arXiv:astro-ph/9807101v1 10 Jul 1998 edopitrqet to requests offprint Send 10 range mass the in CHOs MA- galaxies. of spiral detection im- in the (Paczy´nski is allows curves Microlensing 1986) rotation presence flat whose observed halos, the by galactic plied in matter dark nature the the concerns of astrophysics in problem important An Introduction 1. Clusters Jetzer Philippe Globular by Microlensing Gravitational fno lete l 97 dlk,Kba Szyma´nski & Kubiak Udalski, 1997, al. Palanque-Delabrouille, et 1997c, Albert al. Afonso, et Alves (Al- (SMC) Allsman, 1993), Cloud cock, Magellanic al Br´ehin Small et Allsman, the Bareyre, towards Alcock, event Auburg, one 1997b, 1993, al., al. (LMC) et our et Cloud Alves Allsman Magellanic of have Large Akerlof, bulge events the (Alcock, microlensing or towards 15 found disk than been more halo, now, the Till galaxy. in 1992) Mass´o Jetzer, 1 2 lblrclusters globular words: Key matter. signif- dark a clusters, of contain amount globular clusters icant the these that in therefore, located are suggesting, MACHOs events NGC to microlensing and due some 6528 indeed that NGC evidence find 6522, We NGC 6540. the to clusters close globular microlens- lie which the three bulge, well analyse galactic not the we towards events presently purpose, ing latter is this which To clusters, known. globular of dark content total its constrain better to content. different allowing matter along halo sight, in- galactic of get the could lines of one shape particular, the In on formation results. can useful observations very such to small, lenses lead is as events expected or microlensing the of sight Although rate of stars. background line distant the Compact more for along Astrophysical located (Massive Objects) Halo MACHOs for sources Abstract. accepted Received; 00.,1.71 20.,10.07.2 12.04.1, 12.07.1, 10.08.1, are: codes thesaurus later) Your hand by inserted be (will no. manuscript A&A alShre nttt aoaoyfrAtohsc,CH-5 Astrophysics, for Laboratory Institut, Scherrer Paul ntttfu hoeicePyi e Universit¨at Z¨uric der Physik f¨ur Theoretische Institut oevr ncnas ne h oa akmatter dark total the infer also can on Moreover, tr ngoua lsescnatete as either act can clusters globular in Stars aatchl irlnig-dr atr- matter dark - microlensing - halo galactic 1 , 2 acsStr¨assle Marcus , [email protected] : − 7 M/M < ⊙ 2 < n ruaWandeler Ursula and D R´ujula, (De 1 ,Wnetuesrse10 H85 Z¨urich CH-8057 190, Winterthurerstrasse h, 3 ilgnPSI Villigen 232 fterc akrudo ihrteSCo h galactic the or SMC the front either in of M22 or background Tuc rich 47 the like clusters of globular monitor to is bright the as observation. eases center which not the is stars towards cluster the the concentrated populate of component similarly objects dark light the the Hence, whereas outskirts. towards sink cores, More- to tend cluster 1996). stars the heavy 1995, the Longaretti that et expects & one (Heggie over, Salati of dwarfs Taillet, form white 1996, the or their in al. stars of be low-mass could fraction which dwarfs, large dark, brown a is that 50%) recently (around mass argued clusters. been globular of has mass im- It total give the addition on in information can portant observation an Such glob- clusters. foreground ular in located MACHOs by stars background different between information models. distinguish to halo structure allowing galactic extracted be shown interesting can and way & possibility this Rhoads this that and studied have (1997) (1997) Holder mi- Malhotra & some get Gyuk to events. have order would in crolensing one clusters targets, globular as many compared used monitor be than to can 1991). stars SMC or less (Jetzer LMC much the MACHOs to clusters determine the globular the better in of towards to Since distribution ones allowing way the spatial this to the SMC, addition the in or sight LMC dif- of microlensing probe could fact, lines targets In ferent as problems. clusters these globular using of searches some solve to ful satellites. space of use the to how- require proposed which ever 1997), been (Gould has measurements it parallax lens, degeneracy perform not the this generally break of are To velocity mass known. transverse the and directly distance microlens- its infer single since cannot a one of of duration event, location the ing the from and fact, mass In the lenses. on the particular in open, still are 1997). Guibert & bulge Alard 1994, galactic Szyma´nski, al. Udalski, the et 1997a, Stanek towards al. et events Alves Allsman, 200 (Alcock, about and 1997) h da–soiial rpsdb Paczy´nski (1994)– by proposed originally –as idea The of microlensing for search to is possibility Another use- very respect many in be could clusters Globular questions several events, many the of spite in However, 2 ASTROPHYSICS ASTRONOMY 30.10.2018 AND 2 Gravitational Microlensing by Globular Clusters bulge. In this case, when the lens belongs to the cluster The time dependent magnification of a light source due population, its distance and velocity are roughly known. to gravitational microlensing is given by The velocity is defined by the dispersion velocity of the cluster stars together with the overall transverse velocity u2 + t2/T 2 +2 A(t)= ◦ ◦ , (1) of the cluster as a whole. Knowing approximately the dis- (u2 + t2/T 2)(u2 + t2/T 2)+4 tance and the velocity of the lens would allow to extract ◦ ◦ ◦ ◦ from a microlensing event the mass of the lens with an p with u = dmin/RE, where dmin is the minimal distance accuracy of 30%. ◦ ∼ of the MACHO from the line of sight between the source Due to these reasons, it is important to study in more and the observer. RE is the Einstein radius, defined as detail microlensing by globular clusters either using their stars as sources, or the dark matter contained in them 2 4GMD RE = x(1 x) , (2) as lenses for more distant stars (Wandeler 1995). In this c2 − paper we discuss both aspects in detail. Although the with x = s/D, D and s are the distances to the source mass distribution of the luminous part of the cluster, as and the MACHO, respectively. v is the relative trans- inferred from the observation of the distribution of the T verse velocity of the involved objects. T = R /v is the red giant population, agrees well with a King model, we E T characteristic time for the lens to travel◦ the distance R . do not consider this to be representative for the popu- E lation of light objects. Taillet, Longaretti & Salati (1995, The probability τopt, that a source is found within a 1996) have shown that in an isolated globular cluster ther- radius RE of some MACHO, is defined as malisation between the different populations does occur. 1 4πG However, globular clusters –especially the ones towards τ = ρ(x)D2x(1 x) dx , (3) opt c2 − the bulge– tidally interact with the surrounding material Z0 which might counteract thermalisation, hence the consid- eration of alternative mass distributions should be taken with ρ(x) the mass density of microlensing matter at the into account and variations of the microlensing event rate distance s = xD from us along the line of sight. due to it might give some hints at the dynamical history The microlensing event rate is given by (De R´ujula, of the cluster. Jetzer, Mass´o1991, Griest, Alcock, Axelrod et al. 1991)
We also analyse the microlensing events towards the dn galactic bulge, which are close to the three globular clus- Γ=2uT H D RE(x)vT fT (vT ) dvT dxdµ , (4) dµ ters NGC 6522, NGC 6528 and NGC 6540. These clus- Z ters lie within the observation fields of the MACHO and where µ = M/M and fT (vT ) is the transverse velocity OGLE teams. We find evidence that some microlensing ⊙ distribution. The maximal impact parameter uT H is re- events are indeed due to MACHOs located in the globular lated to the threshold magnification AT H by Eq. (1). In clusters, suggesting therefore that these clusters contain a the following we take AT H = 1.34 which corresponds to significant amount of dark matter. dn(x) uT H = 1. dµ is the MACHO number density, which for The paper is organized as follows: in Sect. 2 we in- a spherical halo is given by troduce briefly the basics of microlensing. In Sect. 3 we 2 2 discuss as an example the globular cluster 47 Tuc, which dn(x) dn a + RGC = ◦ , (5) will then be used to estimate in a very conservative way 2 2 2 2 dµ dµ a + RGC + D x 2DRGC cos α the optical depth and the lensing rate for other clusters − as well. In Sect. 4 we present microlensing using globular here α denotes the angle between the line of sight and the clusters towards the galactic bulge. In particular, we anal- direction towards the galactic center. n is assumed not yse the events as reported by the MACHO and the OGLE to depend on x and is normalized such that◦ collaboration lying within a distance of 30 pc around the centers of NGC 6522, NGC 6528 and NGC 6540. In Sect. 5 dn 3 M M µ dµ ◦ = ρ 7.9 10− ⊙ (6) we conclude with a short summary of our results. ⊙ dµ ◦ ≃ × pc3 Z
equals the local dark matter mass density. RGC 8.5kpc is the distance from the Sun to the galactic center≃ and a 5.6 kpc is the core radius of the halo. 2. Basics of microlensing ≃ For the average lensing duration one gets the following relation For completeness we give here a short summary of the most important formulae of gravitational microlensing; for 2τopt T = u . (7) more details see for instance Jetzer (1997). h i πΓ T H Gravitational Microlensing by Globular Clusters 3
3. The system SMC-47 Tuc: a paradigm for cluster mass of a ”particle”. Taking into account the spherical lensing symmetry of the globular cluster and defining the one- dimensional velocity dispersion σ as In this section we thoroughly discuss the basics of mi- crolensing by a globular cluster. We use the system SMC- k T σ = B , (10) 47 Tuc as an example, but the results, by appropriately m scaling them, are valid for other globular clusters as well. r First, we discuss simple models of 47 Tuc which will Eq. (9) reads then be used for the computation of the optical depth and ρ Φ(r) 1 v2 the microlensing event rate for different geometries of lens f(r, v)= ◦ exp − 2 . (11) (2πσ2)3/2 σ2 and source. The globular cluster 47 Tuc (NGC 104) lies at galactic Here r is the radial distance relative to the cluster center. coordinates l = 305.9◦, b = 44.89◦. For the distance Integrating over all velocities, we get we assume 4.1 kpc, although in− the literature values up to Φ(r) 4.7kpc are quoted (Harris 1996). Our choice will rather ρ(r)= ρ exp . (12) ◦ σ2 underestimate the optical depth. The position is such that it overlaps with a part of the outer region of the SMC, Inserting Eq. (12) into the Poisson equation for the grav- which makes it an interesting object. Globular clusters itational potential, one obtains are small objects compared to the scale of their distance, hence they are well suited for gravitational lensing, since d d ln ρ 4πG r2 = r2ρ . (13) one may assume the distance of their stars to be the same dr dr − σ2 for all practical purposes. A solution of this equation is 3.1. Spatial density and velocity dispersion for 47 Tuc σ2 ρ(r)= . (14) 2πGr2 For the calculation of the lensing rate we need to know the spatial distribution of the dark matter in the globular The mass density given in Eq. (14), together with the inte- cluster. Since this is not known, we will instead discuss grated velocity distribution Eq. (11), defines the singular models for the total mass of the cluster. To get the mass isothermal sphere. density ρd of the dark objects, we then make the simplify- To avoid the singularity at the origin we rescale the ing assumption that ρd is proportional to the total mass variables (˜r = r/rc, ρ˜ = ρ/ρ ) and find a non-singular ◦ density ρ, i.e. ρd is given by solution which can be approximated forr ˜ < 2 by (for details see Binney & Tremaine 1987) Mdark ρd = fρ = ρ , (8) 1 Mtot ρ˜(˜r)= forr< ˜ 2 (15) (1+˜r2)3/2 where Mdark is the total mass of the MACHOs in the clus- ter. We are aware, that this assumption is oversimplified, and forr ˜ >> 2 with the singular function given in however, as long as the content of dark matter in glob- Eq. (14). ular clusters is not known, it is one way to parametrize An observable quantity, connected with the surface our ignorance. Moreover, since the expected event rate for density of stars in a cluster, is the surface brightness. The 47 Tuc is about one event per year or even less, we do luminosity function Φ(M) gives the relative number of stars with absolute magnitude M in the range [M 1/2, not consider multi-mass models for 47 Tuc. We postpone − the discussion of them to Sect. 4, when we discuss lensing M +1/2]. The number surface density of the stars can be by globular clusters towards the bulge in which situation computed from the observed surface brightness. Assuming the model can be tested due to the higher number of mi- that all stars in the cluster have the same mass, we can crolensing events. compare the above densities with surface brightness mea- The simplest model of a globular cluster is a self- surements. The mass surface density Σ is derived from the gravitating isothermal sphere of identical ”particles” mass density by the integration (stars). The equilibrium distribution function in phase- ∞ 2 2 space coordinates is Σ(b)= ρ( l + b )dl , (16) Z−∞ p ρ m(Φ(x) 1 v2) where b is the projected radial distance. For the density f(x, v)= ◦ exp − 2 , (9) 3/2 of the singular isothermal sphere, given by Eq. (14), the 2π kB T kB T m surface density is where Φ(x) is the gravitational potential of the cluster, σ2 Σ(b)= . (17) T the temperature, kB Boltzmann’s constant and m the 2Gb 4 Gravitational Microlensing by Globular Clusters
Similarly, the densityρ ˜(˜r)in Eq. (15) leads to the surface 3. Singular model: the mass density is described by the density isothermal sphere, i.e. by Eq. (14) with the same σ as mentioned above. The total mass of this model within 2 sing 6 Σ(b) , (18) the tidal radius rt is Mtot =2.8 10 M . ∝ 1+ ˜b2 1 × ⊙ 4. 1+r2 - model: the mass density is given by where ˜b = b/r . ρ c ρ(r)= ◦ (23) Since Eq. (15) fits the regular solution of Eq. (13) in 2 1+ r the ranger ˜ < 2, we find that the corresponding surface rc density falls to roughly half of its central value at the core with rc as above and ρ chosen such that the total ◦ radius rc. Knowing rc and σ from observations, the central mass within the tidal radius is the same as for the density ρ is determined in the isothermal model by fitted King model. ◦ The last three mass distributions are cutted discontinu- 9σ2 ously at the tidal radius. Thus the integration range for rc = . (19) s4πGρ the optical depth and the lensing rate is only a sphere ◦ with a radius equal to the tidal radius. For all models the With rc =0.52 pc as given in Lang (1992) and a velocity velocity distribution follows a Maxwell distribution as in dispersion of σ = 10km/s (Binney & Tremaine 1987), Eq. (11) with σ = 10km/s. Eq. (19) for 47 Tuc yields ρ =6.0 104 M /pc3 . King (1962) found that the◦ above× surface⊙ density func- tions (17), (18) fit well star counts up to a limiting radius rt (called tidal radius), where the measured surface den- sity drops sharply to zero. Thus a better fit to the surface density of the cluster is given by
2 4 1 1 Σ(b) . (20) 2 1/2 2 1/2 ∝ [1 + (b/rc) ] − [1 + (rt/rc) ] ! With a spherical symmetric density
rt 1 d db 2 ρ(r)= Σ(b) , (21) −π db (b2 r2)1/2 Zr − the corresponding mass density becomes k 1 1 ρ(r)= arccos z (1 z2)1/2 , πr [1 + (r /r )2]3/2 z2 z − − 0 c t c (22) where 1 + (r/r )2 1/2 z = c 0 10 20 30 40 1 + (r /r )2 t c and k is chosen such that ρ(0) = ρ (King 1962). Inte- grating this density with a tidal radius◦ r = 60.3pc (Lang t Fig. 1. King 5 The four different mass density models used in the 1992), one finds a total mass of Mtot =3.5 10 M for × ⊙ calculations, plotted as a function of the radial distance 47 Tuc. r. For clarity the plot ends at r = 40 pc rather than at To study the dependence of the optical depth and the r = rt = 60.3 pc. lensing rate on different mass distributions, we will use the following four models (see Fig. 1). 1 The 2 -model is considered mainly because the in- 1. Fitted King model: 1+r the density is given by Eq. (22) tegrations can be performed analytically. Hence, we can 4 M⊙ with central density ρ = 6.0 10 3 , core radius ◦ × pc easily compare the analytic result with the ones obtained King 5 rc = 0.52 pc and Mtot = 3.5 10 M as already numerically for the other models. mentioned above. × ⊙ Of course there exist more sophisticated models for 2. Inner isothermal model: the density is given by globular clusters, e.g. the (single or multimass) King- Eq. (15) with rc =0.52 pc and σ = 10 km/s as above, Michie models (King 1966, Gunn & Griffin 1979), that but with ρ chosen such that the total mass within the take into account the finite escape velocity from the clus- tidal radius◦ is the same as in the fitted King model, ter, which naturally leads to a finite extension of the clus- rather than determined via Eq. (19). ter. However, for the study of the influence of different Gravitational Microlensing by Globular Clusters 5 mass distributions on the measured quantities, we con- sider the above mentioned models to be sufficient and, -5 therefore, in this section, restrict our discussion to them. In addition, we remind on the well known scaling proper- ties of the King model, which allow to derive the corre- sponding quantities for a different choice of parameters.
-10 3.2. Optical depth, lensing rate and mean event duration for SMC-47 Tuc We now discuss the different possibilities for microlensing in the system SMC-47 Tuc, i.e. events where the source and the lens are both located in 47 Tuc; the source is in the SMC and the lens in 47 Tuc; the source is in 47 -15 Tuc and the lens in the halo of the Milky Way, or finally the source resides in the SMC and the lens in the Milky Way. For SMC self-lensing we refer e.g. to the paper by Palanque-Delabrouille, Afonso, Albert et al. (1997). At the end we also discuss the dependence on the mass -20 function, which itself is independent of the lensing geom- -60 -40 -20 0 20 40 60 etry. Since we hope to disentangle the different cases, we dΓ also calculate the differential rate dT . Throughout this part we will assume all lenses to have the same mass. The Fig. 2. The optical depth for the four different models as a velocity distribution of the halo objects, as well as those function of the source position. All sources are located on the of the cluster is taken to be a Maxwell function. As al- line of sight from the observer to the cluster center, but at ready mentioned, we define the amplification threshold to different radii relative to the center. The optical depth of the be AT H =1.34. singular model is only shown for sources with Dc − D > 0, since its mass density has a singularity at r = 0. For the plot 3.2.1. Optical depth for source and deflecting mass in the we assume f = 1, and the total mass of the models as given in cluster Sect. 3. For completeness we discuss also this case, although, as we will see, its contribution can be neglected for all practical Here purposes. 2 2 xu = rt b (Dc D) For a pointlike source τopt is, according to Eq. (3), − − − given by q and l = D Dd is the distance from the lens (located at 1 − 4πG Dd) to the source. Hence, τopt is proportional to D Dd 2 − τopt = 2 D x(1 x)ρd(r(x)) dx (24) weighted with the mass density in the globular cluster x c − Z b along the line of sight from the observer to the source. where
2 2 Dc rt b rt 2 3.2.2. Lensing rate for source and deflecting mass in the xb = − − 1 =1 1.4 10− . (25) pD ≃ − D − × cluster D is the distance from the observer to the cluster cen- c We assume the mass function to be independent of the ter and D the one to the source. Hence, the integration position and all lenses are taken to have the same mass µ is cut at the boundary of the cluster. The optical depth (in units of M ). From Eq. (9) we find that the transverse◦ for a source located in the center of the cluster (b = 0) is ⊙ 9 Mdark velocity distribution is given by τ = 9.2 10 5 for the fitted King model. opt − 3.5 10 M⊙ × × The results for the 4 different models are shown in Fig. 2. v 2 2 T vT /vH Of course the isothermal-sphere model differs most, be- fT (vT )=2 2 e− (26) sing 6 vH cause its total mass Mtot =2.8 10 M varies substan- tially from the others. × ⊙ with v2 =2σ2. If the source as well as the deflecting mass Since the lower limit of integration is very close to 1, H are in the cluster, the event rate becomes τopt can be approximated as follows x 1 4πG u √2πσrE Γ = D ρd(r(x)) x(1 x)dx , (27) τopt 2 l ρd(r(l)) dl . ≃ c 0 M √µ xb − Z ⊙ ◦ Z p 6 Gravitational Microlensing by Globular Clusters with r(x)= b2 + (D xD)2 and r defined to be c − E 4G p rE = 2 M D. (28) -10 r c ⊙ We use the same geometry as in the previous subsection. The result for the fitted King model for a source located −15 6.6 10 Mdark at the center of the cluster is Γ = × 5 1/s. √µ◦ 3.5 10 M⊙ × In order to more easily compare between the microlens- -15 ing rates for different locations of the source and the lens, we introduce the quantityn ˜(α): N˜(α) n˜(α) 2 , ≡ (1′) -20 where N˜(α) is the number of microlensing events per unit 2 time in an area of (1′) located at an angular distance α from the cluster center. For simplicity, we will give the raten ˜(α) in units of pc. In the plane perpendicular to the line of sight through the cluster center, 1′ corresponds to -25 0 20 40 60 1.2 pc. Hence, we can define a new quantity 1.2pc 1 n b = α n˜(α) 2 , 1′ ≡ (1.2pc) Fig. 3. The surface density of microlensing events is plotted 2 1 which is in units of pc− unit time− . We call n(b) the as a function of the projected radius b for the four different surface density of microlensing events. To calculate n(b) models calculated in the text, assuming µ◦ = 1. For the plot we have to add up the lensing rates for all stars located we assume f = 1, and the total mass of the models as given in on the line of sight with impact parameter b from the Sect. 3. cluster center. From Eq. (27), we see that the lensing rate Γ depends on D and on b, through r(x) and xb. Taking The results for the four models are shown in Fig. 3. this into account we get The number of lensing events per year in the whole Df cluster is n(b)= n ( b2 + (D D)2) Γ(D,b) dD (29) star c r i − t ZD 2 1 Nstar p 2πn(b)b db =2 10− f 5 (31) where in the ideal situation, which will lead to an upper 0 × µ 4.6 10 Z r ◦ × bound for n(b) we have Df,i = D r2 b2. n (r) c t star for the fitted King model. On the other hand, the number is the number density of stars (in units± of− pc 3) in the p − cluster at distance r from the center and D, b, D are all of lensing events depends also on the telescope resolution. c Assuming a distribution according to Eq. (30), we find in units of pc. To describe the distribution of stars in the that stars with a projected radial distance smaller than cluster, we assume that the number density is proportional to the mass density ρ of the fitted King model, since 0.6 pc from the center cannot be resolved if the limiting King resolution of the telescope is 0.1”, whereas if it is 1” even the mass surface density of this model is proportional to the number surface density of stars in a cluster. Thus n stars with projected radius b smaller than 7.5pc from the star center cannot be resolved. Hence, the number of expected is given by lensing events per year using a telescope with a resolution Nstar of 0.1” is nstar(r)= King ρKing(r) (30) Mtot 2 1 Mdark Nstar 1.1 10 5 5 r − µ◦ 3.5 10 M⊙ 4.6 10 with N = 4.6 105 (Lang 1992). With Eqs.(27) and t × × × star 2πn(b)b db = (29) this leads to × q 0.6pc 3 1 Mdark Nstar Z 2.4 10 5 5 − µ◦ 3.5 10 M⊙ 4.6 10 19 Nstar 1 × × × n(b)=3.6 10− q × Mtot √µ ◦ and the one for a resolution of 1” is Df 3/2 2 2 4 1 Mdark Nstar D ρKing b + (Dc D) 1.4 10− 5 5 × i − rt µ◦ 3.5 10 M⊙ 4.6 10 ZD × × × 1 p 2πn(b)b db = q 2 2 x(1 x)ρd b + (Dc xD) dD dx . 7.5pc 4 1 Mdark Nstar Z 2.3 10− 5 5 × x (D) − − µ◦ 3.5 10 M⊙ 4.6 10 Z b × × × p p q Gravitational Microlensing by Globular Clusters 7
The first line is for the fitted King model, the second for – the distance D of the source is now about 63kpc, cor- 1 the 1+r2 -model. responding to the distance of the SMC from the Sun; The distribution of the microlensing events as a func- – the x-integration goes from x = (D r2 b2)/D, i c − t − tion of their duration T, using the transverse velocity dis- to x = (D + r2 b2)/D; f c t − p tribution as defined in Eq. (26) and the definition of rE in – the velocity distribution gets shifted, because of the p Eq. (28), is given by motion of 47 Tuc relative to the SMC with a velocity of about v = 180km/s. Hence, a velocity drift has to 4 d ∼ dΓ 2Dµ rE 1 ◦ be introduced in the Maxwell distribution, such that = 2 4 dT − M σ T 2 2 2 ⊙ 1 (vx vd) + v + v R2 (x) f(v)= exp − y z , (33) (x(1 x))2 ρ (x)exp E dx. (32) (2πσ2) − 2σ2 × − d −2σ2 T 2 ! Z where we assume the x-axis to be parallel to the veloc- The result for the fitted King model assuming a lens ity drift. The resulting transverse velocity distribution mass µ = 1 is shown in Fig. 4. is then given similarly to Eq. (26). With this, the in- ◦ tegral over dvT in Eq. (4) yields ∞ v f(v )dv = v 180 km/s . (34) T T T d ≃ Z0 Thus, the lensing rate and the optical depth become 4πG xf τ = D2 x(1 x)ρ (r(x))dx , (35) -8 opt c2 − d Zxi xf 2vdrE Γ = D x(1 x) ρd(r(x))dx M √µ x − ⊙ ◦ Z i -10 18 p3/2 xf 5.1 10− D ρd(r(x)) = × x(1 x) 3 dx .(36) M √µ pc x − M /pc ⊙ ◦ i ⊙ Z p -12 For the extreme case of a star in the SMC lying on the line of sight going through the center of the cluster, the optical 4 Mdark depth is τ =1.4 10 5 and the lensing rate opt − 3.5 10 M⊙ × × 11 1 Mdark Γ=2.0 10 5 1/s. − µ◦ 3.5 10 M⊙ -14 × × Since the tidalq radius is smaller than about 200 pc, the quantity x(1 x) as well as x(1 x) does not vary more than 10% over− the integration range,− and the variation of p -16 0 20 40 60 80 100 τopt as well as Γ is directly proportional to the surface density of the cluster. Assuming a tidal radius of 60 pc we find (withx ¯ an average value for x) 4πG Fig. 4. The distribution of the lensing events with duration τopt =x ¯(1 x¯) 2 D Σ(b) f T = RE /vT for different geometries of the lens system. For − c the calculations we used the fitted King model and made the 9 D Σ(b) = (2.32 0.04) 10− f (37) assumption that all lenses have the same mass µ◦ = 1. The ± × 63kpc × (M /pc2) notation is to be understood as follows: the first entry denotes ⊙ the position of the source, the second the one of the lens e.g. 2v r SMC−>Cl means a star of the SMC is lensed by an object Γ(b)= x¯(1 x¯) d E Σ(b) f in the cluster. MW abbreviates Milky Way; b, D and D are − M √µ c ⊙ ◦ defined within the text. For the plot we assume f = 1, and the p 16 D Σ(b) 1 total mass of the model as given in Sect. 3. = (3.25 0.03) 10− f . (38) ± × 63kpc M /pc2 µ s ⊙ r ◦ The mean event duration is 2 τ T = opt = 52 √µ days, 3.2.3. Optical depth and lensing rate for the source in the h i π Γ ◦ SMC and the deflecting mass in the cluster which is nearly independent of the angle between source What changes in the calculation of τopt and Γ is itemized and cluster center, since both τopt and Γ are proportional below: to Σ(b) as given by Eqs.(37) and (38). 8 Gravitational Microlensing by Globular Clusters
Of course the chance to find a lensing event in this for the fitted King model. The expected number of events case depends not only on the optical depth, but also on per year for an observation done with a resolution of 0.1′′ the number of SMC stars in the background of 47 Tuc. is Hesser, Harris, Vandenberg et al. (1987) finds an average 1 1 Mdark nstar 2 1.0 10− 5 of 1000 stars/30(arcmin) with a magnitude in the inter- r µ◦ 3.5 10 M⊙ 50 t × × vall of 0.0 < (B V ) < 0.8, 21 -15 brightness observations of 47 Tuc leads to a total mass of 6 Mtot =1.1 10 M (Meylan 1989), which is about 3 times bigger than× the value⊙ for the fitted King model. In addi- 0 20 40 60 tion, the fitted King model concentrates much more mass in the center, whereas in the singular isothermal sphere model, there is still a lot of mass in the outer regions of the cluster. From these considerations, we see that the un- Fig. 5. The surface density of microlensing events is shown for certainties are quite important and they are mainly due to the different geometries of the lens system. For the calculations the poor knowledge of the dark matter content in globular we used the fitted King model and made the assumption, that clusters. all lenses have the same mass µ◦ = 1. For the abbreviations we refer to Fig. 4. For the plot we assume f = 1, and the total Paczy´nski (1994) proposed to measure the microlens- mass of the models as given in Sect. 3. ing of background stars by MACHOs in globular clusters, because in this way the mass of the lens can be determined more precisely than in the lensing experiments under way. Since the lens is in the cluster, it has the transverse ve- The number of events per year in the whole area of 47 locity and the position of the cluster. To estimate the in- Tuc is then given by herent error of the method we adopt the values below, rt where the transverse velocity vT is assumed to be pre- 0 2πn(b) bdb = cisely known up to the cluster dispersion velocity and the 1 Mdark nstar 1 =0.12 R 5 2 (40) µ◦ 3.5 10 M⊙ 50/(1.2pc) year distance up to the cluster size: vT = 180km/s 10km/s, × ± q Gravitational Microlensing by Globular Clusters 9 D = 4100pc 60pc and D = 63kpc 6kpc. It fol- For the distribution of the lensing events we obtain d ± ± lows then an uncertainty in the mass determination of 4 ∆m dΓ 4Dµ rE 1 0.33 . The uncertainty in the lens mass with the = ◦ m ≤ dT − M v2 T 4 currently reported lensing events is more like a factor 3 or ⊙ H 1 2 even bigger (Jetzer & Mass´o1994, Jetzer 1994). R dxρ (x)[x(1 x)]2 exp E (45) The distribution of the lensing events as a function of × d − −v2 T 2 Z0 H the duration T is: with the MACHO mass density ρ(x) as in Eqs.(5) and 4 xb π dΓ 2Dµ rE 1 2 (6). = ◦ dx dy [x(1 x)] ρ (x) dT πM T 4 σ2 d − xE 0 − dΓ 2 1 ⊙ Z Z =3.0 10− v2 R2 R v dT × T 4 × exp d E + E d cos y , (41) × −2σ2 − 2σ2T 2 σ2T 1 x(1 x) 1 dxρ(x)[x(1 x)]2 exp -2.3 103 − . − × T 2 days2 where y is the angle between v and the projected MA- Z0 d (46) CHO velocity in the plane perpendicular to the line of sight. The numerical results, with the above mentioned The numerical results with vH = 210 km/s and D = values for vd and σ are shown in Fig. 4. 4.1 kpc for µ = 1 are shown in Fig. 4 where the motion of the cluster◦ relative to the Milky Way halo is neglected. 3.2.4. Optical depth and lensing rate for the source in the For a list of globular clusters which might be used as cluster and the deflecting mass in the halo of the targets for a systematic microlensing search, we refer to Milky Way the papers of Gyuk & Holder (1997) and Rhoads & Mal- hotra (1997). To calculate the optical depth and the lensing rate we set the position of the source equal to the position of the 3.3. Optical depth and lensing rate for a source in the SMC cluster. In the reference frame, where the origin is at the and the deflecting mass in the halo of the Milky Way galactic center and the x1 x2 plane is the symmetry plane of the galaxy, the coordinates− of a lens, located on Since this is a standard case we restrict to the presentation the line of sight from the observer to the cluster center, are of the results. With a distance D = 63 kpc and the galactic x1 = xDc cos bT cos lT RGC, x2 = xDc cos bT sin lT , and coordinates of the SMC l = 302.8◦ b = 44.3◦, we get for − o o − x3 = xDc sin bT , where bT = 44.89 and lT = 305.9 are the optical depth and the lensing rate the galactic latitude and longitude of 47 Tuc. With the 7 14 1 number density distribution Eq. (5), we find the optical τ =7.0 10− and Γ=6.6 10− 1/s . 8 × × µ depth (with D = Dc) to be τopt =1.5 10− . r × ◦ Let’s now look at the lensing rate. Since the velocity Therefore, the time scale is Γ is the lensing rate computed under the assumption In this section we discuss the possibility to use dark that⊙ all lenses have mass µ = 1. objects in globular clusters towards the bulge as lenses (see For the limiting cases γ◦ << 1/2 and γ >> 1 this leads also Taillet, Longaretti & Salati 1995, 1996), in front of to a lensing rate rich regions of the galactic center or the spiral arms. There is a large sample of possible clusters which could be used 1 γ >> 1 √ma for this purpose (see Table 1). The core radius of the dark Γ=Γ 1 (49) ⊙ γ << 1/2 . component of the cluster can be larger than the core radius ( √mb obtained by surface luminosity measurements. Since the In Fig. 6 Γ/Γ is shown for different upper and lower lim- size of the cluster is much smaller than its distance, it can its of the MACHO⊙ mass. be assumed to be equal for all its members, moreover, the velocity dispersion inside the cluster is also small, hence globular clusters are well suited for determining the lens 40 mass. 4.1. Estimate of optical depth and event rate 30 In the first part of this section, we give some (conservative) estimates for the optical depth and the event rate due to some globular clusters towards the bulge by scaling the results obtained for 47 Tuc under the assumption that the 20 clusters contain only little dark matter. As was derived in Eq. (39), the optical depth due to a globular cluster is proportional to the surface density of microlensing events and the pure geometrical quantityx ¯(1 x¯). Assuming Σ(b) − to be proportional to the total visual magnitude MV , we 10 can calculate the contribution to the optical depth due to the clusters (see Table 1). Since the galactic bar is not too extended, we fix the distance of the sources to be 8.5 kpc. Compared with the measured average optical depth to- 0 6 -1 0 1 2 3 wards the galactic center τopt 2.4 10− due to the disk and the bar itself (Alcock, Allsman,≃ × Alves et al. 1997a), we see from Table 1 that some of the globular clusters can give a very significant contribution to the total op- Fig. 6. Γ/Γ⊙ is plotted as a funtion of the slope γ. This cor- tical depth or even dominate it along their line of sight, responds to the dependence of the lensing rate on the slope of as in the case of NGC 6553, which lies close to the ideal the mass function. Γ⊙ is the lensing rate for a model where all distance ofx ¯ =0.5. MACHOs have the same mass µ◦ = 1 A rough estimate of the microlensing event rate per year due to globular clusters towards Baades Window is obtained by properly rescaling Eq. (40). We compute the 1 Mdark nstar 1 event rate in units of 5 2 as in 4. Microlensing towards the bulge using dark ob- µ◦ 3.5 10 M⊙ 50/(1.2pc) year × jects in globular clusters as lenses Eq. (40), in order to beq able to easily compare between the different clusters. To that purpose one has to use the One encounters several problems by using globular clus- scaling factor η given by: ter stars as sources. In fact, globular clusters have at most some 105 stars that can be monitored. The angular size v r n (1.2pc)2 Σ (b) db of a globular cluster is small and the stars are highly con- η =0.12 d′ E′ star′ ′ , (50) v r n A Σ(b) db centrated towards the center. Furthermore, most globu- d E star R lar clusters are located towards the galactic center which R restricts the number of objects being well suited for an where the unprimed quantities belong to 47 Tuc and the exploration of the halo. Therefore, microlensing by glob- primed to the cluster under consideration. A is the surface 2 ular clusters only leads to valuable results, if one is able which corresponds to (1′) at the distance of the cluster. to simultanously observe dozens of clusters with a very For the calculation of the numbers, as given in Table 1, high resolution for several years. At present this seems to we assumed vd′ = 30km/s which is indeed a very conser- be a somewhat to demanding task, however this could be vative value and, therefore, we will get a lower limit for feasible in the near future. the eventrate. Gravitational Microlensing by Globular Clusters 11 Mdark 1 Mdark nstar 1 Object Position [l, b] MV x¯ τopt 5 N 5 2 3.5×10 M⊙ µ◦ 3.5×10 M⊙ 50/(1.2pc) year Terzan 1 357.57 1.00 -3.3 0.765 2.h1 × 10−7 i hq 1.1 × 10−5 i Terzan 5 3.81 1.67 -7.9 0.941 4.7 × 10−6 5.5 × 10−4 Terzan 6 358.57 -2.16 -6.8 0.882 3.2 × 10−6 2.2 × 10−4 UKS 1 5.12 0.76 -6.2 0.882 1.8 × 10−6 1.3 × 10−4 Terzan 9 3.60 -1.99 -3.9 0.918 1.6 × 10−7 1.4 × 10−5 Terzan 10 4.42 -1.86 -7.8 0.988 9.0 × 10−7 4.3 × 10−4 NGC 6522 1.02 -3.93 -7.5 0.824 8.6 × 10−6 5.0 × 10−4 NGC 6528 1.14 -4.17 -6.7 0.871 3.2 × 10−6 2.1 × 10−4 NGC 6540 3.29 -3.31 -5.3 0.412 1.9 × 10−6 2.6 × 10−4 NGC 6544 5.84 -2.20 -6.5 0.294 4.9 × 10−6 1.5 × 10−3 NGC 6553 5.25 -3.02 -7.7 0.553 1.8 × 10−5 1.3 × 10−3 NGC 6558 0.20 -6.03 -6.1 0.753 3.0 × 10−6 1.6 × 10−4 Table 1. Optical depth and event rate due to MACHOs in some globular clusters for sources located towards the galactic center. The symbols are defined within the text. Cluster data is adopted from Harris (1996). 4.2. Spatial distribution of microlensing events around decided for a single event, we assume that the events which NGC 6522, NGC 6528 and NGC 6540 lie in the ring from 12 to 30 pc are due to MACHOs in the disk or bulge. This way we certainly overestimate the In the following we present a rough analysis of the mi- ”background”. Moreover, the common events within 30 pc crolensing events around the three globular clusters which around both NGC 6522 and NGC 6528 were counted as lie within the observation fields of MACHO and OGLE. ”background” events for both clusters. This way leading Within a radius of 30pc around NGC 6522 and 6528 also to a higher background rate. The so estimated event we found 7 events for each of them (see Table 2). Since the rate per area is then subtracted from the value in the inner projected areas of these two clusters overlap, 4 events lie 12 pc region. The leftover events should be due to MA- within the 30 pc circles around both clusters. NGC 6540, CHOs located in the globular cluster. We find the follow- which is nearer to us, hosts a total of 15 events within ing values (in parenthesis the ”background”): 2 (1) hence the 30 pc circle. At first sight, since the covered area is a total of 3 events for NGC 6522, 1 (1) event for NGC 6528 about four times larger, this value seems to be lower than and 2 (2) events for NGC 6540. We see that the number of expected. However, NGC 6540 is about 8 times less bright observed events in the inner 12 pc is roughly twice as high than NGC 6522. Therefore, if the total luminosity roughly as one would get due to MACHOs in the disk or bulge scales with the total mass content, we expect NGC 6540 alone. By assuming that all the events in the ring from to contain less dark matter than NGC 6522. 12 to 30 pc are due to MACHOs in the disk or bulge, we Within 12 pc we found 3 events for NGC 6522, 2 events have certainly underestimated the contribution from the for NGC 6528 and 4 events for NGC 6540 (see Table 2). globular cluster, since some of these events might also be The event rate ratio in units of events per square degree associated with the cluster. for the 12 pc circle and the 30 pc ring (excluding the in- nermost 12pc) are 99/25 for NGC 6522, 74/35 for NGC Of course, we must discuss the shortcomings of our 6528 and 33/17 for NGC 6540, respectively. For all three analysis. We tacitly assumed that the product of total ob- clusters the central region shows an increase in microlens- servation time and background star density is the same for ing events. Due to the fact that the 30 pc circles around the 12 and 30 pc regions for a given cluster. This should NGC 6522 and NGC 6528 intersect, we can also calculate at least be well fullfilled for the relatively small regions the event rate in the overlapping region, which we find to around NGC 6522 and NGC 6528. For NGC 6522 and be 70 events per square degree. Within our poor statistics, NGC 6528 we added MACHO and OGLE data, hence this is what one expects for a line of sight crossing twice we use a different normalisation for them than for NGC the region of influence of a globular cluster. It would also 6540. In addition, we did not take into account the dif- give a first hint, that globular clusters can indeed have a ferent efficiencies. Moreover, our evaluation is based upon population of dark objects that reaches out as far as 30pc. a very poor statistics, and thus one must take the above Of course, there is now the problem how to distinguish results with all the necessary caution. However, since all between the events due to MACHOs located in the globu- our estimates were performed very conservatively and for lar cluster and those due to MACHOs in the disk or bulge, all three clusters we get the same behaviour and since which will define our ”background”. Since this cannot be also the overlapping region of NGC 6522 and NGC 6528 12 Gravitational Microlensing by Globular Clusters it is not possible to distinguish them, we have just taken the average value as a first approximation. This should not -2 be far from the true value (as can be seen by inspection of Table 2). The results are given in Table 3. We see that depending on vd the MACHOs can be either Jupiter type objects, brown dwarfs, M-stars or even white dwarfs. -3 -4 Object hMi d ≤ 12 pc hMi 12 Fig. 7. Position of 195 microlensing events towards the bulge in galactic coordinates (taken from the alert list of the MA- Below we compute the optical depth and the event rate CHO collaboration and the corresponding list of the OGLE for four different King models as a function of the distance team). The crosses denote the position of some of the globu- from the cluster center in the lens plane. The parameters lar clusters reported in Table 1, which are located in Baades are tuned such that the average of the values in the in- Window. Only the three clusters NGC 6522, NGC 6528 and terval from 1 to 12pc roughly corresponds to the above NGC 6540 lie within the observation fields. The circles around mentioned values for the optical depth τ and the event the three clusters correspond to a radius of 30 pc around the rate Γ. The tidal radius was assumed to be r = 60.3 pc, cluster center. t the distance to the lens is set to be 3.5 kpc and the one to the source 8.5kpc. For the calculation of the event rate we shows an increase in microlensing events, we think it’s fair again take the rather low value vd = 30km/s. Since we are to state, that globular clusters can –within some pc– at not able to reproduce the mass function, we rather study a least double the optical depth and also the event rate due bi-mass model, hence the cluster consists of a heavy com- to MACHOs in the disk and the bulge; the latter quan- ponent (component 1) and a light one described by one of 6 tities being τ 2.4 10− (Alcock, Allsman, Alves et the other components as defined below. ≃ × 12 1 al. 1997a) and Γ 1.3 10− s− for a typical mass of ≃ × Component 1: the density is given by Eq. (22) with cen- 0.1 M . 4 M⊙ ⊙ tral density ρ =6.0 10 3 core radius rc =0.52 pc Stars in globular clusters can also act as sources for mi- ◦ × pc King 5 crolensing, however, as discussed in Sect. 3.2.4 their con- and M1 =3.5 10 M . For the calculation of the event rate we assumed× a typical⊙ mass M = M . tribution, unless for the very central region of 1 2 pc, ⊙ is very small as compared with lensing due to∼ sources− in Component 2: as Component 1, but with a core radius the bulge. For NGC 6522 and NGC 6528 there is a small rc = 1.56 pc and a typical mass of M = 0.1 M . The total mass of this component is again 3.5 105⊙M . probability that the source lies in NGC 6528 and the lens × ⊙ in NGC 6522. Knowing the distances and the relative mo- Component 3: as Component 2, but with a total mass of the component of 1.75 106 M tion of the two clusters this system might yield the most × ⊙ accurate mass determination for a lensing object. Component 4: as Component 3, but with a core radius It is interesting to note, that the above mentioned rc =2.6 pc event rate ratios scale with the total luminosity i.e. the We find that a population of low mass objects as described brightest cluster NGC 6522 shows also the largest increase by a King model with a total mass of 1.75 106 M can of events towards the center. lead to the desired enhancement of the optical× depth⊙ and For the mean event duration as given in Table 2, we the lensing event rate up to distances of 12pc from the ∼ calculate the typical mass of a MACHO for vd = 30km/s cluster center (see Figs. 9 and 10). Of course, the mod- and vd = 180 km/s. Obviously, one should consider only els and mass values given above have to be taken as an the events due to MACHOs in the cluster. However, since illustration, nevertheless it is clear that the rather high ob- Gravitational Microlensing by Globular Clusters 13 2 Object Position [l, b] MV within 12 pc within 30 pc Duration [days] Mean [days] Events/degree NGC 6522 1.02 -3.93 -7.5 OGLE 5 12.4 97-37∗ 12 97-68 6 10.1 99 OGLE 1∗ 25.9 OGLE 2 45 OGLE 4∗ 14 97-14∗ 21.5 26.6 25 NGC 6528 1.14 -4.17 -6.7 OGLE 1∗ 25.9 97-14∗ 21.5 23.7 74 OGLE 3 10.7 OGLE 4∗ 14 OGLE 10 61.1 97-37∗ 12 95-11 30.5 22 35 NGC 6540 3.29 -3.31 -5.3 96-6, d< 6 pc 19.5 33 96-17 16.5 95-29 12 95-40 8.5 14 33 96-14, d< 18 pc 12.5 95-26, d< 18 pc 19.5 95-31, d< 18 pc 16 20 97-2, d< 24 pc 23.5 97-10, d< 24 pc 8.5 97-24, d< 24 pc 5 95-36, d< 24 pc 12 20 98-1 97.5 97-58 26.5 96-1 76 95-30 33.5 15 30 17 Table 2. Microlensing events within a radial distance of 30 pc around NGC 6522, NGC 6528 and NGC 6540. Values marked with an asterisk lie within 30 pc of both NGC 6522 and NGC 6528. For NGC 6540 we also give the finer binning as used for Fig. 8. Data is taken from the alert list of the MACHO collaboration and the event list of the OGLE team. The event duration follows the OGLE convention i.e. T = RE /vT , which is half the value as reported in the MACHO alert list. served microlensing rate of the clusters imply a substantial the calculation of the optical depth, the microlensing event dark matter component. Although there is a large inherent rate and the average lensing duration for all possible ge- uncertainty in the event rate due to the poor knowledge ometries of the system SMC-47 Tuc-Milky Way. In addi- of vd, it is important to note, that the required optical tion, we studied the dependence of these parameters on depth and event rate cannot be due to the heavy compo- the mass function. nent alone. We have seen that for the case, where the source is a star in the SMC and the lens is a MACHO in 47 Tuc, 5. Summary one can expect an observable eventrate of 0.1 1 per year. However, this result depends crucially∼ on the− total We discussed in detail microlensing by globular clusters. amount of dark matter and its distribution in the cluster, 47 Tuc was taken as an example for which we performed which both are not well known at present. 14 Gravitational Microlensing by Globular Clusters -12 30 -14 -16 25 -18 20 -20 0 20 40 60 15 0 10 20 30 Fig. 10. The microlensing event rate for the four different King models as described within the text. Fig. 8. The microlensing event rate per area as a function of the radial distance from the cluster center for NGC 6540. We then applied these results to study microlensing by globular clusters towards the galactic center, where locally the optical depth can be dominated by dark matter inside clusters. However, since globular clusters are very local- ized objects, the expected number of events as obtained by these scaling arguments is small. A larger event rate is -4 expected, if the average MACHO mass inside the cluster is well below one solar mass, the total amount of dark mat- ter is larger than 3.5 105M or the density of observable × ⊙ -6 stars behind the cluster is significantly higher than the assumed value of 50 stars per (1.2pc)2. Indeed, analysing the event distribution around the three clusters inside the observation fields of MACHO and OGLE, we find an in- -8 crease of the microlensing event rate by at least a factor of about 2, as compared to that expected for MACHOs located in the disk and the bulge. This increase suggests the presence of a substantial amount of dark matter in -10 form of light objects such as brown dwarfs. Given this promising preliminary results it is impor- tant to systematically analyse future lensing data as a -12 function of the position around the mentioned globular clusters. In fact, having few more events at disposal will already be very helpful to draw more firm conclusions and 0 20 40 60 get better limits on the content of dark matter in globular clusters. In addition, we propose to favour observation fields Fig. 9. The optical depth for the four different King models around globular clusters for future campaigns. In partic- as described within the text. ular NGC 6553 is a very promising candidate for lensing by dark objects in globular clusters, since its luminosity is high and also its distance is such that the tidal radius Gravitational Microlensing by Globular Clusters 15 of the cluster corresponds to a relatively large angular size. Moreover, it would be important to have more pre- cise knowledge of vd, which is at present one of the main sources of uncertainty. Acknowledgements. This work is partially supported by the Swiss National Science Foundation. It is a pleasure to thank the referee for his helpful and encouraging remarks. 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