Halo Dark Clusters of Brown Dwarfs and Molecular Clouds

THE ASTROPHYSICAL JOURNAL, 500 :59È74, 1998 June 101998. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

HALO DARK CLUSTERS OF BROWN DWARFS AND MOLECULAR CLOUDS

F. G. PH. AND M.DE PAOLIS,1,2 INGROSSO,3 JETZER,2 RONCADELLI4,5Received 1996 November 26 ; accepted 1998 January 22

ABSTRACTThe discovery of massive astrophysical compact halo objects (MACHOs) in microlensing experiments

makes it compelling to understand their physical nature, as well as their formation mechanism. Withinthe present uncertainties, brown dwarfs are a viable candidate for MACHOs, and the present paperdeals with this option. According to a recently proposed scenario, brown dwarfs are clumped with coldmolecular clouds into dark clustersÈin several respects similar to globular clustersÈthat form in theouter part of the Galactic halo. Here we analyze the dynamics of these dark clusters and address thepossibility that a sizable fraction of MACHOs are binary brown dwarfs. We also point out that Lyaabsorption systems Ðt naturally within the present picture.Subject headings : dark matter È Galaxy : halo È Galaxy : kinematics and dynamics È

gravitational lensing È stars : low-mass, brown dwarfs

1. INTRODUCTION

Observations of microlensing events et al.(Alcock 1993 ;et al. toward the Large Magellanic CloudAubourg 1993)

(LMC) strongly suggest that a substantial fraction of theGalactic halo should be in the form of dark compactobjects, called MACHOs (massive astrophysical compacthalo objects) Ru� jula, Jetzer, & Masso(De 1992).

Actually, the MACHO collaboration has recentlyannounced the discovery of several new events during theirsecond year of observations et al. eight(Alcock 1997) ;microlensing events have been detected so far.6

Although the limited statistics presently available preventsus from drawing clear-cut conclusions from experimentaldata Gyuk, & Turner the evidence for such a(Gates, 1996),discovery is Ðrm and its implications are striking. In fact,under the assumption that MACHOs are indeed located inthe Galactic halo, the inferred halo mass in MACHOswithin 50 kpc turns out to be et2.0~0.7`1.2 ] 1011 M

_(Alcock

al. which is several times larger than the mass of all1997),known stellar components of the Galaxy and represents arelevant portion of the Galactic dark matter. Remarkablyenough, this result is almost independent of the assumedGalactic model. Unfortunately, this circumstance contrastswith the strong model dependence of the average MACHOmass. It has become customary to take the standard spher-ical halo model as a baseline for comparison. Regrettably,because of the low statistics, di†erent data analysis pro-cedures lead to results that are only marginally consistent.

1 Bartol Research Institute, University of Delaware, Newark, DE19716-4793.

2 Paul Scherrer Institut, Laboratory for Astrophysics, CH-5232 VilligenPSI, and Institute of Theoretical Physics, University of Zurich, Win-terthurerstrasse 190, CH-8057 Zurich, Switzerland.

3 Dipartimento di Fisica, Universita di Lecce, Via Arnesano, CP 193,73100 Lecce, Italy, and Instituto Nazionale di Fisica Nucleare, Sezione diLecce, Via Arnesano, CP 193, 73100 Lecce, Italy.

4 Instituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Bassi 6,I-27100, Pavia, Italy.

5 Work partially supported by Dipartimento di Fisica Nucleare eTeorica, Universita di Pavia, Pavia, Italy.

6 It should be mentioned that the MACHO team has found at leastseven more events (which are reported on the Alert list), but a full analysisof them has not yet been published.

SpeciÐcally, within the standard halo model, the averageMACHO mass reported by the MACHO team is 0.46~0.2`0.3

et al. whereas the mass moment methodM_

(Alcock 1997),Ru� jula, Jetzer, & Masso� yields 0.27(De 1991) M

_(Jetzer

1996).What can be reliably concluded from the existing data set

is that MACHOs should lie in the mass range 0.05È1.0 M_(see also Table 9 of et al. but stronger claimsAlcock 1997),

are unwarranted because of the high sensitivity of theaverage MACHO mass to the uncertain properties of theparticular Galactic model under consideration (Evans 1996 ;

Paolis, Ingrosso, & JetzerDe 1996).Mass values of suggest that MACHOs should[0.1 M

_be either M dwarfs or white dwarfs. Observe that thesemass values naturally arise within the standard halo model.

As a matter of fact, the M dwarf option can look prob-lematic upon deeper consideration. The null results ofseveral searches for low-mass stars both in the disk and inthe halo of our Galaxy et al. suggest that the halo(Hu 1994)cannot be mostly in the form of hydrogen-burning main-sequence M dwarfs. Optical imaging of high-latitude Ðeldstaken with the Wide Field Camera of the Hubble SpaceTelescope indicates that less than D6% of the halo mass canbe in this form et al. A more detailed analysis(Bahcall 1994).that accounts for the fact that halo stars are likely to have alower metallicity (with respect to solar), leads to an evenmore stringent upper limit of less than D1% &(Gra†Freese We emphasize that these results are derived1996).under the assumption of a smooth spatial distribution of Mdwarfs, and become less severe in the case of a clumpydistribution In the latter case, as pointed out(Kerins 1997a).by the dynamical limits and HST obser-Kerins (1997b),vations require that the overwhelming fraction of M dwarfs,at least 95%, must still reside in clusters at present. Hisanalysis shows that there exists a wide range of clustermasses and radii that are consistent with these require-ments.

As we said, an alternative explanation for MACHOs canbe provided within the standard spherical halo model bywhite dwarfs, and a scenario with white dwarfs as a majorconstituent of the Galactic halo dark matter has beenexplored et al. Mathews, &(Tamanaha 1990 ; Fields,Schramm & Laughlin Segre-1997 ; Adams 1996 ; Chabrier,tain, & Me� ra However, even this proposal encoun-1996).

59

60 DE PAOLIS ET AL. Vol. 500

ters difficulties. Apart from requiring a rather ad hoc initialmass function (IMF) of the progenitor stars, sharply peakedsomewhere in the range and a halo age larger than1È8 M

_,

D16 Gyr, strong constraints on the number density of halowhite dwarfs arise from present-day metal abundances inthe interstellar medium Olive, & Silk &(Ryu, 1990 ; GibsonMould and from deep Galaxy counts & Silk1997) (Charlot

In any case, future HST deep Ðeld exposures will1995).either Ðnd the white dwarfs or put constraints on their frac-tion in the halo (Kawaler 1996).

Mass values make brown an attractive[0.1 M_

dwarfs7candidate for In fact, these mass values areMACHOs.8supported by several nonstandard halo models. Oneexample is the maximal disk model Albada & Sancisi(van

& Salucci in which more1986 ; Persic 1990 ; Sackett 1997),matter is contained within the disk and the halo is lessmassive than in the standard halo model. The latter condi-tion implies a falling rotation curve, and so a smaller trans-verse velocity of MACHOs. Hence, the microlensingtimescale gets longer for a given MACHO mass, whichmeans a smaller implied MACHO mass for a givenobserved timescale. We stress that a reduced transversevelocity of MACHOs also arises in other nonstandard halomodels. Either a radially anisotropic velocity distributionor a halo rotation, for instance, would do the job et(Alcockal. Paolis et al. We also note1997 ; Evans 1996 ; De 1996).that the EROS collaboration et al. has(Renault 1997)shown that MACHOs in the mass range 10~7È2 ] 10~2

do not contribute signiÐcantly (less than 20%) to theM_halo dark matter (this result is consistent with the MACHO

experiment for objects of mass to0.1 M_

1 M_

).Although present uncertainties do not permit us to make

any sharp statement about the nature of MACHOs, browndwarfs still look like a viable possibility to date, and weshall stick to it throughout.

Even if MACHOs are indeed brown dwarfs, the problemnevertheless remains of explaining their formation, as wellas the nature of the remaining dark matter in galactic halos.

We have previously proposed a scenario in which darkclusters of brown dwarfs and cold molecular clouds, mainlyof naturally form in the halo at Galactocentric dis-H2,tances larger than 10È20 kpc (De Paolis et al. 1995a, 1995b,

Similar ideas have also been put forward by1995c, 1995d).& Silk Dark clusters of brown dwarfs haveGerhard (1996).

been extensively investigated by Ashman & Carr (Ashman& Carr A slightly di†erent picture,1988 ; Ashman 1990).based on the presence of a strong cooling Ñow phase duringthe formation of our Galaxy, has been considered byFabian & Nulsen and leads to a halo made of(1994, 1997)low-mass objects. In addition, Combes, & Marti-Pfenniger,net suggested that clouds may constitute the dark(1994) H2matter in the disk of our Galaxy.

The model in question encompasses the model Ðrst pro-posed by & Rees to explain the formation ofFall (1985)globular clusters, and no substantial additional hypothesisis required. Various resulting observational implicationshave also been addressed. In particular, (1) the c-ray Ñux

7 Brown dwarfs have been discovered quite recently in the solar neigh-borhood and in the Pleiades cluster et al. et al.(Rebolo 1995 ; Nakajima

The idea that MACHOs are brown dwarfs has been contemplated1995).by several authors (see and references therein).Carr 1994

8 We notice that the limit for hydrogen burning, usually quoted asincreases up to for low-metallicity objects, such as a0.08 M

_, 0.11 M

_halo population Hubbard, & Lunine(DÏAntona 1987 ; Burrows, 1989).

arising from halo molecular clouds through the interactionwith high-energy cosmic-ray protons has been estimated(De Paolis et al. (2) an anisotropy in the1995b, 1995c),cosmic background radiation (CBR) is predicted to show upwhen looking at the halo of the M31 galaxy Paolis et al.(De

and (3) the infrared emission from MACHOs1995a),located in the halo of the M31 galaxy should be observablewith the detector on the Infrared Space Observatory (ISO)or with the next generation of satellite-borne detectors (DePaolis et al. 1995a).

We would like to stress that a large proportion ofMACHOs (up to 50% in mass) may well consist of binarybrown dwarfs, formed either by the same fragmentationprocess that produces individual brown dwarfs or later,when the dark clusters start to undergo core collapse.

The aim of the present paper is to discuss in a systematicfashion further aspects of the above scenario. Basically, wetry to Ðgure out the dynamics of dark clusters. More speciÐ-cally, we investigate the constraints that ensure their sur-vival against various kinds of gravitational perturbations.We also demonstrate that because of dynamical friction onmolecular clouds in the dark cluster cores (to be referred toas frictional hardening), the present orbital radius of binarybrown dwarfs that are not too hard turns out to be typicallyof the order of the Einstein radius for microlensing towardthe LMC. As a consequence, and also taking into accountthe adopted selection procedure in the data analysis, weunderstand why they have not been resolved so far ; still, weargue that they can be resolved in future microlensingexperiments with a more accurate photometric observation.Finally, we show that Lya absorption systems naturally Ðtwithin our model.

The plan of the paper is as follows. In we recall the° 2main points of the considered picture of the formation ofdark clusters. Various dynamical constraints are thor-oughly analyzed in paying particular attention to the° 3,phenomenon of core collapse. In we study the process° 4whereby brown dwarfs form close binary systems (as a con-sequence of core collapse), and we investigate the mecha-nism of frictional hardening in great detail. In we turn° 5our attention to the thermal balance in halo molecularclouds. contains a short discussion of the rele-Section 6vance of Lya absorption systems for the present scenario.Our conclusions are o†ered in ° 7.

2. SCENARIO FOR DARK CLUSTER FORMATION

As shown elsewhere (De Paolis et al. the1995b, 1995c),model in question encompasses the model Ðrst consideredby Fall & Rees (1985) for the formation of globular

and relies on the conclusion of Salpeter, &clusters,9 Palla,Stahler hereafter that the lower bound on the(1983 ; PSS)Jeans mass in a collapsing metal-poor cloud can be as lowas provided that certain environmental condi-10~2 M

_,

tions are met.Let us begin by summarizing the ideas of & ReesFall

from the point of view that is most convenient for our(1985)considerations.

After the initial collapse, the protogalaxy (PG) is expectedto reach a quasiÈhydrostatic equilibrium state with a virial

9 A somewhat di†erent extension of the & Rees scheme hasFall (1985)been proposed by Among his motivations was the largeAshman (1990).spread in the age of globular clusters found by However, itLarson (1990).seems now that the stars in the halo of our Galaxy have a small scatter inages Wyse, & Gilmore(Unavane, 1996).

No. 1, 1998 HALO DARK CLUSTERS OF BROWN DWARFS AND CLOUDS 61

temperature D106 K. & Rees have shown thatFall (1985)in such a situation a thermal instability develops ; densityenhancements grow rapidly as the gas cools to lower tem-peratures. In fact, irregularities in the inÑow during the gascollapse and Ñuctuations in the distribution of nonbaryonicdark matter (if present on the Galactic scale) would intro-duce perturbations with a wide range of sizes and ampli-tudes. As a result, randomly distributed overdense regionswill form inside the PG. For reasons that will become clearlater, these overdense regions will be referred to as protoÈglobular cluster (PGC) clouds.

Under the assumption that the plasma in the PG is incollisional ionization equilibrium, it turns out that thecooling rate (as a function of density o and temperature T )has the form

"(o, T ) \ o2L (T ) (1)

(the expression of L (T ) can be found, e.g., in &EinaudiFerrara The cooling time is1991).

tcool \3okB T

2k("[ !), (2)

where the heating rate ! attributable to external heatingsources has been taken into account (here is thek ^ 1.22m

pmean molecular mass of the primordial gas). Since at thehigh temperatures under consideration the heating rate cansafely be neglected, it follows that On the othertcoolD o~1.hand, the free-fall time is

tffD (Go)~1@2 , (3)

so we see that decreases faster than as o increases. Astcool tffthe above quasiÈhydrostatic equilibrium state of the PG ischaracterized by the condition it is clear thattcoolD tff,inside the PGC clouds we have That is to say, thetcool \ tff.PGC clouds cool more rapidly than the rest of the PG. Thisprocess continues until hydrogen recombination occurs,because as soon as this happens, at a temperature of D104K, the cooling rate decreases precipitously, under the ion-ization equilibrium assumption & McCray(Dalgarno

Therefore, the regime should now be estab-1972). tcool[ tfflished even in the PGC clouds, so that the PG can beregarded at this stage as a two-phase medium, with coldPGC clouds in pressure equilibrium with the external(inter-PGC clouds) di†use hot gas.

However, et al. realized that the fast radi-Kang (1990)ative cooling of the PGC clouds (from 106 K to 104 K)implies that the plasma inside these clouds cools morerapidly than it recombines, so that the above ionizationequilibrium assumption is violated. Actually, the out-of-equilibrium recombination results in an enhanced ioniza-tion fraction. This fact does not a†ect the previous conclu-sions for temperatures [104 K, but entails drastic changesat lower temperatures. Indeed, the existence of a sizablenumber of protons and electrons at temperatures \104 Kgives rise to formation via theH2 reactions10

H ] p ] H2`] c , H] e] H~] c (4)

and

H2`] H ] H2] p , H] H~] H2] e . (5)

10 We wish to emphasize that the familiar reaction H ] H ] H2] c(which requires grains in order to be efficient) is presently irrelevant, sinceno dust exists in the metal-poor PGC clouds.

As a consequence, the PGC clouds undergo a furthercooling below 104 K. SpeciÐcally, there is a direct radiativecooling via reactions and a radiative cooling via(eq. [4])excitation of rotovibrational transitions of (observe thatH2is further produced by the reactions in eqs. andH2 [5] [6]below). We stress that the latter process is very e†ective(much more than the former) at temperatures \104 K andplays a crucial role in our considerations. Because now

in the PGC clouds, the collapse goes on and thetcool > tffPGC cloud density rises steadily. When the number densityin the PGC clouds exceeds 108 cm~3, the productionH2increases dramatically thanks to the three-body reactions

H ] H ] H ] H2] H , H ] H ] H2] H2] H2 , (6)

as pointed out by In e†ect, these reactions are soPSS.efficient that virtually all the atomic hydrogen is convertedrapidly to Correspondingly, the cooling of the PGCH2.11clouds is strongly enhanced and their evolution can proceedas in the scenario proposed by PSS.

Still, it goes without saying that can be dissociated byH2various sources of ultraviolet (UV) radiation, such as anactive galactic nucleus (AGN) or a population of massiveyoung stars (Population III ; see Bond, & ArnettCarr, 1984)at the center of the PG. So, the ultimate fate of the PGCclouds strongly depends on (apart from other environ-mental conditions, which will be discussed later) the sur-vival of H2.In the early phase of the PG, an AGN is expected to format its center, along with Population III stars, through thedisruption of central PGC clouds. This indeed happens,since the cloud collision time is shorter than the coolingtime in the central region of the PG.

Thus, will be dissociated at Galactocentric distancesH2smaller than a certain critical value Following theRcrit.analysis of et al. it is straightforward to evalu-Kang (1990),ate Consider Ðrst the case of a UV Ñux arising from aRcrit.central AGN. Then we Ðnd

RcritAGNÂ L AGN2 ] 1042 ergs s~1

B1@2kpc . (7)

For typical luminosities up to ergs s~1,L AGN^ 1045yields kpc. On the other hand, whenequation (7) RcritAGN^ 20

the UV dissociating Ñux is produced by massive young starsmainly located at the center of the PG, the critical Galacto-centric distance turns out to be

Rcrit* ^10~3 kpc~3

n0

L totL*

kpc . (8)

In ergs s~1 is the bolometricequation (8), L*

^ 2 ] 1038luminosity of a single B0 V star. Assuming a total stellarluminosity up to ^2 ] 1045 ergs s~1 and a centralL totnumber density up to ^103 kpc~3, we Ðndn0 Rcrit* ^ 10kpc. In conclusion, should remain undissociated atH2Galactocentric distances larger than 10È20 kpc.

2.1. Globular ClustersAccording to the preceding analysis, in the inner Galactic

halo (that is, for Galactocentric distances smaller than

11 Observe that, di†ering from the reactions given in equations and(4)the reactions given in do not require the presence of(5), equation (6)

electrons and protons as a catalyst.


10È20 kpc), gets thus preventing anyH2 dissociated,12further cooling of the PGC clouds below T D 104 K. There-fore, these clouds remain for a long time in quasi-hydrostatic equilibrium at T D 104 K (namely, we have

during this period). In such a situation a charac-tcool [ tffteristic mass scale is imprinted on the PGC clouds by thegravitational instability, thereby resulting in a stronglypeaked mass spectrum of the PGC clouds. In fact, &FallRees have shown that the Jeans mass of the PGC(1985)clouds after the long permanence at T D 104 K is given (asa function of the Galactocentric distance R) by 13

MPGC(R) ^ 5 ] 105A RkpcB1@2

M_

, (9)

while the PGC radius turns out to be

rPGC(R) ^ 20A RkpcB1@2

pc . (10)

Observe that in the present case, the propagation ofsound waves erases all large-scale perturbations, leavingonly those at small scales.

Surely this is not the end of the story, since the UV Ñux isexpected to eventually decrease. So, after some time a non-trivial fraction of should form in any case, causing aH2sudden drop of the PGC cloud temperature to well belowD104 K (the clouds now enter the regime Whattcool \ tff).happens next is a rapid growth of the small-scale pertur-bations, which lead directly (in one step), because of thethermal instability, to the formation of stars inside the PGCclouds & Lin It goes without saying that we(Murray 1989).expect supernova explosions to have occurred, thus givingrise to shocks inside globular clusters and leading to arather wide stellar IMF. In this way, globular clustersshould form as they are observed today, especially in theinner part of the Galactic halo. Incidentally, the formationof the PGC clouds could have been delayed until after theGalaxy was enriched by metals through Population IIIstars, to explain the absence of globular clusters with pri-mordial metal abundances and the radial gradient of metal-licity in the Galactic halo.

2.2. Dark Clusters of MACHOs and Cold Molecular CloudsAs we have seen, in the outer Galactic halo (namely, for

Galactocentric distances larger than 10È20 kpc), mol-H2ecules are not dissociated, owing to the absence of a signiÐ-cant UV Ñux. Under the assumption of a quietenvironment, this circumstance entails a very efficientcooling of the PGC clouds, whose state is therefore charac-terized by the condition Hence, the gravitationaltcool\ tff.collapse is expected to occur as in the scenario of PSS.SpeciÐcally, both the temperature decrease and the densityincrease cause a substantial drop of the Jeans mass for thePGC clouds. As a consequence, they fragment into smallerand smaller clouds. This process stops when the cloudsbecome optically thick to their own radiation, since coolingthen becomes manifestly ine†ective. have shown thatPSSsuch a situation occurs when the Jeans mass is as low as

12 This indeed occurs for a wide range of UV Ñuxes and PGC clouddensities et al.(Kang 1990).

13 We stress that the R dependence in equations and holds for(9) (10)R\ 20 kpc only (see the discussion in & Pesce while forVietri 1995),R[ 20 kpc the mass of globular clusters tends rather to decrease.

We remark that the presence of rotation and10~2 M_

.magnetic Ðelds in the PGC clouds would allow the Jeansmass to drop even further. Obviously, fragmentation downto low-mass values is favored if the initial gas has beenmetal enriched by Population III stars (see Figure 7 of Palla& Stahler 1988).

The result of the above scenario is the formation of darkclusters containing compact objects with an IMF peaked inthe range This value is close to the peak10~2È10~1 M

_.

mass of the present-day IMF in the diskD0.1 M_

(Miller& Scalo but in our case we expect it to decrease much1979),more rapidly for higher masses and thus to be more narrowaround its mean value. So, the compact objects in questionshould be predominantly brown dwarfs, although a fractionof M dwarfs could be present. Actually, we want to stressthat our considerations still hold true even if a large fractionof MACHOs consists of M (as already mentioned,dwarfs14

has shown that M dwarfs clumped intoKerins [1997b]clusters are a viable possibility). Notice that the expectedlower limit of is consistent with present boundsD10~2 M

_derived from microlensing data, as found by the EROScollaboration et al.(Renault 1997).

What are the environmental conditions (besides H2survival) that permit the mechanism in question to work?First, turbulence should be negligible. Indeed, turbulencee†ects would make fragments collide, which, as shown by

would increase the Jeans mass. Second, no sizablePSS,gravitational perturbations (shocks, supernova explosions,etc.) should be since otherwise they would inducepresent,15gravitational collapse before cooling has succeeded inlowering the fragment Jeans mass down to the above-mentioned values. Because these environmental conditionsare likely to have occurred in the outer halo, the mass of theproduced compact objects is expected to be close to thecorresponding Jeans mass.

In addition to individual brown dwarfs, it seems quitenatural to suppose that in this case, in much the same wayas happens for ordinary stars, the fragmentation processshould also produce a substantial fraction of binary browndwarfs. For reasons that will become clear, these will bereferred to as primordial binaries (we shall come back to thisissue in the next sections). It is important to keep in mindthat the mass fraction of primordial binaries can be as largeas 50% & Mathieu So, we are led to the(Spitzer 1980).conclusion that MACHOs consist of both individual andbinary brown dwarfs in this scenario.

However, we do not expect the fragmentation process tobe able to convert all the gas in a PGC cloud into browndwarfs. For instance, standard stellar formation mecha-nisms lead to an upper limit of at most 40% for the conver-sion efficiency Therefore, a fairly large amount(Scalo 1985).of gas (mostly should have been left over. What is itsH2)fate? At variance with the case of globular clusters, strongstellar winds are now manifestly absent, so this gas shouldremain gravitationally bound in the dark clusters. This con-clusion is further supported by the following arguments(provided that dark clusters comprise a consistent fraction

14 The reader should keep this point in mind, in spite of our focus onbrown dwarfs. Of course, quantitative changes in some results will occur,because M dwarfs are more massive than brown dwarfs (this is especiallytrue concerning discussions in °° and3 4).

15 This circumstance is consistent with our assumption that the stellarIMF in the dark clusters is more sharply peaked than the IMF in theglobular clusters and in the disk.


of the Galactic dark matter). First, the gas cannot havedi†used into the whole halo, for otherwise it would havebeen heated by the gravitational Ðeld to a virial tem-perature D106 K, thereby becoming observable in theX-ray band ; this option is ruled out by the available upperlimits & Lockman Second, the alternative(Dickey 1990).16possibility that the gas wholly collapsed into the disk is alsoexcluded, because the disk mass would then be of the orderof the inferred dark halo mass. Now, the virial theorementails that the temperature of a di†use gas componentTDGinside a dark cluster is

TDG^ 1.1AMDC

M_

B2@3K . (11)

Accordingly, a large fraction of di†use gas would presum-ably give rise to an unobserved radio emission. Thus, weconclude that the amount of virialized di†use gas inside adark cluster must be low (it will henceforth be neglected).This circumstance implies in turn that most of the leftovergas should be in the form of self-gravitating clouds clumpedin the dark clusters (since in this case the virial theoremapplies to individual clouds). As we shall see, there are goodreasons to believe that the central temperature of theT

mmolecular clouds in question should be very low, in factclose to that of the CBR. Accordingly, the molecular cloudmass and median radius are related by the virialM

mrmtheorem as

rm

^ 4.8] 10~2 Mm

M_

pc . (12)

Presumably, the fraction of cluster dark matter in the formof molecular clouds should be a function of the Galactocen-tric distance R, depending on environmental conditionssuch as the UV Ñux and the collision rate for the PGCclouds.

Before proceeding further, an important issue should beaddressed. Given the supposed existence of a large amountof gas in the dark clusters, one would expect a star forma-tion process to be presently operative. However, things arenot so simple. Under the above environmental conditions,only stars of mass smaller than the cloud mass can beformed. Evidently, these stars are either brown dwarfs or Mdwarfs. So, we see that undetected bright stars should notform in the dark clusters to the extent that our assumptionshold true. One might also wonder whether a sizable quan-tity of gas is eventually left over. As already pointed out, weargue that this should be the case, since otherwise it wouldmean that the brown or M dwarf formation mechanismwould be much more efficient than any known star forma-tion mechanism. Moreover, & Silk haveGerhard (1996)shown that the cluster gravitational Ðeld can stabilize theclouds against collapse.

Unfortunately, the lack of any observational informationabout dark clusters would make any e†ort to understandtheir structure and dynamics hopeless, were it not for someremarkable insights that our uniÐed treatment of globularand dark clusters provides us.

In the Ðrst place, it seems quite natural to assume thatdark clusters have a denser core surrounded by an extendedspherical halo. For simplicity, we suppose throughout that

16 Incidentally, the same argument also rules out a halo primarily madeof unclustered brown dwarfs (as well as white and M dwarfs).

the core density proÐle can be taken as constant. Moreover,it seems reasonable to imagine (at least tentatively) thatdark clusters have the same average mass density as globu-lar clusters. Hence, we obtain

rDC ^ 0.12AMDC

M_

B1@3pc , (13)

where and denote the mass and the median radiusMDC rDCof a dark cluster, respectively. In addition, dark clusters(just like globular clusters) presumably stay for a long timein a quasi-stationary phase, with an average central density

slightly lower than pc~3 (which is theo*(0) 104 M

_observed average central density for globular clusters).As a further implication of the present model, we stress

that, at variance with the case of globular clusters, the massspectrum of the dark clusters should be smooth, since themonotonic decrease of the PGC cloud temperature fails tosingle out any particular mass scale. As will be shown in ° 3,dark clusters in the mass range 3 ] 102 M

_[MDC[ 106

should survive all disruptive e†ects, so we restrict ourM_attention to such a mass range throughout.As far as dark clusters are concerned, we have seen that

the brown-dwarf mass is expected to lie in the rangeFor deÐniteness (and with an eye to micro-10~2È10~1 M

_.

lensing results), we imagine that all individual brown dwarfshave the same mass, So, binary brown dwarfsm^ 0.1 M

_.

are twice as heavy. As a consequence, the mass stratiÐcationinstability will drive them into the dark(Spitzer 1969)cluster cores, which then tend to be composed chieÑy ofbinaries. Furthermore, an average MACHO mass some-what larger than can naturally be accounted for.^0.1 M

_Finally, let us consider molecular clouds. Since they alsooriginate from the above-mentioned fragmentation process,we suppose (for deÐniteness) that they lie in the mass range

Correspondingly,10~3 M_

[ Mm

[ 10~1 M_

. equationentails 4.8] 10~5 pc and(12) pc[ r

m[ 4.8] 10~3

2.7] 1010 cm~3, respectively,cm~3Z nm

Z 2.7 ] 106where denotes the number density in the clouds.n

mBefore closing this section, some comments are in order.There is little doubt that the foregoing considerations arequalitative in nature. Nevertheless, they provide nontrivialinsights into several questions that arise in connection withthe discovery of MACHOs. SpeciÐcally, a sharply peakedIMF in the range comes out naturally,10~2È10~1 M

_without having to invoke any new physical process. Thisexplains the observed absence of a substantial quantity ofmain-sequence stars inside the dark clusters. Therefore, theobserved absence of a large number of planetlike objects inthe halo et al. is automatically explained.(Renault 1997)Furthermore, we can understand why brown dwarfsclumped into dark clusters form copiously in the outer halo,but not in the inner halo or in the disk. Indeed, the di†erentstellar content of these regions is here traced back to thedi†erent environments in which the same star formationmechanism operates. It goes without saying that variousissues addressed above require further investigations.

3. DYNAMICAL CONSTRAINTS ON DARK CLUSTERS

As we have seen, MACHOs are clumped into dark clus-ters when they form in the outer Galactic halo. Still, thefurther fate of these clusters is quite unclear. They mighteither evaporate or drift toward the Galactic center. In thelatter case, encounters with globular clusters might have


dramatic observational consequences, and dynamical fric-tion could drive too many MACHOs into the Galacticbulge. So, even if dark clusters are unseen, nontrivial con-straints on their characteristic parameters arise from theobserved properties of our Galaxy. Moreover, in order toplay any role as a candidate for dark matter, MACHOsmust have survived until the present in the outer part of theGalactic halo. Finally, it is important to know whetherMACHOs are still clumped into clusters today, especiallybecause an improvement in the statistics of microlensingobservations permits us to test this possibility (Maoz 1994 ;

& SilkMetcalf 1996).We remark that previous work on dynamical constraints

on clusters of brown dwarfs & Lacey(Carr 1987 ; Carr 1994 ;& Salpeter & CarrWasserman 1994 ; Kerins 1994 ; Moore

& Silk & Silk rests on1995 ; Gerhard 1996 ; Kerins 1997a)the hypothesis of an initial dark cluster distribution thatextends inside the inner part of the Galaxy all the way downto the center. Hence, a novel ab initio analysis is requiredwithin the present scenario.

We begin with the usual assumption that the halo darkmatter density is modeled by an isothermal with asphere17density proÐle

o(R)\ o0a2] R02a2] R2 , (14)

where a ^ 5 kpc is the halo core radius, kpc is ourR0^ 8.5Galactocentric distance, and g cm~3o0 ^ 6.5 ] 10~25denotes the local dark matter density, corresponding to theone-dimensional velocity dispersion p ^ 155 km s~1 of thehalo. Furthermore, we shall suppose for deÐniteness thatthe age of the universe is yr. Our treatment oft0^ 1010encounters rests upon the di†usion approximation, and wefollow rather closely the analysis of & TremaineBinney(1987).

3.1. Dynamical FrictionDark clusters are subject to dynamical friction as they

orbit through the Galaxy, which makes them loose energyand therefore spiral in toward the Galactic center.Assuming (for illustrative purposes) that a dark clustermoves with velocity v on a circular orbit of radius R, thedrag brought about by the background density o(R) is givenby

F(R) \ [0.434nG2MDC2 ln "

v2 o(R) , (15)

where ln" is the usual Coulomb logarithm, the value ofwhich in the present case is

ln "^ lnARtyp v2GMDC

B^ 24.3[ ln

AMDCM

_

B, (16)

where kpc and v^ (2)1@2p. UsingRtyp D 20 equation (14),becomesequation (15)

F(R)^ [5.1] 1014 ln "AMDC

M_

B2 11 ] R52

g cm s~2 , (17)

17 As mentioned in this model may not provide the best description° 1,of the Galaxy. However, it will be used in the present section, since other-wise the ensuing discussion would become exceedingly complicated. Fur-thermore, it should be kept in mind that the various uncertainties a†ectingthe dark cluster properties in any case make the following results reliableonly as order-of-magnitude estimates.

where is in units of 5 kpc. Accordingly, the equations ofR5motion entail that a dark cluster originally at Galactocen-tric distance R will be closer to the Galactic center today bythe amount

*R(R) ^ 1.85] 10~8 MDCM

_

24.3[ ln (MDC/M_)

R5] R5~1 kpc .

(18)

Keeping in mind that in our model R[ 10È20 kpc and(see later discussion), we see thatMDC[ 106 M

_*R[ 5.8

] 10~2 kpc. Therefore, dark clusters are still conÐned to theouter Galactic halo. As a consequence, encounters betweendark and globular clusters as well as disk and bulge shock-ing of dark clusters are dynamically irrelevant, as long asthey move on not too highly elongated orbits (in this way,an e†ective circularization of the orbits is achieved).

3.2. Encounters between Dark ClustersEncounters between dark clusters may, under the circum-

stances to be analyzed below, lead to their disruption. Fororientation, we note that an estimate of the one-dimensional velocity dispersion of MACHOs and molec-p

*ular clouds within a dark cluster is supplied by the virialtheorem and reads

p*

^ 6.9] 10~2AMDC

M_

B1@3km s~1 , (19)

where has been used. Because in the presentequation (13)scenario (see later discussion), we getMDC [ 106 M

_p*

[

6.9 km s~1. Therefore, the one-dimensional velocity disper-sion of dark clusters, which we naturally suppose to be justp, is much larger than Hence, it makes sense to workp

*.

within the impulse approximation, whose range of validityis established more precisely by the condition MDC> 1010

which is evidently always met in our model.M_

,In order to proceed further, we introduce *E as the

change of the internal energy of a dark cluster in a singleencounter. Then (following & Tremaine weBinney 1987)Ðnd that encounters with impact parameter b in the range

increase the clusterÏs energy at the ratebmin¹ b ¹ bmax

E0 (R) ^ JnnDC(R)

p3P0

=dv v3e~v2@4p2

Pbmin

bmaxdb b*E , (20)

where v and are the cluster velocity and numbernDC(R)density (in the halo), respectively. We let c stand for thefraction of halo dark matter in the form of dark clusters, sothat we have with o(R) given bynDC(R) \ co(R)/MDC,

Accordingly, becomesequation (14). equation (20)

E0 (R) ^1

2Jnc

pGMDC

1R2] a2

]P0

=dvv3e~v2@4p2

Pbmin

bmaxdbb *E . (21)

Now, a natural deÐnition of the time required by encoun-ters to dissolve a cluster is provided by t

d(R) \ Ebind/E0 (R),

where the binding energy is expressed in terms of theEbindcluster properties as Ebind^ 0.2GMDC2 /rDC.3.2.1. Distant Encounters

Let us Ðrst consider distant encounters. Correspondingly,*E is to be evaluated in the tidal approximation (Spitzer


and presently reads1987)

*E^4G2MDC3 rDC2

3b4v2 . (22)

We insert this quantity into neglecting theequation (21),term in the ensuing expression. Experience with(bmin/bmax)2a similar treatment for globular clusters suggests that wechoose Correspondingly, we Ðndbmin^ rDC.

td(R) ^

4.7] 1011c

AM_

MDC

B1@3(R52] 1) yr , (23)

where use has been made of Assumingequation (13).R[ 10È20 kpc and keeping in mind that c¹ 1 and MDC[

(see later discussion), we get that for all the dark106 M_clusters in question, exceeds the age of the universe.t

d(R)

3.2.2. Close Encounters

In order to deal with close encounters, dark clusters mustbe regarded as extended objects. As in the case of globularclusters, this task is most simply accomplished by modelingthe dark clusters by means of a Plummer potential withcore radius a. Correspondingly, *E is found to be

*E^G2MDC33a2v2 . (24)

As before, we insert this quantity into equation (21),assuming now and In this way, webmin^ 0 bmax ^ rDC.obtain

td(R) ^

2.4] 1011c

A arDC

B2 pcrDC

(R52] 1) yr . (25)

We proceed by recalling that

a \C 3MDC4no

*(0)D1@3

, (26)

where denotes the dark cluster mass density. Takingo*(r)

pc~3 (as for globular clusters today) ando*(0)^ 104 M

_using we can rewrite asequation (13), equation (25)

td(R) ^

1011cAM

_MDC

B1@3(R52] 1) yr . (27)

Assuming R[ 10È20 kpc and remembering that c¹ 1, weare led to the conclusion that dark clusters are not disruptedby close encounters, provided that MDC[ 106 M

_.18

3.3. EvaporationVarious dynamical e†ects conspire to make dark clusters

evaporate within a Ðnite time. Relaxation via gravitationaltwo-body encounters leads to the escape of MACHOsapproaching the unbounded tail of the cluster velocity dis-tribution. Tidal truncation by the Galactic gravitationalÐeld enhances this process. A more substantial e†ect iscaused by the gravothermal instability, when the inner partof the dark clusters contracts (core collapse) and theenvelope expands.

Below, we shall address these issues separately. As is wellknown, a key element in such analyses is the relaxation

18 It should hardly come as a surprise that close encounters yield amore stringent bound on than distant encounters.MDC

time,

trelax(r) \ 0.34p*3

G2mo*(r) ln (0.4N)

, (28)

where N is the number of MACHOs per cluster. As in thecase of globular clusters, is expected to vary by variouso

*(r)

orders of magnitude in di†erent regions of a single darkcluster ; this dependence obviously shows up in trelax(r).Therefore, for reference purposes, it is often more conve-nient to characterize a dark cluster by a single value of therelaxation time. This goal is achieved by introducing themedian relaxation time & Hart(Spitzer 1971),

trh\ 6.5] 108ln (0.4N)

A MDC105 M

_

B1@2 M_

mArDC

pcB3@2

yr . (29)

Explicitly, using together withequation (13) equation (19)and takes them^ 0.1 M

_, equation (28) form19

trelax(r) ^ 5 ] 107 MDCM

_

CM_

pc~3o*(r)

D 11.4] ln (MDC/M_

)yr .

(30)

In the same fashion, becomesequation (29)

trh^ 8 ] 105 MDCM

_

11.4] ln (MDC/M_

)yr . (31)

3.3.1. Spontaneous Evaporation

As is well known, any stellar association evaporateswithin a Ðnite time as a result of relaxationspontaneously20

via gravitational two-body encounters. SpeciÐcally, a singleclose encounter between two MACHOs can leave one ofthem with a speed larger than the local escape velocity. So,the MACHO under consideration gets ejected from thedark cluster. We Ðnd for the ejection time (He� non 1969)

tej ^ 1.1] 103 ln (0.4N) trh^ 9 ] 108 MDCM

_

yr . (32)

Alternatively, several more distant, weaker encounters cangradually increase the energy of a given MACHO until afurther weak encounter is sufficient to make it escape fromthe cluster. In this case, the evaporation time turns out to be

& Thuan(Spitzer 1972)

tevap ^ 300trh^ 2.4] 108 MDCM

_

11.4] ln (MDC/M_

)yr .

(33)

Since is in any case longer than we focus our atten-tej tevap,tion on the latter quantity. By demanding that shouldtevapexceed the age of the universe, we conclude that dark clus-ters with are not yet evaporated.MDCZ 3 ] 102 M

_3.3.2. T idal Perturbations

Dark clusters, just like globular clusters, are tidally dis-rupted by the Galactic gravitational Ðeld unless isrDC

19 For simplicity, we here neglect the fact that binaries have masseslarger than individual brown dwarfs. Furthermore, given the logarithmicN-dependence, we can safely take N D MDC/mD 10(MDC/M_

).20 We use this terminology for the evaporation process that is neither

induced by external perturbations nor speciÐc to the gravitational inter-actions.


smaller than their tidal radius. So, the survival conditionreads

rDC \A MDC3M

G(R)B1@3

R , (34)

where R should be understood here as the perigalactic dis-tance of the dark cluster and denotes the GalaxyM

G(R)

mass inside R. From we obtainequation (14),

MG(R)^ 5.5] 1010R5(1 [ R5~1 arctan R5) M

_, (35)

and from equation becomes(13), equation (34)

R5(1[ R5~1 arctan R5)~1@2 [ 0.047 , (36)

which is always satisÐed for R[ 10È20 kpc. Thus, the darkclusters under consideration are not tidally disrupted by theGalactic gravitational Ðeld.

3.3.3. Core Collapse

Core collapse plays an important role in the consider-ations that follow. It is by now well established that theinitial stage of this process is triggered by evaporation,which leads to a shrinking of the core as a consequence ofenergy conservation. Numerical Fokker-Planck studies ofthe early phase of core collapse have shown that thedynamics of the cluster is correctly described by a sequenceof King models However, once the cluster(Cohn 1980).density reaches a certain critical value, core collapse is dra-matically accelerated by the gravothermal instability

& Wood Indeed, the(Antonov 1962 ; Lynden-Bell 1968).negative speciÐc heat of the core implies that the internalvelocity dispersion increases, thereby enhancing evapo-p

*ration, as the average kinetic energy decreases throughevaporation itself. Moreover, the unbalanced gravitationalenergy makes the core contract, so its density rises byseveral orders of magnitude in a runaway manner. Numeri-cal simulations show that the central velocity dispersionand the number of stars in the core, scale asN

*,

p*

D o*(0)0.05 (37)

N*

D o*(0)~0.36 , (38)

respectively & Tremaine Incidentally, the(Binney 1987).somewhat surprising slow rise of in is duep

*equation (37)

to the large mass loss from the core, as follows fromequation (38).

When does the gravothermal instability show up? Unfor-tunately, a clear-cut answer does not exist, since the corre-sponding time depends on how clusters are modeled astGIwell as on their concentration Manifestly,(Quinlan 1996).the lack of observational data for dark clusters makes aprecise determination of impossible. The best we can dotGIis to suppose that dark clusters behave like globular clustersas far as core collapse is concerned. In this way, we are ledto the order-of-magnitude estimate & Tremaine(Binney1987)

tGI^ 3trh ^ 2.4] 106 MDCM

_

11.4] ln (MDC/M_

)yr . (39)

Comparing with the age of the universe, we concludetGIthat dark clusters with are expected toMDC[ 5 ] 104 M_have started core collapse.

As we have said, the central density grows dramaticallyduring the second stage of core collapse, so the central

relaxation time gets shorter and shorter. Detailed studies ofthe gravothermal instability have shown that if nothingopposes the collapse, the time needed to complete core col-lapse starting from an arbitrary time t proceeds astcollwith the latter quantity computed for the particulartrelax(0),value taken by at t. Computer simulations of globularo

*(0)

cluster dynamics entail tcoll^ 330trelax(0) (Cohn 1980).Because of the huge increase in the central density, close

two-body encounters lead to the formation of bound binarysystems by converting enough kinetic energy into internalenergy (tidal capture). As we will see in binary brown° 4,dwarfs are produced in this way in the dark cluster coresduring the early phase of core collapse. These binaries,which will be referred to as tidally captured binaries, andhappen to be very hard, play a crucial role in this context,since they ultimately stop and reverse the collapse. Sche-matically, the argument goes as follows. Because hardbinaries necessarily get harder in collisions with individualstars the internal binding energy released by(Heggie 1975),a binary is transformed into kinetic energy of both the starand the binary. Actually, the exchanged energy is so largethat they both leave the cluster. However, as is not the casewith evaporation, the kinetic energy (per unit mass) of thecluster is una†ected, while mass ejection obviously increasesthe potential energy. That is, the binding energy given up bythe binaries ultimately becomes gravitational energy of thecore. As a result of the unbalanced kinetic energy, the corestarts expanding. Moreover, because of the negative speciÐcheat, the increased potential energy makes decrease,p

*thereby slowing down mass ejection. In this manner, corecollapse is halted and reversed (Spitzer 1987).

As a matter of fact, the presence of binaries in appreciablequantities can also modify to some extent the standard sce-nario of core collapse as outlined above. Indeed, numericalFokker-Planck simulations have shown that in this case thecollapse is also driven by the mass stratiÐcation instability.As a consequence, the collapse proceeds faster than andstarts before the few-binary case (the latter point makes eq.

more plausible than it might appear at Ðrst sight). This[39]phenomenon is found to occur for both tidally capturedbinaries Ostriker, & Cohn and primordial(Statler, 1987)binaries & Mathieu(Spitzer 1980).

3.4. DiscussionWhat the above analysis shows is that dark clusters

within the mass range 3 ] 102 M_

[MDC[ 106 M_should have survived all disruptive e†ects arising from

gravitational perturbations and are at present expected topopulate the outer part of the Galactic halo. In addition,clusters with should3 ] 102 M

_[MDC[ 5 ] 104 M

_undergo core collapse. Unfortunately, it is practicallyimpossible to predict the further fate of those dark clustersthat are in the postcollapse phase today. A priori it seemsnatural to imagine that bounce and subsequent reexpansionshould follow core collapse & Hut &(Cohn 1984 ; HeggieRamamani Perhaps also a whole series of core con-1989).tractions and expansions may take place, giving rise to theso-called gravothermal oscillations & Sugimoto(Bettwieser

However, this conclusion crucially depends on the1984).unknown model that correctly describes dark clusters. Forinstance, in tidally truncated models the cluster is com-pletely destroyed within a Ðnite time (Stodolkiewicz 1985 ;

Statler, & Lee Moreover, what certainlyOstriker, 1985).happens in either case is that the number of MACHOs in


the core monotonically decreases with time. So, anunclustered MACHO population is expected to coexistwith dark clusters in the outer Galactic halo (unless all darkclusters have detection of unclusteredMDCZ 5 ] 104 M

_) ;

MACHOs would therefore not rule out the present sce-nario.

4. MACHOS AS BINARY BROWN DWARFS

As has already been pointed out, it seems natural tosuppose that a fraction of primordial binary brown dwarfs,possibly as large as 50% in mass, should form along withindividual brown dwarfs as a result of the fragmentationprocess of the PGC clouds. Subsequently, because of themass stratiÐcation instability, primordial binaries will con-centrate inside the dark cluster cores, which are thereforeexpected to be chieÑy composed of binaries and molecularclouds. In addition, as far as dark clusters with MDC[ 5

are concerned, a population of tidally captured] 104 M_binary brown dwarfs ought to form in the dark cluster cores

because of the increased central density caused by core col-lapse. Thus, a large fraction of binaries should be presentinside the dark cluster cores at a late stage of their evolu-tion. Below, we will try to make the discussion of this issueas quantitative as possible.

4.1. Survival and Hardness of Binary Brown DwarfsThe Ðrst question to be addressed is whether a binary

brown dwarf, produced by whatever mechanism long ago,survives until the present. To this end, we recall that abinary system is ““ hard ÏÏ when its binding energy exceedsthe kinetic energy of Ðeld stars (otherwise it is ““ soft ÏÏ). In thepresent case, binary brown dwarfs happen to be hard whentheir orbital radius a obeys the constraint

a [ 1.4] 1012AM

_MDC

B2@3km . (40)

As is well known, soft binaries always get softer, whilehard binaries always get harder because of encounters withindividual stars So, if individual-binary(Heggie 1975).encounters were the only relevant process, we would con-clude that hard binary brown dwarfs should indeed survive.However, binary-binary encounters also play an importantrole in the dark cluster cores, where binaries are expected tobe far more abundant than individual brown dwarfs. Now,in the latter process, one of the two binaries is often dis-rupted (which cannot happen for both binaries, given thatthey are hard, while Ñy-bys are rather infrequent), therebyleading to the depletion of the binary population. We willaddress this e†ect in where we Ðnd that for realistic° 4.3,values of the dark cluster parameters, the binary breakupdoes not take place.

4.2. T idally Captured Binary Brown DwarfsAs far as globular clusters are concerned, it is now well

known that the most efficient mechanism for late binaryformation is dissipative tidal capture in the core (Fabian,Pringle, & Rees & Teukolsky &1975 ; Press 1977 ; LeeOstriker Hence, we expect a similar situation to1986).occur in dark clusters.

Let us now analyze this phenomenon in a quantitativefashion. As a Ðrst step, we observe that the radius of ar

*brown dwarf of mass is km^0.1 M_

r*

^ 0.7] 105et al. Furthermore, the Safronov number(Saumon 1996).

& Tremaine is(Binney 1987)

# \ Gm2p

*2 r

*^ 2 ] 107

AM_

MDC

B2@3, (41)

which turns out to be much larger than 1. Within thisapproximation, the time for brown dwarf tidal capture canbe written as & Ostriker(Lee 1986)

ttid ^ 1012 105 pc~3nIBD(0)

A p*

100 km s~1B1.2AR

_r*

B0.9AM_

mB1.1

yr ,

(42)

where is the number density of individual brownnIBD(0)dwarfs in the Obviously, we havecore.21 nIBD(0)^

where denotes the mass fraction of individ-fIBD o*(0)/m, fIBDual brown dwarfs in the core. From we seeequation (37),

that increases very slightly, and so core-collapse e†ectsp*on can safely be neglected. Accordingly, we can rewritep

* asequation (42)

ttid ^1.6] 1014

fIBD

M_

pc~3o*(0)

AMDCM

_

B0.4yr , (43)

using Comparing with the age of the uni-equation (19). ttidverse, we see that practically all individual brown dwarfs inthe core are tidally captured into binaries provided that22

o*(0)[

3.2] 104fIBD

AMDCM

_

B0.4M

_pc~3 . (44)

According to the above assumptions, we expect o*(0)^ 104

pc~3 just before core collapse. Therefore, we see thatM_tidal capture requires an increase of by a factor in theo

*(0)

range corresponding to in the range31/fIBDÈ242/fIBD, MDCThus, the formation of tidally cap-3 ] 102È5 ] 104 M_

.tured binaries would occur during the early phase of corecollapse (the same conclusion was reached in a di†erent wayby Ostriker, & Cohn for globular clusters).Statler, [1987]However, this conclusion depends on the fractional (mass)abundance, of primordial binaries in the core, sincefPB, fIBDnecessarily becomes small for large fPB.Next, we compute the (average) orbital radius of tidallycaptured binary brown dwarfs, following the procedureoutlined by et al. Correspondingly, we ÐndStatler (1987).a ^ 2.5] 105 km (this value is practically independent of

As a consequence, we see that they are so hard thatMDC).the condition of is always abundantly met.equation (40)Let us Ðnally try to estimate the fractional abundance of

tidally captured binary brown dwarfs in the dark clustercores soon after their formation, namely, when the inequal-ity of just starts to be satisÐed (their totalequation (44)number at this stage will be denoted by Thanks toNTCB).we easily getequation (38),

NTCB ^ 3.3f IBD1.36AMDC

M_

B0.86 Mc

M_

, (45)

where denotes the core mass just before core collapseMc

21 Although has been derived for main-sequence stars, itequation (42)seems plausible that it may also apply to brown dwarfs, since it is not verysensitive to the particular stellar model & Ostriker(Lee 1986).

22 More precisely, we are demanding that the rate for tidal capturetimes the age of the universe should exceed 1. Note that the former quan-tity is one-half of so we require that should be smaller than 1010ttid~1, 2ttidyr (we are actually following the same procedure used by & Teu-Presskolsky 1977).


[corresponding to pc~3]. Further deÐningo*(0)^ 104 M

_as the total number of individual brown dwarfs in aNIBDtotdark cluster before core collapse, we Ðnd

NTCBNIBDtot ^ 0.33f IBD1.36

AM_

MDC

B0.14 Mc

MDC. (46)

Realistically, even in the extreme case of a fully baryonichalo we expect (of course, a large would implyfIBD[ 0.3 fPBa considerably smaller than that). Moreover,fIBD M

c/MDCsensitively depends on how the core is deÐned ; the analogy

with globular clusters entails, however, that it should in anycase be less than 20%. Accordingly, we conclude that thefractional abundance of tidally captured binary browndwarfs should not exceed 1% (again in remarkable agree-ment with the result of et al. for globularStatler [1987]clusters), and so they are irrelevant from the observationalpoint of view.

4.3. Primordial Binary Brown DwarfsPrimordial binaries are a very di†erent story. Not only

are they expected to be much more abundant than tidallycaptured binaries, but they are also presumably much lesshard, since all values for their orbital radius consistent withthe conditions of should in principle be con-equation (40)templated. So, hardening e†ects ought to play a crucial rolefor primordial binaries (as may be guessed intuitively,e†ects of this kind turn out to be totally negligible for tidallycaptured binaries ; this will be demonstrated later on).

4.3.1. Collisional Hardening

Let us begin by considering collisional hardening,namely, the process whereby hard binaries get harder inencounters with individual brown dwarfs. We recall that theassociated average hardening rate & Mathieu(Spitzer 1980)presently reads

E0 ^ [2.8G2m3nIBD(0)

p*

. (47)

From the deÐnition of andnIBD(0) equation (19), equationbecomes(47)

E0 ^ [1.7] 1032fIBDAM

_MDC

B1@3 o*(0)

M_

pc~3 ergs yr~1 . (48)

Observe that a characteristic feature of collisional hard-ening is that is independent of hardness, and so is timeE0independent.

As is well known, the internal energy of a binary isE\ [Gm2/2a, which yields

E0 \Gm22a2 a5 . (49)

Hence, by combining equations and and inte-(48) (49)grating the ensuing expression, we get

kma2

^kma1

] 1.3] 10~20fIBDAM

_MDC

B1@3 o*(0)

M_

pc~3t21yr

,

(50)

where is the initial orbital radius and is the orbitala1 a2radius after a time t21.Assuming momentarily that no other hardening mecha-nism is operative and taking equal to the age of thet21universe, we Ðnd that the present orbital radius of a binary

brown dwarf is given by

kma2

^kma1

] 1.3] 10~10fIBDAM

_MDC

B1@3 o*(0)

M_

pc~3 . (51)

Of course, collisional hardening works to the extent that a2becomes considerably smaller than Correspondingly, wea1.see from that this is indeed the case, providedequation (51)that

a1 Z 8 ] 109f IBD~1AMDC

M_

B1@3 M_

pc~3o*(0)

km . (52)

Using in turn yieldsequation (51), equation (52)

a2 ^ 8 ] 109f IBD~1AMDC

M_

B1@3 M_

pc~3o*(0)

km . (53)

Physically, the emerging picture is as follows. Only thosebinaries whose initial orbital radius obeys the conditiongiven in undergo collisional hardening, andequation (52)their present orbital radius turns out to be almost indepen-dent of the initial value. We can make the present discussionsomewhat more speciÐc by noticing that our assumptionsstrongly suggest pc~3, in which case botho

*(0)[ 104 M

_and acquire the formequation (52) equation (53)

a1,2 Z 8 ] 105f IBD~1AMDC

M_

B1@3km . (54)

Evidently, very hard primordial binaries, which fail tomeet the condition given in do not su†erequation (54),collisional hardening, and the same is true for tidally cap-tured binaries.

4.3.2. Frictional Hardening

As we will show, the presence of molecular clouds in thedark cluster cores, which is indeed the most characteristicfeature of the model in question, provides a novel hardeningmechanism for binary brown dwarfs. Basically, this isbrought about by dynamical friction on molecular clouds.

It is not difficult to extend the standard treatment ofdynamical friction & Tremaine to the relative(Binney 1987)motion of the brown dwarfs in a binary system that movesinside a molecular cloud. For simplicity, we assume thatmolecular clouds have a constant density proÐle In theo

m.

case of a circular the equations of motion implyorbit,23that the time needed to reduce the orbital radius a of at21(0)binary that moves all the time inside molecular clouds from

down to isa1 a2t21(0)^ 0.17

AmGB1@2 1

om

ln "(a2~3@2 [ a1~3@2) , (55)

where the Coulomb logarithm reads

ln "^ lnAr

mvc2

GmB

^ lnAr

ma1

B, (56)

where is the circular velocity (approximately given byvcKeplerÏs third law). Manifestly, the di†usion approximation

upon which the present treatment is based requires that theorbital radius of a binary should always be smaller than themedian radius of a cloud. As we are concerned henceforthwith hard binaries, must obey the condition given ina1On the other hand, is the larger value forequation (40). a1the orbital radius in So, we shall take forequation (55).

23 Indeed, the circularization of the orbit is achieved by tidal e†ectsafter a few periastron passages (Zahn 1977).


deÐniteness (in the Coulomb logarithm only) a1^ 1.4km. In addition, from we] 1012(M

_/MDC)2@3 equation (12)

have

om

^ 2.5Apcrm

B2M

_pc~3 . (57)

Hence, putting everything together we obtain

Akma2

B3@2Âkm

a1

B3@2] 2 ] 10~26$~1Apcrm

B2 t21(0)yr

, (58)

having set

$4C3 ] ln

Arm

pcB

] 0.7 lnAMDC

M_

BD~1. (59)

SpeciÐcally, the di†usion approximation demands $[ 0,which yields in turn

rm

[ 5 ] 10~2AM

_MDC

B0.7pc . (60)

Observe that for this constraintMDC [ 2.1 ] 104 M_

,restricts the range of allowed values of as stated inr

m, ° 2.2.

Were the dark clusters completely Ðlled by clouds,would be the Ðnal result. However, the dis-equation (58)

tribution of the clouds is lumpy. So, if we want to know theorbital radius after a time we must proceed asa2 t21,follows. First, we should compute the fraction of thet21(0)time interval in question, spent by a binary inside thet21,clouds. Next, we need to reexpress in int21(0) equation (58)terms of This will be achieved by the procedure outlinedt21.below.

Keeping in mind that both the clouds and the binarieshave average velocities (for simplicity, wev^ (3)1@2p

*neglect the equipartition of kinetic energy of the binaries), itfollows that the time needed by a binary to cross a singlecloud is

tm

^rm

J2v^ 5.6] 106 r

mpcAM

_MDC

B1@3yr . (61)

As an indication, we notice that for pcrm

^ 10~3 (Mm

^ 2and we Ðnd] 10~2 M

_) MDC^ 105 M

_, t

m^ 1.2 ] 102

yr. Therefore, frictional hardening involves many clouds.SpeciÐcally, during the time the number of cloudst21,crossed by a binary is evidently

Nm

^t21(0)tm

^ 1.8] 10~7 pcrm

AMDCM

_

B1@3 t21(0)yr

. (62)

Let us now ask how many crossings of the core are neces-sary for a binary to traverse clouds. To this end, weN

mestimate the number of clouds encountered during oneNccrossing of the core. Describing the dark clusters by a King

model, we can identify the core radius with the King radius.The cross section for binary-cloud encounters is so wenr

m2 ,

have

Nc^C 9p

*2

4nGo*(0)D1@2

nCL(0)nrm2 , (63)

where is the cloud number density in the core. FromnCL(0)we can writeequation (12),

nCL(0)^ fCLo*(0)

Mm

^ 4.8] 10~2fCLpcrm

o*(0)

M_

pc~3 pc~3 ,

(64)

where denotes the fraction of core dark matter in thefCLform of molecular clouds. Correspondingly, equation (63)becomes

Nc^ 0.13] fCL

C o*(0)

M_

pc~3D1@2AMDC

M_

B1@3 rm

pc, (65)

through So, the total number of core cross-equation (19).ings that a binary has to make in order to traverseNcc N

mclouds is

Ncc ^N

mN

c^ 1.4] 10~6f CL~1

CM_

pc~3o*(0)

D1@2Apcrm

B2 t21(0)yr

.

(66)

Because the core crossing time is

tcc ^C 9p

*2

4nGo*(0)D1@2 1

v^ 7 ] 106

CM_

pc~3o*(0)

D1@2yr , (67)

it follows that the process under consideration takes a totaltime However, this time is by deÐnition just So,Ncc tcc. t21.we get

t21^ Ncc tcc^ 9.8f CL~1 M_

pc~3o*(0)

Apcrm

B2t21(0) , (68)

which is the desired relationship between and Wet21 t21(0).are now in position to reexpress in terms ofequation (58)Accordingly, we gett21.

Akma2

B3@2Âkm

a1

B3@2 ] 2.1] 10~27fCL$~1 o*(0)

M_

pc~3t21yr

.

(69)

In order to quantify the e†ect of frictional hardening, wemay proceed in much the same way as in the case of col-lisional hardening, neglecting, however, the latter e†ect forthe moment. SpeciÐcally, taking in to bet21 equation (69)equal to the age of the universe, we Ðnd that the presentorbital radius of a binary brown dwarf is given by

Akma2

B3@2Âkm

a1

B3@2 ] 2.1] 10~17fCL$~1 o*(0)

M_

pc~3 .

(70)

Manifestly, frictional hardening is operative to the extentthat becomes considerably smaller than Accordingly,a2 a1.from we see that this is indeed the case, pro-equation (70)vided that

a1Z 1.3] 1011f CL~2@3$2@3CM

_pc~3

o*(0)

D2@3km . (71)

From entails in turnequation (71), equation (70)

a2^ 1.3] 1011f CL~2@3$2@3CM

_pc~3

o*(0)

D2@3km . (72)

Physically, only those binaries whose initial orbital radiussatisÐes the condition of are a†ected by fric-equation (71)tional hardening, and their present orbital radius turns outto be almost independent of the initial value. We can makethe present discussion somewhat more speciÐc by noticingthat our assumptions strongly suggest o

*(0)[ 104 M

_pc~3, in which case both equations and acquire the(71) (72)


form

a1,2 Z 2.8] 108f CL~2@3$2@3 km . (73)

Evidently, very hard primordial binaries, which violate thecondition of do not su†er frictional hard-equation (73),ening, and the same is true for tidally captured binaries.

4.3.3. Present Orbital Radius of Primordial Binaries

Which of the two hardening mechanisms under consider-ation is more e†ective? A straightforward implication ofequations and is that frictional hardening is more(53) (72)efficient than collisional hardening whenever it happensthat

fIBD\ 6.3] 10~2f CL2@3$~2@3AMDC

M_

B1@3CM_

pc~3o*(0)

D1@3.

(74)

Unfortunately, the occurrence of various dark clusterparameters in the condition given by does notequation (74)permit a sharp conclusion to be drawn, but in the illustra-tive case where pc,MDC^ 105 M

_, r

m^ 10~3 fCL ^ 0.5,

and pc~3, we get As we expect ino*(0)^ 103 M

_fIBD \ 0.5.

the cores and we see that frictional hard-fIBD > fPB fCL^ fPB,ening plays the dominant role. Moreover, from equationsand it also follows that the e†ectiveness of col-(53) (72)

lisional hardening decreases for smaller values of fIBD.Finally, the fairly slow dependence of the conditions of

on and makes our conclusionequation (74) MDC o*(0)

rather robust.Having shown that collisional hardening can e†ectively

be disregarded, we see that the present orbital radius ofprimordial binary brown dwarfs is actually given by

as long as the condition of isequation (72), equation (71)met. Taking again the above particular case as an illustra-tion, we Ðnd km, which is of the same order asa2^ 7 ] 108the Einstein radius for microlensing toward the LMC

& Gould Therefore, we argue that primordial(Gaudi 1997).binaries that are not too hard can be resolved in futuremicrolensing experiments with a more accurate photo-metric observation, the signature being small deviationsfrom standard microlensing light curves (Dominik 1996).

4.3.4. Gravitational Encounters

We are now in position to take up the question of thesurvival of binary brown dwarfs against gravitationalencounters. As already pointed out, individual-binaryencounters are harmless in this respect, since hard binariesare considered throughout, so we shall restrict our attentionto binary-binary encounters.

As a Ðrst step, we recall that their average rate ! (Spitzer& Mathieu can be written as1980)

!^ aGmap*

, (75)

with a ^ 13. Correspondingly, from the reac-equation (19)tion time in the dark cluster cores turns out to be

treact^ 1019b M_

pc~3o*(0)

AMDCM

_

B1@3 kma

yr , (76)

with b ^ 6.7/fPB.Now, if no hardening were to occur, which means thatthe orbital radius a would stay constant, the binary survivalcondition would simply follow by demanding that treactshould exceed the age of the universe. However, hardening

makes a decrease, so increases with time. This e†ecttreactcan be taken into account by considering the average valueof the reaction time over the time interval in ques-StreactTtion (to be denoted by T ), namely,

StreactT 41TP0

Tdt treact . (77)

In order to compute the temporal dependence a(t)StreactT,of the binary orbital radius is needed. Because frictionalhardening plays the major role, the latter quantity is evi-dently supplied by Setting for notational con-equation (69).venience and yieldst 4 t21 a(t) 4 a2, equation (69)

a(t) ^CAkm

a1

B3@2 ] 2.1] 10~27$~1fCL

]o*(0)

M_

pc~3tyrD~2@3

km . (78)

Combining equations and and inserting the(76) (78)ensuing expression into we getequation (77),

StreactT ^ 1.9] 1046f PB~1f CL~1$AMDC

M_

B1@3 yrTAkm

a1

B5@2

]CM

_pc~3

o*(0)

D2GC1 ] 2.1] 10~27$~1

] fCLo*(0)

M_

pc~3TyrA a1kmB3@2D5@3[ 1

Hyr . (79)

Let us now require to exceed the age of the universeStreactT(taking evidently T ^ 1010 yr). As is apparent fromis shorter for softer binaries. Hence, inequation (76), treactorder to contemplate hard binaries with an arbitrary orbital

radius, we need to set km ina1^ 1.4] 1012(M_/MDC)2@3Correspondingly, the binary survival condi-equation (79).

tion reads24

fPB [ 8.2] 10~5f CL~1$AMDC

M_

B2CM_

pc~3o*(0)

D2

]GC

1 ] 35fCL$~1 o*(0)

M_

pc~3M

_MDC

D5@3 [ 1H

. (80)

Although the presence of various dark-cluster parametersprevents a clear-cut conclusion to be drawn from equation

in the illustrative case where and(80), MDC^ 105 M_entails, e.g., forfCL ^ 0.5, equation (80) fPB [ 0.3 o

*(0)^ 3

] 103 pc~3. Thus, we infer that for realistic values ofM_the parameters in question, a sizable fraction of primordial

binary brown dwarfs survive binary-binary encounters inthe dark cluster cores.25

5. THERMAL BALANCE IN HALO MOLECULAR CLOUDS

As far as the energetics of halo molecular clouds is con-cerned, we can identify two main heat sources. One isenergy deposition from background photons, and the otheris of gravitational origin (coming from frictional hardeningof primordial binary brown dwarfs). Although we discuss

24 Applying the same argument to tidally captured binaries shows thatno depletion occurs in this way.

25 If the initial value of fails to satisfy the condition offPB equation (80),primordial binaries start to be destroyed in binary-binary encounters untiltheir fractional abundance is reduced down to a value consistent with thecondition given in equation (80).


them separately, it should be kept in mind that they bothact at the same time.

5.1. Energy from Background PhotonsWe proceed to estimate by momentarily neglectingT

mgravitational e†ects. To this end, we need to know theheating rate (due to external sources) and the cooling rate(due to the molecules). In the Galactic halo, the dominantheat source for molecular clouds is expected to be ioniza-tion from photons of the X-ray background, whose spec-trum in the relevant range 1 keV\ E\ 25 keV (see below)can be parametrized in terms of the energy E (expressed inkeV) as I(E) \ 8.5E~0.4 cm~2 s~1 et al.sr~1(OÏDea 1994).The ionization rate per molecule (taking secondary ion-H2ization into account) is

mX \ 26PEmin

Emax4nI(E)pX(E)dE s~1 , (81)

where cm2 is the absorptionpX(E) \ 2.6] 10~22E~8@3cross section in the above energy range for incoming X-rayson gas with interstellar composition & McCam-(Morrisonmon In fact, decreases faster as E increases1983). pX(E)when the gas metallicity is lower (see Fig. 1 of &MorrisonMcCammon So, the expression quoted above for1983).

is expected to yield an upper bound to the crosspX(E)section for X-rays on halo molecular clouds. The integra-tion limits (below which X-rays are absorbed) andEmin Emax(above which X-rays go through the whole cloud withoutbeing absorbed) depend on the cloud column density. Thus,taking as an orientation in the range 104È108 cm~3, wen

mget 1.25 keV and 10 keV.keV \Emin\ 7 keV \Emax \ 20It turns out that is rather insensitive to the upper limit,mXand we obtain s~1 for keVmX \ 2.2 ] 10~19 Emin\ 1.25and s~1 for Since eachmX \ 5.6] 10~21 Emin\ 7 keV.26ionization process releases an energy of ^8 eV, the heatingrate per molecule ! turns out to beH2

3.5] 10~32 ergs s~1H2

\ ! \1.5] 10~30 ergs s~1

H2. (82)

Unfortunately, the cooling rate for the halo clouds inquestion is not well known, owing to the lack of detailedinformation about their chemical composition. Neverthe-less, by merely considering the cooling rate due to asH2given by & Langer the equality betweenGoldsmith (1978),cooling and heating rate per molecule leads to K.T

m^ 10

More complete models for the cooling rate, which includethe contribution from HD and heavy molecules, imply thatthe cooling efficiency is substantially enhanced, and thusmake it very plausible that halo molecular clouds clumpedinto dark clusters should have a temperature close to that ofthe CBR, namely, K (see & Silk forT

m^ 3 Gerhard 1996

similar conclusions).

5.2. Energy from Primordial Binary Brown DwarfsAs the analysis in shows, dynamical friction transfers° 4.3

a huge amount of energy from primordial binary browndwarfs to molecular clouds, so it looks compelling to inves-

26 It is straightforward to verify that the ionization rate due to themcrhalo cosmic rays is less important in this context. In fact, assuming theionization rate per molecule typical for disk cosmic raysH2 m0\ 5] 10~17 s~1 (see, e.g., Dishoeck & Black and rescaling for thevan 1986)di†usion of the cosmic rays in the Galactic halo Paolis et al. we(De 1995c),infer s~1.mcr D 10~23

tigate (at least) the gross features of the ensuing energybalance.

Let us start by evaluating the energy acquired by molecu-lar clouds in the process of frictional hardening. Recallingthat the traversal time for a single cloud is given by

entails that after a binary withequation (61), equation (58)initial orbital radius has crossed N clouds, its orbitala1radius is reduced down to

aN`1^

C1.1] 10~19N $~1 pc

rm

AM_

MDC

B1@3

]Akm

a1

B3@2D~2@3km . (83)

Accordingly, we see that the orbital radius remains almostconstant until N reaches the critical value

N04 9 ] 1018$ rm

pcAMDC

M_

B1@3Akma1

B3@2, (84)

whereas it decreases afterward. Because the energy acquiredby the clouds is just the binding energy given up by primor-dial binaries, this information can be directly used tocompute the energy gained by the Nth cloud tra-*E

c(N)

versed by a binary whose initial orbital radius was a1.Manifestly, we have

*Ec(N) \ 1

2Gm2

A 1aN`1

[ 1aN

B. (85)

From a straightforward calculation showsequation (83),that stays practically constant,*E

c(N)

*Ec^ 9.8] 1032$~1 pc

rm

AM_

MDC

B1@3A a1kmB1@2

ergs , (86)

as long as while it decreases subsequently. So, theN [ N0,amount of energy transferred to a cloud is maximal duringthe early stages of hardening. Now, since the binding energyof a cloud is

Ec^ 7.7] 1042 r

mpc

ergs , (87)

it can well happen that (depending on*Ec[ E

crm, MDC,and which means that the cloud would evaporate unlessa1),it manages to efficiently dispose of the excess energy.

A deeper insight into this issue can be gained as follows(we focus on the early stages of hardening, when the e†ectunder consideration is more dramatic). Imagine that aspherical cloud is crossed by a primordial binary thatmoves along a straight line, and consider the cylinder *traced by the binary inside the cloud (its volume beingapproximately Hence, by cm~2na2r

m). n

m^ 62.2(pc/r

m)2

(which follows from the average number of mol-eq. [12]),ecules inside * will be

N*^ 5.8] 1030A a1kmB2 pc

rm

. (88)

Physically, the energy is Ðrst deposited within * in the*Ecform of heat. Neglecting thermal conductivity (which will be

discussed later), the temperature inside * accordinglybecomes

T*^23

*Ec

N* kB^ 8.1] 1017$~1

AM_

MDC

B1@3Akma1

B3@2K ,

(89)


being the Boltzmann constant. Taking into accountkBequations and yields(40) (72), equation (89)

0.5$~1AMDC

M_

B2@3K [ T*[ 16.2fCL$~2

]o*(0)

M_

pc~3AM

_MDC

B1@3K , (90)

which in the illustrative case where fCL^ 0.5, o*(0)^ 3

pc~3, and entails in turn] 103 M_

MDC^ 105 M_

,5.3] 103 K. As a consequence of theK [ T*[ 1.3] 104increased temperature, the molecules within * will radiate,thereby reducing the excess energy in the cloud. In order tosee whether this mechanism actually prevents the cloudfrom evaporating, we notice that the characteristic timeneeded to accumulate the energy inside * is just the*E

ctraversal time Therefore, this energy will be totally radi-tm.

ated away provided that the cooling rate per molecule "exceeds the critical value given by the equilibrium condi-"0tion

N*"0 tm

^ *Ec

. (91)

SpeciÐcally, yieldsequation (91)

"0^ 10~12$~1Apcrm

BAkma1

B3@2ergs s~1 molecule~1 , (92)

taking into account equations and Moreover,(61), (86), (88).in the present case, in which most of the molecules are H2,the explicit form of " is (see, e.g., et al.OÏDea 1994 ; Neufeld,Lepp, & Melnick 1995)

"^ 3.8] 10~31AT*

KB2.9

ergs s~1 H2~1 , (93)

which, from becomesequation (89),

"^ 3.4] 1021$~2.9Akm

a1

B4.35ergs s~1 H2~1 (94)

(note that " is almost independent of Now, fromMDC).equations and it follows that the condition(92) (94) "Z"0implies

a1[ 6 ] 1011$~0.7Ar

mpcB0.35

km . (95)

For a wide range of dark cluster and molecular cloudparameters, it turns out that is fulÐlled forequation (95)hard primordial binaries.

Thus, we conclude that the energy given up by primordialbinary brown dwarfs and temporarily acquired by molecu-lar clouds is efficiently radiated away, so that the clouds arenot dissolved by frictional hardening.

As a Ðnal comment, we stress that our estimate for T*should be understood as an upper bound, since thermalconductivity has been neglected. In addition, the aboveanalysis implicitly relies upon the assumption that T*\ 104K, which ensures the survival of Actually, in spite of theH2.fact that the condition given by suggests thatequation (90)this may well not be the case, our conclusion neverthelessremains true. This is because higher temperatures wouldlead to the depletion of which correspondingly isH2,replaced by atomic and possibly ionized hydrogen. As iswell known, in either case the resulting cooling rate wouldexceed that for and so cooling would be even moreH2,efficient than estimated above & Hensler(Bo� hringer 1989).

6. Lya ABSORPTION SYSTEMS

It is well known that quasar Lya absorption lines providedetailed information on the evolution of the gaseous com-ponent of galaxies (see, e.g., Hogan, & PeeblesFukugita,

and references therein). These lines are seen for a1996neutral hydrogen column density ranging fromNH ID3 ] 1012 cm~2 (the detection threshold) to D1022 cm~2.

At the upper limit of this range cm~2),(NH I º 2 ] 1020the lines are classiÐed as damped Lya systems (mostlyassociated with metal-rich objects), and it is generallybelieved that they are the thick progenitors of galactic disks.The H I distribution in damped Lya systems is usuallyÑatter than the corresponding surface brightness of theoptical disks and extends to much larger radii (up to D40kpc in giant galaxies). In the outer galactic regions, Lyasystems show sharp H I edges at a level of NH ID 1017cm~2, the so-called Lyman limit. Damped Lya systems areobserved up to redshift zD 3.5, and the survey resultssuggest that the average mass of neutral hydrogen perabsorption system decreases with time, in agreement withthe hypothesis of gas consumption into stars (Lanzetta,Wolfe, & Turnshek 1995).

Within our picture, it is tempting to identify damped Lyasystems with the PGC clouds in the inner Galactic halo,where they undergo disruption.

Below the Lyman limit (i.e., for 3] 1012 cm~2\ NH I\3 ] 1015 cm~2 ; see, e.g., Fig. 2 in theCristiani 1996),absorption lines are classiÐed as Lya forest and are gener-ally ascribed to a large number of intervening clouds (insome cases extending up to D200 kpc from a centralgalaxy) along the line of sight to distant quasars. Studies ofLya forest lines have made rapid progress recently, andsome observational trends are Ðrmly established. In partic-ular : (1) The evolution of the comoving number density ofsystems per unit interval in redshift shows that the numberof Lya forest clouds decreases rapidly with time for z[ 2,while it is approximately constant for z\ 2. (2) The corre-lation between the thermal Doppler parameter b and thenumber of clouds can be Ðtted with a Gaussian distributionof median SbT ^ 30 km s~1 and dispersion ^10 km s~1,corresponding to a temperature of a few 104 K.

Within our model, it seems natural to identify Lya forestclouds with the molecular clouds clumped into dark clus-ters located in the outer halo. Indeed, we expect that theclouds contain in their external layers an increasing fractionof H I gas and that the outer regions are even ionized,because of the incoming UV radiation. Observe that so farwe have been assuming that halo molecular clouds have aconstant temperature. This was a good approximation asfar as the previous analysis was concerned, but would betoo poor in the present discussion. An H I column density ofD1014 cm~2, corresponding to a layer of about 10~6 pc, issufficient to shield the incoming radiation. Remarkablyenough, cm~2 corresponds to the average valueNH I D 1014of the observed Lya forest distribution (Cristiani 1996).Since we expect the UV Ñux to decrease as the Galactocen-tric distance increases, clouds lying at a larger distance maythus have a smaller H I column density, while clouds closerto the Galactic disk may have a higher column density. Thisfact explains the observed distribution of the columndensity.

Moreover, the number of clouds is expected initially todecrease with time because of both MACHO and for-H2


mation, in this way explaining the observed evolution ofLya forest clouds according to the above-mentioned point(1). As a Ðnal comment, we mention that very recently,

Miley, & Van Ojik by considering theRo� ttgering, (1996),Ðlling factors and the physical parameters derived fromtheir Lya forest observations, pointed out that galactichalos may contain of neutral hydrogen gas andD109 M

_are typically composed of D1012 clouds, each of size D40light days. It seems remarkable that these are the typicalparameters of molecular clouds dealt with in this paper.

A thorough quantitative analysis requires, however,further investigations that are beyond the scope of thepresent paper.

7. CONCLUSIONS

Looking back at what we have done, it seems to us fair tosay that present-day results of microlensing experimentstoward the LMC are amenable to a very simple explanationin terms of what has repeatedly been proposed as a naturalcandidate for baryonic dark matterÈnamely, browndwarfs. Once this idea is accepted, a few almost obvioussteps follow. First, given the fact that ordinary halo starsform in (globular) clusters, it seems likely that brown dwarfsalso form in clusters rather than in isolation (this circum-stance has been repeatedly recognized in the last few years).Of course, whether brown dwarfs are still clumped intodark clusters today is a di†erent and nontrivial question ; wehave seen that core collapse can liberate a considerablefraction of brown dwarfs from the less massive dark clus-ters. Second, in much the same way as happens for ordinarystars, a consistent fraction of binary brown dwarfs (up to50% in mass) is expected to form. Third, since no knownstar formation mechanism is very efficient, it is natural toimagine that a substantial quantity of primordial gas is leftover. Although we cannot be sure about the subsequent fateof this gas, which should be mostly cold it is likely toH2,remain conÐned within the dark clusters, since stellar winds

are absent. So, cold self-gravitating clouds should pre-H2sumably also be clumped into dark clusters, along withsome residual di†use gas.

We have shown that within the considered model, pri-mordial binary brown dwarfs that are not too hard turn outto have an orbital radius that is typically of the order of theEinstein radius for microlensing toward the LMC. There-fore, they are not so easily resolvable in the microlensingexperiments performed so far, but we argue that they couldbe resolved in future microlensing experiments with a moreaccurate photometric the signature beingobservation,27small deviations from standard microlensing light curves

& Hirschfeld Notice that(Dominik 1996 ; Dominik 1996).such a procedure complements the detection strategy forbinaries suggested by & GouldGaudi (1997).

Many physical processes certainly occur in the dark clus-ters, and the (likely) presence of a large amount of gasmakes them even more difficult to understand than globularclusters. Moreover, the lack of any observational informa-tion makes any attempt to Ðgure out the physics of darkclusters almost impossible, were it not for some remarkablestructural analogies with globular clusters suggested by themodel upon which our discussion is based. What should inany case be clear is that a host of di†erent phenomena canoccur. Some of them have intentionally been neglected here,in order not to make the present paper either excessivelylong or too speculative. Yet many others are likely to occur,that we have not even been able to imagine.

We would like to thank B. Bertotti, B. Carr, and F.DÏAntona for useful discussions. F. D. P. has been partiallysupported by the Dr. Tomalla Foundation and by INFN.F. D. P. and G. I. acknowledge some support from ASI(Agenzia Spaziale Italiana).

27 See, e.g., the ongoing experiments by the GMAN and PLANETcollaborations (Proc. 2nd International Workshop on GravitationalMicrolensing Surveys, Orsay, 1996).

REFERENCES

F. C., & Laughlin, G. 1996, ApJ, 468,Adams, 586C., et al. 1993, Nature, 365,Alcock, 6211997, ApJ, 486,ÈÈÈ. 697

V. A. 1962, Vestn Leningrad-Gross. Univ., 7,Antonov, 135K. M. 1990, MNRAS, 247,Ashman, 662K. M., & Carr, B. J. 1988, MNRAS, 234,Ashman, 219E., et al. 1993, Nature, 365,Aubourg, 623

J., Flynn, C., Gould, A., & Kirhakos, S. 1994, ApJ, 435,Bahcall, L51E., & Sugimoto, M. 1984, MNRAS, 208,Bettwieser, 439

J., & Tremaine, S. 1987, Galactic Dynamics (Princeton : PrincetonBinney,Univ. Press)

H., & Hensler, G. 1989, A&A, 215,Bo� hringer, 147A., Hubbard, W. B., & Lunine, J. I. 1989, ApJ, 345,Burrows, 939

B. 1994, ARA&A, 32,Carr, 531B. J., Bond, J., & Arnett, W. 1984, ApJ, 277,Carr, 445B. J., & Lacey, C. G. 1987, ApJ, 316,Carr, 23

G., Segretain, L., & Me� ra, D. 1996, ApJ, 468,Chabrier, L21S., & Silk, J. 1995, ApJ, 445,Charlot, 124

H. 1980, ApJ, 242,Cohn, 765H., & Hut, P. 1984, ApJ, 277,Cohn, L45

S. 1996, ESO preprintCristiani, 1117A., & McCray, R. A. 1972, ARA&A, 10,Dalgarno, 375F. 1987, ApJ, 320,DÏAntona, 653

Paolis, F., Ingrosso, G., & Jetzer, Ph. 1996, ApJ, 470,De 493Paolis, F., Ingrosso, G., Jetzer, Ph., Qadir, A., & Roncadelli, M. 1995a,De

A&A, 299, 647Paolis, F., Ingrosso, G., Jetzer, Ph., & Roncadelli, M. 1995b, Phys. Rev.De

Lett., 74, 141995c, A&A, 295,ÈÈÈ. 5671995d, Comments Astrophys., 18,ÈÈÈ. 87

Ru� jula, A., Jetzer, Ph., & Masso� , E. 1991, MNRAS, 250,De 3481992, A&A, 254,ÈÈÈ. 99

J. M., & Lockman, F. J. 1990, ARA&A, 28,Dickey, 215M. 1996, Ph.D. thesis, DortmundDominik, UniversityM., & Hirschfeld, A. C. 1996, A&A, 313,Dominik, 841

G., & Ferrara, A. 1991, ApJ, 371,Einaudi, 571N. W. 1996, in Aspects of Dark Matter in Astro and ParticleEvans,

Physics, ed. H. V. Klapdor-Kleingrothaus (Singapore : World ScientiÐc),in press

A. C., & Nulsen, P. E. J. 1994, MNRAS, 269,Fabian, L33A. C., Pringle, J. E., & Rees, M. J. 1975, MNRAS, 172,Fabian, 15

S. M., & Rees, M. J. 1985, ApJ, 298,Fall, 18B. D., Mathews, G., & Schramm, D. N. 1997, ApJ, 483,Fields, 625

M., Hogan, C. J., & Peebles, P. J. E. 1996, Nature, 381,Fukugita, 489E. J., Gyuk, G., & Turner, M. 1996, Phys. Rev. D, 53,Gates, 4138B. S., & Gould, A. 1997, ApJ, 482,Gaudi, 83

O. E., & Silk, J. 1996, ApJ, 472,Gerhard, 34B. K., & Mould, J. R. 1997, ApJ, 482,Gibson, 98

P. F., & Langer, W. D. 1978, ApJ, 222,Goldsmith, 881D. S., & Freese, K. 1996, ApJ, 456,Gra†, L49D. C. 1975, MNRAS, 173,Heggie, 729D. C., & Ramamani, N. 1989, MNRAS, 237,Heggie, 757M. 1969, A&A, 2,He� non, 151

E. M., Huang, J. S., Gilmore, G., & Cowie, L. L. 1994, Nature, 371,Hu, 493Ph. 1996, Helvetica Phys. Acta, 69,Jetzer, 179H., Shapiro, P. R., Fall, S. M., & Rees, M. J. 1990, ApJ, 363,Kang, 488

S. D. 1996, ApJ, 467,Kawaler, L61E. J. 1997a, A&A, 332,Kerins, 7091997b, A&A, 328,ÈÈÈ. 5E. J., & Carr, B. J. 1994, MNRAS, 226,Kerins, 775

K. M., Wolfe, A. M., & Turnshek, D. A. 1995, ApJ, 440,Lanzetta, 435R. B. 1990, in Dynamics and Interactions of Galaxies, ed.Larson,

R. Wielen (Berlin : Springer), 1H. M., & Ostriker, J. P. 1986, ApJ, 310,Lee, 176

74 DE PAOLIS ET AL.

D., & Wood, R. 1968, MNRAS, 138,Lynden-Bell, 495E. 1994, ApJ, 428,Maoz, L5

R. B., & Silk, J. 1996, ApJ, 464,Metcalf, 218G. E., & Scalo, J. M. 1979, ApJS, 41,Miller, 513B., & Silk, J. 1995, ApJ, 442,Moore, L5

R., & McCammon, D. 1983, ApJ, 270,Morrison, 119S. D., & Lin, D. N. C. 1989, ApJ, 339,Murray, 933

T., et al. 1995, Nature, 378,Nakajima, 463D. A., Lepp, S., & Melnick, G. J. 1995, ApJS, 100,Neufeld, 132P. E. J., & Fabian, A. C. 1997, MNRAS,Nulsen, submittedC. P., et al. 1994, ApJ, 422,OÏDea, 467

J. P., Statler, T. S., & Lee, H. M. 1985, BAAS, 16,Ostriker, 946F., Salpeter, E. E., & Stahler, S. W. 1983, ApJ, 271, 632Palla, (PSS)F., & Stahler, E. E. 1988, in Galactic and Extragalactic Star Forma-Palla,

tion, ed. R. E. Pudeitz & M. Fich (NATO ASI Ser. C-232) (Dordrecht :Kluwer), 519

M., & Salucci, P. 1990, MNRAS, 247,Persic, 349D., Combes, F., & Martinet, L. 1994, A&A, 285,Pfenniger, 79

W. H., & Teukolsky, S. A. 1977, ApJ, 213,Press, 183G. D. 1996, New Astron., 1,Quinlan, 35

R., Zapatero Osorio, M. R., & Martin, E. L. 1995, Nature, 377,Rebolo, 129C., et al. 1997, A&A, 324,Renault, L69

H., Miley, G., & Van Ojik, R. 1996, ESO Messenger, 83,Ro� ttgering, 26

M., Olive, K., & Silk, J. 1990, ApJ, 353,Ryu, 81P. D. 1997, ApJ, 483,Sackett, 103D., et al. 1996, ApJ, 460,Saumon, 993

J. M. 1985, in Protostars and Planets II, ed. D. C. Black & M. S.Scalo,Mathews (Tucson : Univ. Arizona Press), 201

L. 1969, ApJ, 158,Spitzer, L1391987, Dynamical Evolution of Globular Clusters (Princeton : Prin-ÈÈÈ.

ceton Univ. Press)L., & Hart, M. H. 1971, ApJ, 164,Spitzer, 399L., & Mathieu, R. D. 1980, ApJ, 241,Spitzer, 618L., & Thuan, T. X. 1972, ApJ, 175,Spitzer, 31T. S., Ostriker, J. P., & Cohn, H. N. 1987, ApJ, 316,Statler, 626

J. S. 1985, IAU Symp. 113, Dynamics of Star Clusters, ed.Stodolkiewicz,J. Goodman & P. Hut (Dordrecht : Reidel)

C. M., Silk, J., Wood, M. A., & Winget, D. E. 1990, ApJ, 358,Tamanaha,164

M., Wyse, R., & Gilmore, G. 1996, MNRAS, 278,Unavane, 727Albada, T. S., & Sancisi, R. 1986, Phil. Trans. R. Soc. London A, 320,van

447Dishoeck, E. F., & Black, J. H. 1986, ApJS, 62,van 109

M., & Pesce, E. 1995, ApJ, 442,Vietri, 618I., & Salpeter, E. E. 1994, ApJ, 433,Wasserman, 670

J. P. 1977, A&A, 57,Zahn, 383

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Halo Dark Clusters of Brown Dwarfs and Molecular Clouds

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