Astroparticle Physics

ELSEVIER A%tropartlclc Physic\ 4 ( 1995) 87-97

Baryonic dark matter in globular clusters Richard Taillet ‘, Pierre-Yves Longaretti b, Pierre Salati ‘A’

Kcce~ved 6 blarch I995

Abstract

A persistent rumor has it that globular clusters camlot contain dark matter. We show here the opposite. Because stars arc densely packed. thermal relaxatmn ohtams. Heavy Stars tend to sink towards the cluster cores, whereas light objects

populate the outskirts. Just like icebergs, globular clusters could therefore contain large amounts of unseen material in the

form of light and faint objects. Their properties, as traced for instance by the red giant stars, would not be much affected.

We have characterited the presence of heavy and light species with two-component King models. The amount of low-mass

stars turns out to he badly constrained from observations of the luminous component. However, the dominance of low-mass

stars in the outer regions makes them detectable hy the forthcoming infrared telescopes. Those light objects should also

induce quite a few gravitational microlensings on the more distant stars of the background. Observation of such events

could shed some light on the dark \ide of globular clusters, and on the missing mass of the galaxy.

1. Introduction

A significant fraction of the mass in the universe is invisible. Flat rotation curves around spiral galaxies

point towards the presence of large haloes of unseen

material whose nature is still unresolved. in spite of

the recent observations. If this dark matter is made up of brown dwarves or very low-mass stars. it has been suggested by Pac/yrisky [ I ] that their presence could be detected through their statistical lcnsing effect on the stars of the Large Magellanic Cloud (LMC), Such

gravitational microlensings have been observed [ 2.31, indicating that the halo of our galaxy contains light objects, with mass ranging from 0.03 to 0.3 M,;. How- ever, the rate of events towards the LMC is lower than

expected. The low-mass lenses detected by the EROS

and MACHO collaborations could account for a frac- tion only of the halo dark matter. Sahu [4] has even claimed that these objects are mere stars within the LMC itself. Hu et al. [ 51 have carried out a moderately deep search for very low-mass main-sequence stars at high galactic latitudes. They conclude that faint stars contribute very little to the galactic halo dark matter. This observation is supplemented by a proper-motion survey of high-velocity stars in the solar neighbour- hood where M subdwarves are found to comprise less than a tenth of the halo [ 61. On the other hand, stel- lar counts in the spheroid lead to the very steep mass

functiondN/dm x m-(‘+.‘) wherex = 3.5~!~1..2down to 0.14 II&~, with no evidence of a turn-over [7]. A faint luminous halo has been found around the edge- on spiral galaxy NGC 5907. The light intensity falls smoothly in the direction perpendicular to the galac- tic plane and is compatible with the distribution of

dark matter as infcrrcd from the rotation curves of that galaxy [ 81.

Faint low-mass stars arc potential infrared emitters. Observation of their diffuse emission could unravel their prcscnce. However, in the galactic halo. the var- ious stellar populations arc intimately mixed and the infrared glow of light stars is desperately swamped in the luminosity of their heavier companions 191. Alternatively. the prcscnce of low-mass stars in large amounts can be more easily detected in globular clus- ters. Heavy stars tend to sink towards the cluster cores. whereas light stars populate the outskirts. Based on a deep imaging in the I handpass [ 1 (J-121, the stellar mass function of six globular clusters seems to turn up sharply below 0.4 A4.. None of these functions Hatten before the limit of the data is reached. These observations may therefore be interpreted as if globu- lar clusters comprised two distinct stellar populations: visible massive stars and faint low-mass species. Hou ever, a recent stellar count performed in one of those clusters, NGC 6397, does not lind evidence lhr a large population of M subdwarves [ 13 ] The mass function essentially vanishes at the H burning limit where the luminosity function shows a steep decline to Lero. Oh- viously, more observations are needed at that point. with deeper and fainter surveys. Quite exciting is the detection by the MACHO and OGLE collaborations 01‘ w 40 microlensing events in the direction of the galac- tic bulge [ 14 ]. Such a large signal may be explained by the presence of a bar at the center of the Milky way ] 1.5 1. It also points towards a mass-to-light ratio larger than expected. and is the indirect signature ot an excess of low-mass stars in the bulge. The latter is believed to result from the coalcsconce of a primeval population of globular clustcra [ I6 ] The presence 01 faint low-mass stars in the remaining clusters is there- fore a relevant and still open question.

Globular clusters other a unique opportunity: mass segregation plays here a key role in separating light stars from their heavier companions. In Section 2, we have modelled the presence of both species with two- component King models. Since the dynamical mass is comparable to the visible mass, a persistent rumor has it that the mass of globular clusters is well deter- mined. In fact. the amount of low-mass stars turns out to be badly constrained from observations of the lumi- nous component. Only measurements of the velocity dispersion profile may unravel the presence of‘ darh

matter, up to the condition that the velocity ellipsoid of the heavy stars does not flatten away from the cen- ter of the cluster. Evaporation of both stellar species through two-body relaxation is shown to be inefficient. In Section 3, we investigate the various signatures of the underside of globular clusters. First, their infrared glow is estimated. The potential dominance of low- mass stars in the outer regions makes them detectable by the Infrared Space Observatory if they contribute more than 80% to the total mass. Brown-dwarves are detectable by the Space Infrared Telescope Facility if they dominate the cluster mass. Then, we compute the rate of gravitational microlensing of distant back- ground stars by the light species surrounding globular clusters. The optical depth of those low-mass objects may reach a few parts in 106. If globular clusters be- have indeed as icebergs, a dozen of such events are ex- pected each year in the direction of the galactic bulge. Finally, in Section 4, we discuss our results in the light of stellar counts and conclude.

2. Globular clusters as icebergs

Recent stellar counts in some galactic globular clus- ters point towards a sharp turn up of the mass func- tion below 0.4 MO [ 121. Even stellar counts per- formed by Paresce et al. with the Hubble space tele- scope could lead to a bipolar mass function depend- ing on the mass-to-lighi relationship of low-metallicity light stars. Therefore, globular clusters can be de- scribed with a good approximation by simple two- component self-gravitating King models where heavy stars of mass mt are mixed with lighter companions of mass m2.

The phase-space distribution for species i is given by the King function

(1)

where E = @(r) + c’/2 denotes the energy per unit mass and c, is the one-dimensional velocity disper- sion. This King function is merely a Maxwellian dis- tribution. It is truncated at some critical energy @, which corresponds precisely to the gravitational po- tential @,( r,) at the boundary of the cluster. Above Qil, stars have enough energy to escape from the system and he pulled away by the gravitational field of the

galaxy. The globular cluster as :I wjhole as well as the local velocity distribution are assumed to he spheri- cally symmetric. The mass density at distance r from the center may be derived by summing up the distri- bution function ( I ) over the velocities

may he readily cxpresscd w>ith the error function erl‘

Thcrcli~re, for each stellar spccica i. the variation of the density p, with respect to It:, central value is

p,,/pi, = F{(4J, ~ @):‘tr;}:‘F{4&hT;}. (5)

Here. the gravitational potential is conveniently set equal to (D(0) = 0 at the center. The overall mass distribution determines in turn the potential through the Poisson relation

M=337G{p,(r) -//,;I,)]. (6)

Thi\ equation ix supplcrncntcd hy the requirements that both 9 and (D’ vanish at the center of the clus- ter. Our two-component toy-model depends therefore on live parameters : the dispersion velocities m,. the central mass densities p(, and the potential @, at the tidal radius. Thermalisation results from two-body rc- laxation and implies that the velocity dispersions 01. the various stellar spccic\ arc related through

Throughout this article, the propcrtics of the heavy population. i.e., the most luminous stars, have been set equal to the typical values ~11 = I M .~for the stellar mass, CT1 = 7 km/s for the velocity dispersion and pC t = 8000 M,. /PC’ for the central mass density. WC take the remaining free parameters to be the density contrast A = pC-/pC i between light and massive stars at the center. and the ratio W = @,/~f of the depth of the cluster potential well to the velocity dispersion of

heavy stars. The mass of the lighter species has been set equal to m2 = 0.1 Mo, a value which corresponds to M subdwarves. In the case of our two-component model. the Poisson equation for the reduced potential 4 = @( r ) /gf may be written as

(8)

and must be solved numerically for each set of param- eters A and W. The ratio a:/~: = tnz/mt is denoted by B while the radius r = uz is expressed in units of the typical length

-4.3pc(&) (, ;,;pci)-“2. (9)

In our case. u E 0.34 pc. The radius r, of the globular cluster is determined by the requirement that

$(r=rr) =W (10)

The total masses in heavy and light stars Mt and A42 are integrated directly. Our discussion is based on the typical value Mt = 3 x lo’ M,, while M2 is varied from 0 (no light component) up to 9 x IO” Ma (dark matter dominates the globular cluster) in live models labeled from (a) to (e), for which the ratio M*/Mt is respectively 0, I, 3, IO and 30. The corresponding positions in the (A, W) plane are given in Table 1, together with the cluster radii.

The surface mass density of heavy stars

L‘,(r) = J

PI(.F)dS (11)

actually describes the distribution of light as seen on the tield of view. In Fig. 1, 2‘1 (v) is expressed in units of its central value XtC = 1.35 x IO4 MO pcp2 and is plotted as a function of the radius r, expressed in pc. The agreement between these various profiles is very good down to pi /pCt 2 lo-“, implying that low-mass objects do not affect the distribution of heavy visible stars. In the case of the globular cluster Ml 3, the struc- ture of which has been fitted with multi-component models [ 171, the very light stars only dominate the

90

Table I

Some properties of the two-component toy-model discussed in Sectton 2 are presented here. The light-to-heavy components mass ratio

Mz/Mt vartes from 0 t no dark matter) to 30 (the globular cluster IS dominated by low-mass stars). in five models labelled from (a) to (e). The parameters A and W are defncd in the text. The tidal radius I, is in pc while the overall density (Mt + Ml)/r: is expressed in

units of MI~pc -’ Should the heavy stars be virialized. their kinetic energy K would be half their apparent gravitational energy U,. Such

a Wuatton is approximately reached as a result of two-body thermal relaxation, even when the cluster is dominated by low-mass stars.

Finally. the evaporation timescales t i,AP of heavy and light 5tar.s arc expressed in units of 10’ yr

Model

(a) (b) CC) Cd) (e)

ttdal radiu\ I,

( M / + M: :,-,3

2K:‘UR ( thermalisation )

?K!Lt,? (no thermalisation)

tLb.,,, ( heavy stars )

t;up ( hght <tars)

I)

0

7 X23

60 7

I 34

I 04

I 04

I1 03

30

0.0101

24.219

171.8

1.83

1.55

41.4

9.42 x lo6

I.3 77 34.s5 142.9 711.s

I 3

0.024 I 0.0253

9.739 I I .996

67.4 80.4

I .Yh 2.31

I .32 I .S?

2.21 5.14

5s 04 293.7

10

0.0168

16.389

112.2

2.34

I.58

16.2

9493

Fig. I. The surface mass dcn\tty \ I of heavy stars, expressed tn units of its central value \I< = I.315 x IO’ M,~pc~‘, ts plotted a’;

a function of the radius t-. expressed tn pc The diagram features

the five models (a) to (e) dtscussed in Section 2.

mass at large radii and do not have much effect upon the cluster properties as traced by the giants. Because of thermalisation, heavy stars concentrate at the cores. Hence, they directly determine their own distribution while light objects populate mostly the periphery and have little influence on the central regions according to Gauss theorem. The apparent tidal radius ft of the

heavy species is therefore fairly constant. It differs signilicantly from the true tidal radius rt of the global cluster when h42 is large. From model (a) to (e), rt increases from 60.7 up to I7 I pc. Moreover, whatever MI, the average apparent density Mt/F; is approxi- mately the same as the overall density (Mt + M2) /r;. The magnitude of the latter is determined by the grav- itational field of the galaxy

(12)

where TC; is the galactocentric distance. Therefore, the reaction of a globular cluster to the galactic tidal field does not depend much on the amount M2 of hidden matter. Finally, let K denote the kinetic energy of mas- sive stars alone

i, J ‘3

K= 2p1 (r)6f4m2dr. (13)

0

while -l.& is their apparent gravitational energy inside the cluster

” = s

I’

‘GMdr)~dr)~~~2~~,

r i)

(14)

Those integrals run up to the point where the density contrast pl/pcl falls below, say, IO-“. As is clear in

91

Table I. the ratio ZK/lJ, varies from I (no dark mat- ter) up to I .6 (for M?/Mt = 30). The virial theo- rem is indeed approximately satisfied by the luminous component, irrespective of the amount of faint stars hidden in the cluster. This peculiar result holds as long as thermalisation is achieved, because the dark mat- tcr potential is rather flat in the central regions where the visible stars concentrate, and contributes little to their mechanical equilibrium. If equipartition of vc- locities, not of energies, is assumed, as is the case for the haloes of galaxies, that ratio varies from I up to - SO. In this case. cquipartition of velocities oh- tains by imposing ut = ~2, P~~/P~, = Mz/M, and by properly tuning W so that Ml = 3 x 10 M,::. Ther- malisation prevents therefore dark matter from being dynamically detected. The visible part of such a two- component cluster may be interpreted as if low-mass stars were absent. It would look approximately viri- alized, with an apparent tidal cut-off consistent with the gravitational field of the galaxy. Visible stars are not good tracers of the matter distribution in globular clusters.

However, the velocity dispersion ~1 of heavy stars exhibits different profiles as the dark mass M2 is vat- icd. At radius r. this one-dimensional velocity average depends on the potentional Q(r) through

iy- 2G{(@P, -m/u;1 CT; - 53{(@, -@,/CT;‘}’ (15)

where the function

may also be expressed with the error function erf

G(II) = ~vG~” erf( L;; ) $ V’G

I - ill i : 2 I 5’2 -5” (17)

The velocity I:t, averaged along the line-of-sight and expressed in km/s, is plotted in Fig. 2 against the ra- dius r, for the five models (a) to (e). Thermalisation tends to fatten the velocity dispersion profile of heavy stars. An accurate determination of this velocity dis- tribution would betray the presence of any low-mass component. The flatter the velocity curve, the larger M?. As a matter of fact, observations are spoilt by large errors so that M? can only be determined up to

Fig. 2. The line-of-sight velocity dispersion i;, of the visible component. expressed in km/s, is presented as a function of the radius r, for the live models (a) to (e).

a factor - 5. In the case of Ml3, velocities are mea- sured up to 3096, with an approximately flat distribu- tion allowing for a significant fraction of the cluster in the form of low-mass stars ] 171. Such velocity pro- tiles have not yet been measured for most of the clus- ters. Note that any flattening of the velocity ellipsoid far from the center would have the same effect on the velocity distribution 61 as the absence of dark matter. Future analysis should therefore deal with the fact that the outskirts of globular clusters tend to be populated by stars ejected from the center on elongated orbits.

WC have finally estimated the evaporation of both heavy and light stars. Thermalisation of test stars of mass m with field stars of mass m, tends to restore locally a Maxwellian distribution on a time-scale tR set by two-body encounters

t,‘(r) = 1 6,rr2G2 ~ In ii C(rn + m,)m,

3 LI x ,f;,(r,c =O). (18)

The distribution functions fO( r, u = 0) are given by expression ( I ) where E = e(r). A fraction

A(r) = 1 - 4 F{(@, - @)/u2}

J;;exp{(Or -@)/c+*} - 1 (19)

of the stars has a velocity larger than the limit set by the potential at the tidal radius and escape. The local rate of mass-loss is therefore

dM A(r) ----ZZ dt

----.A4 [R(r)

( 20 )

For each stellar component. that mass-loss is inte- grated all over the globular cluster to yield the escape rate hi, and the evaporation timescale Ml/&l,. Light stars always evaporate more efficiently than the heavy species. In our case. the evaporation time-scale of the low-mass population varies from I4 Gy (M*/Ml = I ) up to N 700 Gy (Mz/Mi = 30). The presence nowa- days of an important fraction of light objects in glob- ular clusters is therefore plausible. We also checked that the central relaxation time is a fraction only of the Hubhle time. so that thermalisation can occur.

3. Exploring the dark side of globular clusters

Because of thermalisation, low-mass objects do not have much effect on the inner dynamics of globular clusters. We discuss here specific observations which would signal their presence in large amounts.

Light stars are potential infrared emittors as a re- sult of their cold surfaces. They separate from their heavier companions and tend to populate the periphery of globular clusters. Should they be numerous, they would generate a characteristic infrared glow around clusters. This aura, swamped in the luminosity of mas- sive stars in the central regions, should be dominant at the outskirts. We have computed the infrared emis- sion of both massive and light stars embedded in the globular cluster whose structure has been discussed in the previous section. Two distinct situations have been considered as regards the low-mass objects. The latter were first assumed to be hydrogen-burning light stars. with surface temperature Tl = 3000 K and lumi- nosity L2 = 10e7 L,:; typical of M subdwarves [ 181. Then, the special case of brown dwarves was inves- tigated with cooler surface temperature T* = 1500 K and fainter luminosity t:! = 2.8 x IO-” L,,. In both situations, the surface temperature Tl = 5760 K and the luminosity LI = L,:; were assumed for the heavy species. The fluxes expected for the various popula- tions have been estimated in the wavelength band LW2 of the Infrared Space Observatory Camera (ISOCAM

( 19]), extending from 5 ( v,,,,,) to 8.5 (V”,in) microns. Finally, the various stellar emissions were assumed to follow a black-body spectrum, with intensity given by the Planck law

B,, = 2hv’ p{exp(g) -l}-‘, c2 (21)

where T denotes the surface temperature. The fraction 71~ of the total luminosity Lho, which a star radiates in the frequency range extending from v,,,, to v~,<,~ is therefore

(22)

where (T is Stefan constant. Located at distance r, this star yields at the earth the energy flux

bx~l ?'lR dIR = -- 477r2 Av' (23)

per unit of bandwidth Av = (v,,,, ~ v,,,i,,). The in- tensity of this signal is conveniently expressed in Jan- sky ( I Jy = 10ez3 erg s-’ Hz-’ cm-*). For a dis- tance r = 10 pc, solar-type objects produce a flux of = 5.3 /.~uJy while low-mass stars respectively yield an infrared emission of 30 nJy (M subdwarves with 1122 = 0.1 MC; ) and 4 nJy (brown dwarves with rn2 = 0.08 M,). The overall infrared glow obtains by sum- ming up the contribution of each individual star along the line-of-sight across the globular cluster. Each stel- lar species contributes a signal (per unit of solid an- gle)

Lb01 v/R S,R = __-

497 Au J' n(s)ds, (24)

the magnitude of which is proportional to its surface density. In Figs. 3 and 4, the infrared intensities ex- pressed in units of Jy/arcsec2 are presented as a func- tion of the radius r. Also shown are the detectability thresholds of the ISOCAM instrument on board the Infrared Space Observatory (ISO) and of the Space Infrared Telescope Facility (SIRTF [ 201). They cor- respond to a 3g detection limit, one hour of exposure time and a noise reduction over 100 pixels for IS0 and 400 pixels for SIRTF, so that they respectively reach a level of 50 and 16 nJy/arcsec*.

Fig. .i The Infrared etntrsion of low-tnas~ hydrogen-bumtng stat\

(M subdwarve~ with IQ = (1 I M 1 ts displayed for the four

models (b) to (e) discussed tn Section 2 (thin curves). The bold

line stands for the etnission of the heavy stars. Also plotted are

the detection thresholds for IS0 and SIKTF (see text).

I

Ftg. 4. Same curves a> in Fig 3. but with brown dwarve\

(WI? = 0.08 MC.: ) instead of light main-sequence stars.

For each stellar species, the column mass density, and therefore the magnitude of the associated luminos- ity, decreases outward. As a result of mass segregation, low-mass stars dominate the infrared emission at large radii, typically in our example for r > 20 pc for M sub- dwarves (Fig. 3) and r > 30 pc for brown dwarves (Fig. 4). The same trends appear in both plots: the larger the mass MT, the stronger the luminosity. The infrared glow of low-mass main-sequence stars is de- tcctable by IS0 and SIRTF as long as Mz/M, exceeds 3. In the case of model (c) for which M2/Ml = 3, it is swamped in the emission of heavy stars, but is still barely detectable by SIRTF at the very edge of the cluster. The situation is slightly less optimistic for brown-dwarves. Their signal could only be detected by IS0 in the extreme case where M2/M1 reaches 30 while SIRTF could detect them as long as that ra- tio exceeds 10. However, the next generation of in- frared telescopes should easily reach down a detection threshold of - I nJy/arcsec2, allowing for sensitive surveys of the infrared glow of globular clusters which could unravel the presence of any low-mass popula- tion (see for instance Bock et al. [21] ). If there are indeed many such stars, infrared images of globular clusters should extend much farther than pictures in the U or V bands. Cameras with an angular aperture of - a degree are particularly well suited in so far as the infrared glow in the vicinity of globular clusters is expected to vary smoothly on the entire field of view, allowing for a better rejection of the background.

The compact objects potentially hidden inside glob- ular clusters may also induce the gravitational mi- crolensing of distant stars. If a low-mass object inter- venes between the observer and a background star, the luminosity of the latter is enhanced [ 11 by the factor

(25)

The ratio ~/RE is denoted by u, where b is the impact parameter of the lens with respect to the line-of-sight, while RE is the radius of the Einstein ring which would appear, should the alignment be perfect

4Gm R;: = -

C-2 (26)

The distances to the deflector and to the remote source are respectively denoted by Dd and D,, while m stands

for the lens mass. If a background star is lensed with II < 1, its luminosity is enhanced by a factor of A > 1.34 and its magnitude is decreased by more than dmag = 2.5 log,,, A = 0.3. The optical depth for grav- itational microlensing is defined as the probability that a given background star undergoes such a magnitica- tion at any given time

I

7 7= TRFII ( s ) ds. (27)

The contribution of each stellar species along the linc- of-sight may be expressed as

i- (2x1

where 2, denotes the surface density. The total optical depth 7 towards gravitational microlensing obtains by adding up the contributions of heavy and light stars

where D = Dd ( D, - l?,, ) /D,. It is plotted in Fig. 5 as a function of radius r, for the same globular cluster as before. The position of M22 has been assumed here. with a distance of D,, = 3. I kpc while the galactic center. i.c.. the bulge of the Milky Way. plays the role of the background field with D, = 9 kpc. The optical depth dramatically incrcascs with the ratio M!/MI. owing to the presence of light species at the outskirts of the system. The globular cluster is furthermore a\- sumcd to bc dragged by the rotation of the galaxy. Its transverse velocity. with respect to the line-of-sight towards the galactic center. is ~‘1~ - 200 km/s. In- dividual stellar motions inside the cluster are indeed negligible in so far as ~1 is much smaller than I’ L. Therelhrc, the typical duration rc) of a gravitational magnification with lens mass 111 is

(30)

The square-root \/;;I of the deflector mass must hc avcragcd along the line-of-sight. In thecase ofour two- component model. it may be expressed as a function of the surface densities

Fig 5. The optical depth 7 for gravitational microlensing towards

the galactic center is featured for the five models (a) to (e)

discussed in section 2. The position of M22 has been assumed

here, with a distance to the deflectors of Dd = 3.1 kpc while the

remote background stars are in the bulge. with D, = 9 kpc.

( FFl

to yield

r() 1 35 days

(31)

(32)

This timescale to is featured in Fig. 6 against the radius r. It is 3.5 days for model (a) with heavy stars only. In all other cases, low-mass objects are associated to a smaller period of order I I days.

The probability that a given background field star undergoes a gravitational microlensing with A > I .34, per llF7it tinzr, is

I. = I

2REcln( s)ds. (33)

Because the deflectors have a transverse motion, the drift in time of their Einstein rings results into strips which fill up precisely a fraction f of the sky. Each stellar species contributes the amount

2 Ti I‘, = -- 77 to,

(34)

95

Fig. 6. The average duration I ,, ot the event\ i\ plotted agam\t Fig. 7. The probability I‘ that a given bulge star undergoes, per radius I Fw model (a 1. heavy \tar< are alone. hence a 1ypIcal

nmcscale of h unit time. a gravitational magnification with A > I .34 is presented

15 d. In all other Casey the duration IS smaller. as a function of radius r, expressed in arcdegrees. The five models with 10 - I I cl. (a) to (cl are featured.

to the overall microlcnslng rate

I‘ -” 2.8 x IO ‘) ( lodk;rl’s) (6) ‘I2

The variations of I’ with radius I- are presented in Fig.

7. A bulge star. located within I degree I’rom the clus-

ter center. has a non-vanishing chance to be magnified

if M,/M, exceeds -r 3. Ohservations will eventually

yield the total number of’ events, per year, occurring

in specific regions located in the vicinity of’ the glohu~

lar cluster. Note that background ticlds are extremely

rich in the direction of the ,@actic center. The limit 01

resolution i\ 1 400 stars per arcmin’. Assuming that

hackground fields in the direction ot’ M22 (/ = A IO-,

II = -7”30’) have a density of I00 stars per arcmin’.

we have estimated the amount /VP,. of events that ;I

microlensing experiment would obtain each year in-

side a circular field ot‘ view of angular aperture I’.

centered on the cluster. In Fig. 8, IV,). is plotted as a

function of r. expressed in arc dcprees. When heavy

stars arc alone. the microlcnsing signal is completely

negligible ( - 0.3 event per year). When M2jMI in-

.! ) I

-c

Fig. 8. The number Np~ of events which a microlensing experiment would obtain, each year, inside a circular field of view centered

on the cluster. is displayed as a function of the angular aperture r, for the five models (a) to (e). A density of 100 stars per arcmin’ is assumed for the background field.

creases, events due to lowmass compact objects don- inate. Their contribution to the microlensing signal of the whole cluster is

N,,, E 0.93 event 2 i J

The core of the cluster is fairly concentrated. There- fore. the fraction of the sky which cannot be monitored because it is too luminous is negligible. For a I0 angu- lar aperture. h I. I x IO6 bulge stars must be watched. For models (c) to (cl, N,+l. reaches respectively 3. 9 and 20 events per year. This signal must be com- pared to what is expected from the stars in the galactic disc itself. Using the Bahcall and Soncira model [ 22 I. Paczynski has obtained [ 231 an optical depth towards M22 of ‘T -v 2 x lo-‘. resulting into - 2 events per year for a bulk of lOh monitored stars. Note that if the saturation limit was reached for bulge stars, the con- tribution of the disc would be - 8 events per year. to bc compared to an additional 5 events induced by the low-mass stars potentially concealed in M22, for M2 = Mt The structure of M22 is actually quite sim- ilar to what has been assumed in this article (~1 = 9 km/s. ~~1 = 10’ M,, PC-’ and Mt = 5 x IO’ n;l. ). We conclude that a close follow-up of such a cluster would provide valuable informations on the amount of low-mass compact ob.jccts which it may hide.

4. Discussion and prospects

In this analysis, we have shown that numerous low- mass objects may be concealed inside globular clus- tcrs without affecting much their properties as traced by the heavy and visible stars. We are currently in- vestigating the more realistic case of a multi-mass model with a complete mass function. Our main con- clusions should nevertheless hold, even though the in- frared fluxes and microlensing rates presented above should slightly change. We conclude that the dark side of globular clusters cannot be properly studied on the basis of dynamical arguments only. Searches for the infrared aura of clusters as well as surveys of the grav- itational magnifications of remote background stars located on the line-of-sight of those systems offer, on the contrary, useful probes.

Here. we have adopted a brown-dwarf temperature of 1500 K. that should strictly correspond to a IO”-year

old, populationII,O.O8 Me star. It should be noted that this temperature is quite uncertain. First, it strongly depends on the stellar age, dropping from 2500 to 800 K between IO9 and IO” years, for a 0.08 Ma star [ 181. Therefore, the infrared signal of brown dwarves should be easier to detect in the case of a young clus- ter. Then, the flux radiated in a given wavelength band varies with the metallicity of the star. The spectrum of the latter has been claimed to deviate significantly from a blackbody emission for low-temperature and low-metallicity stellar atmospheres, conditions which are indeed relevant to population II brown dwarves [ 241. Note that our understanding of low-mass stars is not completely achieved. Such uncertainties should be kept in mind when analysing the results from stel- lar counts. Even though the luminosity functions of a few globular clusters have been determined by the Hubble space telescope with unprecedented accuracy, the proper mass-luminosity relationship for the stellar populations under study must be correctly determined before any conclusion is drawn as regards the mass functions of such objects. An alternative method for deriving the mass distribution of globular clusters has been recently proposed [ 25 1. It points towards a sharp rise of the mass function at low masses, an exciting conclusion in disagreement with the results from the stellar counts by Paresce et al. [ 131.

The dark component of globular clusters could in- duce a significant rate of gravitational amplifications. The interpretation of these microlensing events is much easier than in the case of the galactic halo, be- cause the distance to the lenses, i.e., to the cluster, is known accurately. The mass function should therefore be obtained directly from the timescale distribution of the events [ 261.

Finally, we find that the evaporation due to inter- actions between stars inside the cluster is negligible. However, another source of evaporation is the so- called disc-shocking mechanism that occurs each time the cluster passes through the galactic disc. We have not considered here this kind of evaporation, which can be neglected for remote systems. However, this effect could be important for clusters that are very close to the galactic center, as they cross more often the dense regions of the disc. Such an enhanced evaporation, or even disruption in some cases, could well account for the presence of many light stars in the spheroid. The exploration of the low-mass objects content of glob-

ular clusters will give precious int’ormations on the

structure of our whole galaxy and exciting clues on

the nature of its missing msss.

Acknowledgements

This work has been carried out under the auspces and with the linancial support of the Human Capi-

tal and Mobility Programme of the European Eco-

nomic Community, under contract number CHRX-

CT!450120 (DC; I2 COMA).

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