþ 1

2X2r2 �Mg

a1þ 1

2ac

a

� �2� �

; ð36Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pis the distance to the axis of rotation, and

rc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� acÞ2 þ y2 þ z2Þ

qand rg ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ agÞ2 þ y2 þ z2

qdenote the

distances of the star to the center of mass of the cluster and, respec-

tively, the host galaxy. The latter have distances ac ¼ a Mg

MgþMcand

ag ¼ �a McMgþMc

to the center of mass of the cluster-galaxy system,

here at the origin of (x,y,z). We use units in which Newton’s con-stant is equal to 1. Figs. 3 and 4 illustrate the tidal perturbationsabout the critical escape velocity, showing the critical onset of evap-oration from the tidal sphere as a function of stellar velocity andanisotropy produced by the tidal field. There is further a gradualtrend towards reduced anisotropy with increasing velocity.

The tidal interaction generally lowers the barrier for stars to es-cape from the tidal sphere by an energy perturbation d that de-pends on the direction, as d(r,h,/) has period p in h(2p in /)with a pronounced amplitude in h and minor variations in / shownin Fig. 5. The Virial Theorem for the cluster (20) now becomes

2Ek þU ¼ 0; ð37Þ

where U ¼ U � �d with U and d defined in (4), respectively, (36). Itfollows that U ¼ �2Ek þ d. The condition on the kinetic energy ek

of a star to escape becomes ek þ u ¼ ek þ 2U � d > 0, giving a critical

kinetic energy 4Ek 1� d4Ek

� �reduced by the ratio of d relative to the

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

10

20

30

40

50

60

70

80

90

100

fract

ion

(%)

1−(v/vc)2

Fig. 5. (Left.) Shown is the fraction of escaping stars from a cluster with mass Mc=Mg ¼galaxy, computed according to Figs. 3 and 4. The initial velocities v are normalized to the crthe reduction in the potential energy barrier for escape due to the tidal field. The results shof less than 12% for a 50% probability of escape. This sensitivity of a 50% change in the fluresults summarized in Fig. 1. (Right.) Shown is the orientation averaged hdi ¼ �d kinetic eneescape if the cluster were isolated. For typical cluster-to-galaxy mass ratios Mc/Mg� 1, t

kinetic energy 4Ek required for escape if the cluster were isolated.As the core of a globular cluster is much smaller than rt, the escapeof high velocity stars from it is effectively isotropic and the net es-cape rate obtains from averaging over all directions. The average ofd over the tidal sphere satisfies

�d ¼ 14p

Z p

0

Z 2p

0dðrt ; h;/Þ sin hdhd/ ¼ C

Mc

rt; ð38Þ

where the gray-body factor of the evaporation process satisfies

C � 0:11� 0:12 ð39Þ

in the limit of small ratios Mc/Mg as shown in Fig. 5. With

D ¼ 2� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� d

4Ek

qand (4), we arrive at

D ’�d

4Ek

’ CRrt: ð40Þ

We are now in a position to calculate the modified evaporationrates due to a finite tidal radius rt.

6.1. The evaporative lifetime of a cluster in the tidal field of its hostgalaxy

For the globular clusters of the Milky Way at the present epoch,the tidal ratio 1 assume rather moderate values broadly distributedaround 0.1 (Baumgardt et al., 2009). For this reason, we purse aregular perturbation of the evaporative lifetime in this tidal ratio,as opposed to a fully nonlinear treatment that would include tidalratios of order one (Gieles and Baumgardt, 2008).

Following the previous section, we integrate the ordinary differ-ential equations for mass and energy (10) and (11) with closure(fN, fH) given by (28) and (29) and D in (1) expressed by (40). For(31), we have, by the leading order solution (26),

fH

fN¼ 4:71 1� 0:097yþ 0:0031y2 þ Oðy3Þ

� �¼ a 1� by0l

5þ3a3 þ Oðy2Þ

� �; ð41Þ

where we take into account only the linear dependence on y,a = 4.71, b = 0.097, and y0 refers to the initial value. Then the leadingorder solution for the energy evolution is

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 00.1

0.15

0.2

0.25

0.3

0.35

Mc/Mg

0 1 2 3 4 5 6−0.2

0

0.2

0.4

0.6

0.8

θ

<δ>×

r t/Mc

<δ>×

r t/Mc

φ=0φ=π/4φ=π/2

10�5ð�d� rt=M ¼ 11:86%Þ and tidal radius rt/a = 1.5, where a denotes the to the hostitical velocity for escape vc if the cluster were isolated, whereby 1 � (v/vc)2 expressesow a reduced threshold in the kinetic energy for escape by about 14% and a reductionx of stars escaping in response to a 2% change in D is consistent with the theoreticalrgy for escape from a cluster bound to a host galaxy, relative to the kinetic energy forhe resulting gray-body factor C ¼ �d� rt=Mc asymptotes to about 11%.


H ¼ H0l�ae3ab

5þ3al5þ3a

3 ’ H0NN0

� ��4:71

1� cNN0

� �a�13

" #; ð42Þ

where c = 0.072y0, and similar to (26) we find

TT0¼ N

N0

� ��c

1� cNN0

� �a�13

" #;

RR0

¼ NN0

� �a

1þ cNN0

� �a�13

" #: ð43Þ

Similar to (26) and (27), the correlations (43) are independent of thechoice of evolution time scale F.

As when deriving (32) in the previous section with l = N/N0, theevaporation time relative to the initial relaxation time follows byintegration from (10) with (6) and, now, (43) instead of (26),

lq�1 dlds’ �f 0

N 1þ ln lln N0

� 3c2

la�13

� �1þ 0:64y0l

23 þ 1:6y2

0l43

� �; ð44Þ

where f 0N ¼ 1

135.Fig. 6 shown the result of numerical integration of (44) using a

canonical value N0 = 106, where we infer a weak logarithmicdependence on N0 by comparing results with N0 = 50. With a poly-nomial fit to the tidal dependence, we arrive at (2), where wedropped the subscript 0 to initial values of y and N, and wherewe dropped a factor (1 + 0.002lnN6)�1 with extremely weak depen-dence on N = N6 � 106. We emphasize that (6) refers to finite-timesingularity solutions of single-mass clusters, as a regular tidal per-turbation of isolated clusters in their pre-collapse state (e.g. Fig. 1in Baumgardt et al. (2002)). The detailed shape of the graph ofl(t) = N(t)/N0 in case multimass star clusters can be quite differentwith a continuously evolving mass function due to relativelyfast evaporation of low-mass stars (e.g. Fig. 1 in Baumgardt andMakino (2003)).

0 2 40

0.2

0.4

0.6

0.8

1

μ

y0=0..1(N0=106)y0=0..1(N0=50)

0 0.1 0.2 0.3 0.44

6

8

10

12

14

τ ev

Fig. 6. (Top.) Evolution of the number of stars l = N/N0 for various initial tidal parameterradius R. (Bottom) The slope to sev is �0.94 at y0 = 0, showing strong tidal sensitivity, herN0, showing that the process of relaxation-limited evaporation is essentially independen

The numerical result shows a tidal sensitivity

d ln sev

dy0

y0¼0¼ �1:9;

d lnsev

dy0

y0¼1¼ �0:70 ð45Þ

giving rise to reductions up to a factor of a few when y0 approaches1 as shown in Fig. 6. Very similar results follow from ðf �N; f �HÞ. Foryoung clusters with possibly y0 > 4 such that D > 0.2, (44) acquirescontributions from D to higher order than those given in (28) and(29), which would require a further refinement of numericalintegration.

6.2. Comparison with numerical simulations

We may compare the above with existing approximations forthe evaporation rate of stars in tidally truncated clusters (Spitzer,1987; Takahashi et al., 1997), that are focused on fN with point wiseextensions of the potential barrier at L1, as opposed to sphericalaveraging (38) over the tidal sphere. This generally over-estimatesthe sensitivity to tidal interactions by a factor of about 4 given theamplitude of about 0.5 in rt

M d as shown in Fig. 5. Furthermore, (45)may be compared with that resulting from the fit

tev ¼ 6:67rh

rt

� ��1

ð46Þ

to numerical simulations in terms of the half-mass radius rh (Spitzerand Chevalier, 1973; ?), satisfying

d ln tev

dy0

y0¼1¼ �1: ð47Þ

Our result given by (45) shows a generally y-dependent tidalsensitivity of the evaporation lifetime, that is overall consistentwith the fit (47). The resulting evaporation lifetime (2) connects

6 8 10 12

τ

0.5 0.6 0.7 0.8 0.9 1

y0

numerical integration (N0=106)polynomial fitnumerical integration (N0=50)

s y0 according to (44), where s denotes the time in units of trelax defined by the viriale by numerical integration of (44). The results are essentially the same for differentt of N.


y0 = 1 smoothly to the finite lifetime for isolated clusters (y0 = 0). Itwill be noted that the fit (46) is restricted to a neighborhood ofy0 = 1, as it predicts an infinite lifetime for isolated clusters or,equivalently, clusters with very small tidal parameter (1).

Gieles and Baumgardt (2008) discuss a polynomial approxima-tion to the instantaneous evaporation rate as a function of tidal ra-dius. Beyond their cut-off of y = 0.05, their Eq. (7) predicts alogarithmic derivative of �1.5 for the evaporative lifetime to vari-ations in the tidal ratio, similar to (45) derived above.

7. The remaining evaporative lifetimes of globular clusters ofthe Milky Way

For globular clusters of the Milky Way, we recall that most haveKM morphology, all satisfy (16) and y� 1, by which (2) is applieswith rather minor dependency on tidal effects. Independent argu-ments pointing to the same are given by McLaughlin and Fall(2008) and Prieto and Gnedin (2008).

Clearly, the tev of the globular clusters exceed a Hubble time inagreement with their ages, where tev = 14 Gyr separates the half-mass parameters on the right and the core-parameters (Nc = (Rc/R)3N,Rc) on the left. Overall, the distribution follows a trend alongthe paths of steepest descent of trelax, where the core-collapsedclusters continue this trend well below the model limit N = 2.

Fig. 7 also shows that there is no correlation between the X-rayluminous binaries mentioned in the introduction and globularclusters with or without prior core-collapse, except that clusterswith these X-ray sources tend to be of high mass for a given cluster

10−1 10010−3

10−2

10−1

100

101

102

103

104

105

106

107

35Myr

3.5Gy7G

R

N

Fig. 7. Shown is the mass-radius distribution of 119 clusters in the Milky Way (from Gnedat half mass Mh, differentiated in morphology for King Model (KM, circles) versus PostNGC106 and NGC4499). We add the distribution of inferred core parameters (Nc,Rc) in tassociated tev. Curves of constant tev are shown for 35 Myr, 3.5 Gyr, 7 Gyr and 14 Gyrconsistent with the present age of the Universe and the age of the clusters. The average te

of (N,R) and (Nc,Rc) follows the path of steepest descent (thick blue line, slope 2.64) in trelax

below the N = 2 model limit. A list of X-ray sources (Verbunt and Lewin, 2005) is includedare between the curves of 14 and 28 Gyr, that will evaporate in the next Hubble timinterpretation of the references to colour in this figure legend, the reader is referred to

radius. If (compact) primordial binaries contribute to a PCC mor-phology (e.g., Gao et al. (1991)), which is unclear at present, theyare unlikely to contribute to the X-ray sources. Instead, the X-raysources are likely produced in the course of stellar relaxation by ti-dal captures (Pooley et al., 2003; Benacquista, 2006) with no needfor a primordial binary population.

8. Cluster-tail correlations produced by evaporation

Evaporation of stars modulated by tidal fields give rise to aniso-tropic outflows with bipolar morphology, commonly referred to astidal tails. Relaxation limited evaporation provides a launchingmechanism that predicts a correlation between the velocity ofgrowth of these tidal tails and the velocity dispersion in the clusteras defined by its temperature (Küpper et al., 2008a,b, 2009). We herequantify this correlation suitable for detailed observational studies.

Following (10) and (11), the mean kinetic energy �ekðrÞ of theescaping stars at distance r are related to the mean kinetic energyof the stars in the cluster by

�ekðrÞ ¼fH

fN� 4

� �Ek þ

mMR

yrt

r; ð48Þ

where we encounter the evaporation efficiency

fH

fN� 4 ¼ 0:71 1� 0:097yþ 0:0031y2 þ Oðy3Þ

� �ð49Þ

by (41).

101 102

ryr

14Gyr

28Gyr

[pc]

(N,R),KM (N,R),PCC (Nc,Rc),KM (Nc,Rc),PCCwith XRS

in and Ostriker (1997)) in terms of (N,R), where N = Mh/M� and R denotes the radiusCore-Collapse (PCC, squares) (from Chernoff and Djorgovski (1989), adding PCC tohe Ansatz Nc = (Rc/R)3N (KM,>;PCC,⁄) which, for PCC, defines a lower bound on the. The 14 Gyr line separates the (N,R) from (Nc,Rc) without regard to morphology,v for the clusters cores is about 1% of that of the clusters. The combined distributions(two curved black lines), where the core-collapsed clusters continue this trend well(red diamonds), showing some preference towards high N for a given R. Four clusterse consistent with the evaporative destruction rate of Aguilar et al. (1988). (For

the web version of this article.)


Clusters with y < 1 will produce a bipolar outflow of stars that isamenable to detailed observations as mentioned in the introduc-tion. These tidal tails carry a record of the most recent evaporationprocess by the number and kinetic energies of their stars as mea-sured relative to the center of mass of the cluster. The plannedand upcoming LSST, BigBOSS, JASMINE and GAIA offer unique pros-pects for extracting kinematic information in a wide field of viewaround globular clusters with angular resolutions around 10l as,resolving velocities of individual stars to about 1 km/s at visualmagnitudes typical for globular clusters. To test (48), we propose(1) resolving the kinetic energy in tidal tails outside the tidalsphere, (2) the velocity dispersion in the cluster and (3) comparingfor the ratio of the two with the predicted value

�ek P 0:71 1� 0:097yþ Oðy2Þ� �

Ek; ð50Þ

where equality is approached at large distances away from the clus-ter, when the gravitational binding energy to the cluster can be ne-glected. Following Section 2, R = 1.30rh in NEk ¼ GM2

c4R gives a mean

(three-dimensional) velocity dispersion in the cluster,

r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:19GMc

rh

sð51Þ

and a scale for bipolar outflow

Vt PffiffiffiffiffiffiV2

t

q¼ 0:84ð1� 0:049yþ ðy2ÞÞr ð52Þ

by (50). The velocity of the tidal front may be larger than Vt , as it isdetermined by the fastest escaping stars. For instance, Leon et al.(2000) report for NGC 6254 a projected expansion velocity 7 km s�1

of tidal material at a distance of r = 150 pc (D = 4.1 kpc) and a pro-jected velocity dispersion 6.6 km s�1 in the cluster, were it is notedthat the gravitational binding energy to the cluster is negligible atthis distance. Evidently, an expansion velocity exceeding the veloc-ity dispersion in the cluster satisfies (52). The correlation (50) is rel-atively insensitive to various cluster parameters, such as the totalmass, which otherwise is not well defined in view of an uncertainbut possible substantial population of white dwarfs (Baumgardtand Makino, 2003).

Table 1Data on selected globular clusters derived from the catalogue of Harris (1996) (revision 201of escaping stars n per one-quarter orbital period by (53) and the projected angular separat(54) using positional data. Where available, a comparison with observational data is inclu

Namea Db kpc Mc 106 M� rt pc yh trh Gyr

NGC 104i 4.5(7.4) 1.45 64 0.065 3.5Pal 4i 109(111) 0.0541 100 0.162 2.6Pal 5i 23.2(18.6) 0.0284 101 0.182 6.6Pal 14i 73.0(71.6) 0.0200 118 0.219 10NGC 5139i 5.2(6.4) 2.64 104 0.073 12NGC 5904i 7.5(6.2) 0.834 52 0.074 2.5NGC 4590i 10.3(10.2) 0.306 89 0.051 1.9NGC 5466i 16.0(16.3) 0.133 162 0.066 5.7NGC 6254i 4.4(4.6) 0.225 30 0.084 0.8NGC 6121i 1.7d(5.9) 0.225 19 0.112 0.9NGC 6752j 4.0(5.2) 0.364 52 0.042 0.7NGC 7078j 10.4(10.4) 0.984 62 0.049 2.1

a NGC 104 = 47 Tuc, NGC 5139 = x Centauri, NGC 5904 = M5, NGC 4590 = M68, NGC 6b Distances to the Sun (galactic center) with uncertainties of 6% (Chaboyer, 2007).d Richer et al. (2004).e Model predictions for projected angular separation between leading and trailing ti

assuming circular motion and Mg = 3 � 1011 M�.f Leon et al. (2000).g Odenkirchen and Grebel (2004), Belokurov et al. (2006) and Fellhauer et al. (2007).h Jordi and Grebel (2010).i King morphology (Chernoff and Djorgovski, 1989).j Post core-collapse morphology (Chernoff and Djorgovski, 1989).

In Table 1 we give an illustrate sample of our model predictions,where we assume a mean stella mass �m ¼ 1

3 M� to derive N ¼ Mc= �mfor each cluster. We focus on tidal tails created over the periodT = T7106 yr of a few times ten million years since the last diskcrossing, to circumvent the uncertain fate of extended tidal tailsshown in Fig. 2 following the most recent disk crossing. Scaled toone-quarter of the period P = P6 Myr of the orbit around the galac-tic center, the associated outflow of evaporating stars satisfies

n ’ PNfN

4trelax¼ 0:275� N5P6

trh

109 yr

!�1

; ð53Þ

with N = 105N5.By angular momentum conservation, bipolar outflows emanat-

ing from the two Lagrange points L1 and L2 deflect into leading andtrailing streams inside and outside the orbit of the cluster (Heggie,2001). We express it by a correlation of the radial separation to thetemperature of the cluster according to an angular separation

a ¼ DrDgc¼ 3:4

rDgcX

¼ 0:55r1D1=24 deg; ð54Þ

where X ¼ffiffiffiffiffiMg

D3

qwith Mg = 3 � 1011 M� (Odenkirchen et al., 1997)

denotes the orbital period of the cluster about the galactic centerat a distance Dgc = D410 kpc and r = r11 km s�1. The projectedangular separation follows from the known positions of the globularclusters, shown in the last column of Table 1 in the approximationof circular motions.

9. Conclusions and observational outlook

Stellar clusters are open systems that radiate stars and energyby evaporation in a singular perturbation of an essentially thermalstate, mostly by diffusion in momentum space subject to a singularperturbation by rare but finite jumps due to hard 2- or possibly 3-body interactions. We identify a parameter range for relaxation-limited evaporation applicable to clusters in their pre-collapsephase (generally of KM morphology) with N� Nc and small tomoderate tidal ratios y < 1, and develop a model for the evaporativelifetime for single-mass clusters. For a Maxwell velocity distribution

0) and model predictions for the outflow velocity Vt from r given by (51), the numberion a0 between the leading and trailing tails inside and outside the orbit derived fromded.

A rh pc r km/s Vt km/s n aoe arcmin

3.7 4.1 16 14 140 120(� 100)f

2.0 16 1.6 1.3 161 12.1 18 1.1 0.9 6 0.4(<30)f,h

1.5 26 0.8 0.6 23 0.061.2 7.6 16 14 64 75(� 100)f

5.3 3.9 13 11 87 43 < 60)h

6.4 4.5 7.3 6.1 82 22(<60)h

2.5 11 3.1 2.6 29 4(<60)g

18 2.5 8.4 7.0 51 70(� 100)f

23 2.1 9.0 7.6 93 32417 2.2 11 9.5 96 467.1 3.0 16 13 301 8.7

121 = M4, NGC 6254 = M10, NGC 7078 = M15.

dal streams (observed values, if known) from the position of the globular cluster


that extends beyond the escape velocity, evaporation is bright ascharacterized by an inequality between the Kelvin–Helmholtz(18) and Ambartsumian–Spitzer time scales (9), satisfying

tKH < tAS ð55Þ

due to fH > fN, wherein evaporation is largely driven by energy-lossrather than mass loss. The evolution isolated clusters, as a referenceto more realistic modeling, is hereby described by a finite-time sin-gularity solution

fN;R; Tg / 1� ssev

� �f0:0905;0:6351;�0:5405g

: ð56Þ

It shows that the temperature and the radius evolutions (56) arerelatively less shallow than (33), but still evolve only moderatelyaway from the time of complete evaporation according to (56).Here, the indices 0.0905 and 0.6351 in (56) are close to the values0.085 and 0.695, respectively, observed in N-body simulations(Heggie and Hut, 2003). We do not attribute the relatively smalltev (relative to tAS) to a radial stratification of the relaxation timewith fH = 0 as proposed in (Johnstone, 1993). It also correct an olderclaim that fH < fN (with commensurably tKH > tAS) by von Hoerner(1958) and King (1958a,b). Similarly, stars escaping by ejectionsfrom last scattering events from within the cluster should be con-trasted with low velocity evaporation arising from diffusive escapethrough a low temperature outer boundary, commonly used in theFokker–Planck approximations as if the cluster were opaquethroughout up to its tidal boundary (Spitzer, 1987).

The agreement of (2) with numerical simulations supports theidea that energy loss in accord with (55) is important. For aMaxwell velocity distribution, (2) is shorter than that based onparticle loss alone by a factor of about three due energy loss(24). It would be of interest to identify the various scatteringprocesses that may give rise to a relatively large fH , see e.g.Petrovskaya (1970); Retterer (1979); Johnstone (1993); Meylan et al.(2001); Ashurov (2004), and determine the implied shortening ofthe evaporative lifetime following Section 5.1. These studies falloutside the scope of the present work, however. Instead, we focuson some observational tests in Section 8.

The presence of a tidal field modifies the net rate of evaporationby two effects. It generally enhances evaporation by lowering thethreshold on the kinetic energy for stars to spill over the tidal ra-dius, but the escape probably is attenuated by the generally aniso-tropic tidal field as defined by the Lagrange points L1 and L2

described by the gray body factor (39). For moderate strengths ofthe tidal field, the resulting evaporation rate can be expanded ina regular perturbation in the tidal ratio (1).

In studying the globular clusters of the Milky Way, we note thatthey (1) amply satisfy the criterion for relaxation-limited evapora-tion (16), (2) their tidal rations satisfy y� 1, (3) most of them areof KM morphology indicative of a pre-core collapse state. For thisreason, the perturbative expansion (2) for their remaining relaxa-tion-limited evaporative lifetimes is believed to be applicable,and it is found to be consistent with their current ages.

The energy of the escaping stars is directly correlated to thetemperature of the cluster (54), and the outflows thus produceddevelop a bi-polar S-shaped morphology in the plane of the La-grange points L1 and L2 and the orbit of the cluster. For those tidaltails where the S-shaped tail can be resolved observationally, thede-projected orbital displacement of the tails is representativefor the outflow velocity by the correlations (52)–(54).

If periodically destroyed by disk crossings, these tails hereby re-appear on a time scale of 10 Myr with a length scale of a few timesthe tidal radius for clusters that are not too small in mass andvelocity dispersions. Detailed observational studies are proposedto test (52) on a wide field of view around globular clusters using

upcoming high resolution photometric and spectroscopic stellarsurvey instruments. The results may also be used for comparisonwith alternative mechanisms for inducing tidal tails, such by tidalheating or shocks in response to disk crossings (Heggie, 2001) andvelocity anisotropy (Takahashi and Lee, 2000). We remark thatdestruction by tidal fields of an initial distribution might have beenmore severe around spiral galaxies than around the relatively lar-ger elliptical galaxies, which tend to have relatively large numbersof globular clusters in proportion to their luminosity (Harris andvan den Bergh, 1981; Burkert and Tremaine, 2010).

The upcoming large field of view photometric and spectroscopicinstruments LSST, BigBOSS, GAIA and JASMINE seem to be ideallysuited for detailed measurements on tidal tails in energy (52), stel-lar count (53) and morphology (54) to advance an observationaltest of the present thermodynamic model.

Acknowledgment

The author gratefully acknowledges detailed constructive com-ments from the anonymous referee, J. Scalo and H. van Beijeren.

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