Can Supermassive Black Holes Form in Metal-Enriched High-Redshift Protogalaxies ?
aa r X i v : . [ a s t r o - ph ] J u l Can Supermassive Black Holes Form in Metal-EnrichedHigh-Redshift Protogalaxies ?
K. Omukai , R. Schneider and Z. Haiman ABSTRACT
Primordial gas in protogalactic dark matter (DM) halos with virial temper-atures T vir ∼ > K begins to cool and condense via atomic hydrogen. Providedthis gas is irradiated by a strong ultraviolet (UV) flux and remains free of H and other molecules, it has been proposed that the halo with T vir ∼ K mayavoid fragmentation, and lead to the rapid formation of a supermassive blackhole (SMBH) as massive as M ≈ − M ⊙ . This “head–start” would helpexplain the presence of SMBHs with inferred masses of several × M ⊙ , poweringthe bright quasars discovered in the Sloan Digital Sky Survey at redshift z ∼ > T vir ∼ K are likely already enrichedwith at least trace amounts of metals and dust produced by prior star–formationin their progenitors. Here we study the thermal and chemical evolution of low–metallicity gas exposed to extremely strong UV radiation fields. Our results,obtained in one–zone models, suggest that gas fragmentation is inevitable abovea critical metallicity, whose value is between Z cr ≈ × − Z ⊙ (in the absenceof dust) and as low as Z cr ≈ × − Z ⊙ (with a dust-to-gas mass ratio of about0 . Z/Z ⊙ ). We propose that when the metallicity exceeds these critical values,dense clusters of low–mass stars may form at the halo nucleus. Relatively mas-sive stars in such a cluster can then rapidly coalesce into a single more massiveobject, which may produce an intermediate–mass BH remnant with a mass upto M ∼ < − M ⊙ . Subject headings: cosmology: theory — galaxies: formation — stars: formation National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan; [email protected] INAF–Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Florence, Italy; raff[email protected] Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA;[email protected]
1. Introduction
The discovery of bright quasars at redshifts z ∼ > × M ⊙ were already assembled when theage of the universe was less than ≈ HubbleSpace Telescope images (Richards et al. 2004) essentially rules out the hypothesis that mostof the sources experienced strong magnification by lensing (Comerford et al. 2002; Keetonet al. 2005).Relatively little time is available for the growth of several × M ⊙ SMBHs prior to z ∼
6, and their seed BHs must be present as early as z ∼
10 (e.g. Haiman & Loeb2001). As the SMBHs grow from high–redshift seed BHs by accretion, they are expectedto encounter frequent mergers. A coalescing BH binary experiences a strong recoil due togravitational waves (GWs) emitted during the final stages of their merger. The typical recoilspeed is expected to be v recoil ∼ >
100 km s − (and may be as large as 4 ,
000 km s − for specialBH spin configurations; see, e.g. Campanelli et al. 2007 and references therein), significantlyexceeding the escape velocity ( ∼ <
10 km s − ) from typical DM halos that exist at z ∼
10. Asa result, SMBHs are often ejected from their host halos at high redshift. The repeated loss ofthe growing seeds makes it especially challenging to account for the several × M ⊙ SMBHsat z ∼ > T vir ∼ > K, leadingto the rapid formation of SMBHs with a mass of M ≈ − M ⊙ . As primordial gas fallsinto these halos, it initially cools via the emission of hydrogen Ly α photons. Provided the gasis free of H molecules, its temperature will remain near T vir ∼ K. Bromm & Loeb (2003,hereafter BL03) performed hydrodynamical simulations of a metal– and H –free halo, witha mass of ∼ M ⊙ collapsing at z ∼
10, corresponding to a 2 σ Gaussian overdensity and to T vir ∼ K. Under these conditions, which may apply to some dwarf galaxies collapsing closeto the epoch of reionization, the primordial gas is marginally able to collapse and remains 3 –nearly isothermal. BL03 found that during the evolution, fragmentation of the gas cloudis very inefficient, leading at most to binary formation even with some degree of rotation.Thus, a super–massive star is expected to form, and evolve into a SMBH with a mass ashigh as M ≈ − M ⊙ . Oh & Haiman (2002) and Lodato & Natarajan (2006) have alsoshowed that if H formation is inhibited, a primordial-gas disk is stable to fragmentation anda single massive object is formed in accordance with BL03’s conclusion. Volonteri & Rees(2005) arrived at similar conclusions, by considering Bondi accretion onto a stellar seed BH,which can significantly exceed the Eddington rate at the gas density and temperature in asimilar halo. Finally, Begelman et al. (2006) and Spaans & Silk (2006) proposed differentmechanisms to form similarly massive BHs by the direct collapse of primordial, atomic gas.For reference, we note that the total (DM+gas) mass of halos with T vir = (1 − × K at z = 10 is M tot ≈ − M ⊙ , so that such SMBHs would represent ≈ . −
20% of the gasmass in these halos. We also note that in the WMAP5 cosmology, the age of the universeat z = 10 and z = 6 . ∼ . ∼ . × years (assuming Eddington accretion, and a radiative efficiency of 10%; see, e.g.,Haiman & Loeb 2001), a seed BH of M ≈ M ⊙ at z ∼ could easily grow to a supermassive BH of M ≈ × M ⊙ at z ∼ .
5, if fed uninterruptedly.
A crucial assumption in all of the above proposals is that H molecules cannot form asthe gas cools and condenses in the DM halo . This assumption can be justified in the presenceof a sufficiently strong far ultraviolet (FUV) radiation, so that molecular hydrogen (or theintermediary H − necessary to form H ) is photodissociated. The relevant criterion is thatthe photodissociation timescale is shorter than the H –formation timescale; since generically, t diss ∝ J and t form ∝ ρ , the condition t diss = t form yields a critical flux J ∝ ρ . In DM haloswith T vir ∼ < K, whose gas can not cool in the absence of H , the densities remain lowand H can be dissociated even when background flux is as low as J − ∼ − (e.g. Haiman,Rees & Loeb 1997; Mesinger et al. 2006; here J − is the flux just below 13 . − erg cm − sr − s − Hz − ). However, if a gas cloud is massive enough and has avirial temperature higher than ≈ α cooling. Even if the FUV field is initially above the critical value, molecularhydrogen can form, and dominate the gas cooling at a later stage during the collapse (Oh &Haiman 2002); the H –formation rate is furthermore strongly boosted by the large out–of–equilibrium abundance of free electrons in the collisionally ionized gas in these halos (Shapiro& Kang 1987; Susa et al. 1998; Oh & Haiman 2002). The critical flux required to keep thegas H –free as it collapses by several orders of magnitude therefore increases significantly;for halos with T vir ∼ K the value has been found to be J − ≈ − , depending on theassumed spectral shape (Omukai 2001, hereafter O2001; BL03). In halos exposed to suchextremely intense UV fields, the gas cloud is still able to collapse only via atomic hydrogen 4 –line cooling, namely Ly α and H − free–bound (f-b) emission (O2001).One possible source of such an intense UV field is the intergalactic UV backgroundjust before the epoch of cosmic reionization (BL03). The ionizing photon flux J +21 can beevaluated from the number density of hydrogen atoms in the intergalactic medium (IGM)and the average number of photons needed to ionize a hydrogen atom N γ , which, in general,is >
1, owing to recombinations in an inhomogeneous IGM. Using the escape fraction ofionizing radiation f esc , the flux J − just below the Lyman limit is given by J − = J +21 f esc ≃ f esc hc π N γ Y H ρ b m H ≃ × (cid:18) N γ (cid:19) (cid:18) f esc . (cid:19) − (cid:18) z (cid:19) , (1)where Y H = 0 .
76 is the mass fraction of hydrogen, m H is the proton mass, and ρ b is the baryondensity (assumed here to correspond to Ω b h = 0 . J − can approach the critical value at z ∼ >
10, provided f esc is small, and N γ islarge; f esc /N γ ∼ < − . Although the value of f esc is quite uncertain, in low–mass minihalos,the expectation is f esc ≈
1, as these halos are easily self–ionized, and most of their ionizingradiation escapes into the intergalactic medium (Kitayama et al. 2004; Whalen, Abel &Norman 2004). Observations of nearby star forming galaxies indicate lower values f esc < . T vir ∼ K, M ∼ M ⊙ halos forming close tothe epoch of reionization are built from lower–mass progenitors that had collapsed earlier.Many, and perhaps all of these halos should therefore be enriched with at least some traceamount of metals. Furthermore, is it unlikely that the strong critical FUV flux could begenerated and maintained without efficient star formation at higher redshifts (if J ∼ > was produced by accreting BHs, this would significantly overpredict the present–day softX–ray background; Dijkstra et al. 2004). It is well–known that adding metals and dust intoa primordial gas, even at trace amounts as low as Z ∼ − Z ⊙ , can significantly affect itscooling properties (Schneider et al. 2003; Omukai et al. 2005; Schneider et al. 2006). Twoindependent hydrodynamic simulations (Tsuribe & Omukai 2006; Clark, Glover, & Klessen2008) recently studied the fragmentation of metal–enriched collapsing protogalactic clouds,in the absence of an external FUV field, and found efficient fragmentation for Z ∼ > − Z ⊙ . 5 –In the present paper, our goal is to answer the following question: can cooling andfragmentation be avoided in metal–enriched T vir ∼ > K halos, irradiated by a strong FUVflux?
If so, this would suggest that supermassive black holes may form, similar to the metal–free case, in the more likely case of metal–enriched high–redshift protogalaxies. To investigatethis possibility, we here study the thermal and chemical evolution of low–metallicity gas,exposed to extremely strong UV radiation fields. We will evaluate the critical metallicity,above which fragmentation becomes unavoidable in the presence of a strong FUV flux.In §
2, we describe our one–zone modeling procedure. Our results are presented anddiscussed in §
3, first for the metal–free ( § § § §
4, we summarize our results and offer our conclusions.
2. Model2.1. Basics
We use the one–zone model described in Omukai (2001) to follow the gravitationalcollapse of gas clouds. The model includes a detailed description of gas–phase chemistry andradiative processes, and the effect of dark matter on the dynamics in a simplified fashion. Inaddition, in the present version of the model we have implemented the contribution of metallines and dust to gas cooling.In what follows, all physical quantities are evaluated at the center of the cloud. The gasdensity increases as dρ gas dt = ρ gas t col . (2)where the collapse timescale, t col , is taken to be equal to the free-fall time, t col = t ff ≡ r π Gρ , (3)and ρ is the sum of the gas and dark matter density. The dark matter density follows theevolution of a top–hat overdensity, ρ DM = 9 π (cid:18) z ta − cos θ (cid:19) Ω DM ρ crit (4)with 1 + z = (1 + z ta ) (cid:18) θ − sin θπ (cid:19) − / (5) 6 –(e.g., Chapter 8.2 of Padmanabhan 1993), where the turn-around and the virialization cor-respond to θ = π and 2 π , respectively. Although, strictly speaking, this is correct only inthe Einstein-de Sitter universe (Ω = 1), it does not cause a significant error in the high- z universe ( z &
10) we consider.The initial epoch of calculation is taken at the turn–around at redshift z ta = 17. Fromequation 5, the virialization and turn-around redshifts have the relation 1 + z vir = 2 − / (1 + z ta ); thus z vir ≃
10. In our calculation, the dark matter density is kept constant after reachingits virialization value 8 ρ DM ( z ta ). The initial values of the gas number density, temperature,ionization degree, and H fraction have been assumed to be n H = 4 . × − cm − , T = 21 K, y ( e ) = 3 . × − and y (H ) = 2 × − , respectively, to reflect conditions at the turn–aroundat z ta = 17. Some runs with initial temperature ten times higher (210K) are also performedto confirm independence of our main results from the initial temperature. The cosmologicalparameters are Ω DM = 0 .
24, Ω b = 0 .
04, and h = 0 . α emission, the central region whose evolution we intend to follow does not experience thevirialization shock in the spherically symmetric case (Birnboim & Dekel 2003). In more real-istic calculations, the outer regions can experience shocks and the temperature and electronfraction become higher than in our case. In addition, recent numerical calculations (e.g.Kereˇs et al. 2005) show that low-mass galaxies, especially at high-redshifts, obtain their gasthrough accretion predominantly along the large-scale filaments. Three-dimensional effectssuch as asymmetric accretion might affect the evolution at low densities. However, since weare considering halos with T vir ≃ K, which can marginally collapse by Ly α cooling, theshock is not strong: the temperature increase is modest and the electron fraction reachesat most . − (see Figures 5a and 5c in BL03). This additional electrons alter the earlyevolution for the J = 0 case. However, in the irradiated clouds, where H formation issuppressed, during the collapse by the Ly α cooling recombination proceeds until the freeelectron fraction reaches x e ≃ . × − n − / , the value set by the balance between therecombination and the collapse time t rec ∼ t col at 8000K. Thus, our results for moleculeformation and cooling are hardly affected.We adopt t ff as the collapse time scale just because it has been widely used in otherstudies (e.g., Palla et al. 1983). Note that the free–fall time (3) is the time for density of aninitially static cloud to reach infinity, while the dynamical timescale t col = ρ/ ( dρ/dt ) in thefree–fall collapse is t col , ff = 1 √ πGρ (6)in the limit where the density has become sufficiently larger than the initial value. Thus therate we adopted (3) is 3 π/ . t ff as the e–folding timefor density increase mimics the pressure effect. The assumption of nearly free-fall collapseis invalidated, and the collapse is slowed down, once the cloud becomes optically thick tocontinuum radiation. However, our result on the thermal evolution is not altered: with littleradiative cooling, the temperature is now determined by the adiabatic compression and thechemical cooling by dissociation and ionization, both of which are independent of the collapsetimescale. Moreover, the evolution after the cloud becomes optically thick is not relevant toour argument on fragmentation, which occurs at much lower density, in the optically thinregime.The overall size of the collapsing gas cloud (or of the roughly uniform density centralregion) determines its optical depth, and is therefore important for its thermal evolution.Here we assume the size equals the Jeans length, λ J = s πkT gas Gρ gas µm H , (7)where T gas is the gas temperature, µ is the mean molecular weight. Similarly, its mass isgiven by the Jeans mass M J = ρ gas λ . (8)Specifically, we assume that the radius of the cloud is R c = λ J / τ ν = κ ν R c = κ ν (cid:18) λ J (cid:19) . (9)In addition to dust absorption (see § κ ν (Table 1 of O2001): the bound-free absorption of H, He, H − ,H +2 , free-free absorption of H − , H, collision-induced absorption of H -H and H -He, Rayleighscattering of H, and Thomson scattering of electrons.The temperature evolution is followed by solving the energy equation: dedt = − p ddt (cid:18) ρ gas (cid:19) − Λ net ρ gas , (10)where e is the internal energy per unit mass e = 1 γ ad − kT gas µm H , (11) p is the pressure, γ ad is the adiabatic exponent, and Λ net is the net cooling rate per unitvolume. In addition to cooling and heating processes for the primordial gas, which include 8 –continuum emission, as well as emission by H and H lines, and chemical heating/cooling,the net cooling rate includes emission by C and O fine–structure lines Λ metal , by dust grainsΛ gr , and heating by photoelectric emission of dust grains Γ pe . Cooling by fine–structure linesof [CII] and [OI] is included as in Omukai (2000). Dust processes are described below in § , e , H + , H +2 ,H − , He, He + , and He ++ . We do not explicitly include the chemical reactions involving metals.Instead, all the carbon and oxygen is assumed to be in the form of CII and OI, respectively.Having a lower ionization energy (11.26 eV) than hydrogen, carbon remains in the formof CII in the atomic medium owing to photoionization by the background radiation. Wemaintained this assumption even in J = 0 runs, although carbon is expected to recombineand become neutral in these cases. The cooling rates by CII and CI fine–structure linesare within a factor of ≃ T & + + H ↔ H + + O , (12)keeps its ionization degree equal to that of hydrogen. In fact, the coefficient of the rightwardreaction being 6 . × − cm / sec, these reactions reach equilibrium only in ∼ n − yr.In a cold ( . a few 100K) and dense ( & − cm − ) environment, molecular coolantssuch as CO and H O may become important (Omukai et al. 2005). Since we are interestedhere in metal effects on warm ( & a few 1000K) atomic clouds, we neglect the contribution tocooling of metals in molecules. This simplification does not affect the early evolution of gasclouds, when the effects of metals induce a deviation from the primordial evolutionary trackat several 1000K. It is true that it may alter the predicted thermal behavior at later stages,when the gas has cooled significantly ( . Dust in the local interstellar medium (ISM) originates mainly from the asymptoticgiant–branch (AGB) stars, whose age is & z &
6. Athigher redshifts, supernovae (SNe) are considered to be the major dust factories. Indeed,the observed extinction law of high– z quasars and gamma–ray bursts can be well reproduced 9 –by this scenario (Maiolino et al. 2004, Stratta et al. 2007). Dust grains produced in SNejecta are more effective in cooling and H formation because of their smaller size and largerarea per unit mass (Schneider et al. 2006). However, their composition and size distributionare still affected by many uncertainties, such as the degree of mixing in the ejecta and theefficiency of grain condensation and their destruction by the reverse shock (Nozawa et al.2007, Bianchi & Schneider 2007).To be conservative, in this work the properties of dust, such as grain composition andsize distribution, are assumed to be similar to those in the solar neighborhood and its amountis reduced in proportion to the assumed metallicity of the gas clouds. Specifically, we adoptthe dust opacity model developed by Semenov et al. (2003). This model partly follows thescheme proposed by Pollack et al. (1994), which was used in Omukai et al. (2005), assumingthe same dust composition, size distribution and evaporation temperatures, but uses a newset of dust optical constants. Overall, the opacity curves of the two models are in goodagreement, the largest difference being at most a factor of two (see Semenov et al. 2003for a thorough discussion). The main dust constituents include amorphous pyroxene ([Fe,Mg]SiO ), olivine ([Fe, Mg] SiO ), volatile and refractory organics, amorphous water ice,troilite (FeS) and iron. The grains are assumed to follow a size distribution modified fromthat by Mathis, Rumpl, & Nordsieck (1977) with the inclusion of large (0.5 - 5) µ m grains.At each density and gas temperature, the dust is assumed to be in thermal equilibrium,and its temperature T gr , which is followed separately from the gas temperature, is determinedby the energy balance equation4 π Z κ a ,ν B ν ( T gr ) dν = Λ gas → dust + 4 π Z κ a ,ν J in ν dν. (13)Here Λ gas → dust is the energy transfer rate per unit mass from gas to dust due to gas–dustcollisions, which we take from Hollenbach & McKee (1979), κ a ,ν is the absorption opacityof dust, and J in ν is the mean intensity of the radiation field inside the cloud. Note thatΛ gas → dust also represents the net cooling rate of the gas, caused by the presence of dustgrains at temperature T gr . We model the external radiation field assuming a diluted thermalspectrum (i.e. a blackbody spectrum, scaled by an overall constant representing a meangeometrical dilution). Its shape is then fully described by only two free parameters, J , themean intensity at the Lyman limit ( ν H ) and T ∗ , the color temperature, J ex ν = J − [ B ν ( T ∗ ) /B ν H ( T ∗ )] erg cm − sr − s − Hz − . (14)In the following, we will consider two possible values for the radiation color temperature, T ∗ =10 K and 10 K, representing “standard” Population II stars and very massive PopulationIII stars, respectively. Given the mean intensity of the external radiation field J ex ν , the field 10 –inside the gas cloud is obtained as (see O2001), J in ν = J ex ν + ξ ν x ν S a ,ν ξ ν x ν , (15)where 1 − ξ ν is the single–scattering albedo, S a ,ν is the source function, x ν = max[ τ ν , τ ν ] , (16)and the optical depth τ ν includes both dust and gas opacities.The gas is heated by photoelectrons ejected from dust grains after absorption of FUVphotons. Following Bakes and Tielens (1994), we compute the photoelectric heating rate as,Γ (net)pe = Γ pe − Λ pe (17)= [10 − ǫG n H − . × − T . ( G T / /n ( e )) β n ( e ) n H ] Z/Z ⊙ (18)where ǫ = 4 . × − [1 + 4 × − ( G T / /n ( e )) . ] + 3 . × − ( T / − ) . [1 + 2 × − ( G T / /n ( e ))] , (19)and β = 0 . /T . . The Habing parameter G is defined as G = 4 π Z FUV J ν dν . . × − , (20)where the integral is over the FUV radiation in the range (5.12-13.6) eV. In the above formula(eq. 18), small ( < formation coefficient as, k gr = 6 . × − ( T / / f a Z/Z ⊙ . × − ( T + T gr ) / + 2 . × − T + 8 . × − T (21)where f a = [1 + exp(7 . × (1 / − /T gr ))] − . (22)
3. Results
In what follows, we will first discuss the results obtained for the thermal evolution ofmetal–free gas clouds, and then describe the effects induced by the presence of metals anddust grains. 11 –
The thermal evolution of metal–free clouds irradiated by a FUV radiation backgroundis expected to change with radiation temperature T ∗ and intensity J . The models witha radiation temperature of T ∗ = 10 K (10 K) are shown in Figure 1 (2, respectively) fordifferent values of intensity J .Initially, i.e. at low densities, the temperature increases adiabatically, because there isnot enough H to activate cooling. In the no radiation case, when the density is ∼ − and the temperature is ∼ is formed and, as a result, the temperaturedecreases. It is to be noted that the relatively low temperature where this condition is metdoes not contradict previous results (BL03). In fact, the predicted temperature of each fluidelement in the simulation of BL03 shows a large scatter at low densities. This scatter reflectsthe radial temperature gradient, and the central value, which we calculate here, correspondsto the lower boundary of the scattered points and it is in agreement with our result. Weexpect that the central temperature of the gas cloud does not reach the virial temperatureof the host dark matter halo since the innermost region starts to cool and collapse duringthe adiabatic compression and does not experience the virialization shock.As the external radiation intensity J increases, the onset of H cooling is delayedbecause higher densities and temperatures are required for H formation to compensate forthe photodissociation. If the UV intensity is below a threshold value, J , thr , which we findto be in the range 10 − for T ∗ = 10 K and (1 − × for T ∗ = 10 K, there is alwaysa density at which H cooling starts to become effective. The temperature then decreasesand eventually reaches the no–radiation evolutionary track, along which it evolves thereafter.On the other hand, if the radiation is stronger than the threshold value, H cooling neverbecomes important. In this case, atomic hydrogen cooling by H excitation (for . cm − )and H − free-bound (f-b) emission (for & cm − ), are the main cooling channels (see Figure3). In Figure 1, runs with higher initial temperature (210 K) are also shown (dotted lines).During the initial adiabatic phase, the temperature at a given density is proportional to itsinitial value, and thus higher in runs with higher initial temperature. However, after theonset of efficient radiative cooling, these initially different thermal evolutionary tracks soonconverge. At higher densities, the results are independent of the initial temperature (seeFigure 1).As it can be inferred from Figs. 1 and 2, we find that the threshold value, J , thr , islower for a radiation temperature of T ∗ = 10 K than for T ∗ = 10 K. Thus, for comparableradiation intensities, J , the lower T ∗ radiation has a stronger impact on the cloud evolution. 12 –To understand why this is the case, in Figure 4 we show the H and H − photodissociationrates, for the same intensity J = 1. The dilution factor W , defined by J ν ≡ W B ν ( T ∗ ),which was used in Omukai & Yoshii (2003), is also shown for reference. As the figure shows,the H − photodissociation rate decreases steeply with T ∗ , while the H photodissociation rateremains nearly constant. The H and H − photodissociation rate coefficients are k H ph = 1 . × J ν (12 . k H − ph = Z . πJ ν hν σ ν dν (24)where the lower limit on the latter integral, 0.755 eV, is the threshold energy for H − pho-todissociation. Since the radiation field is normalized at the Lyman limit (13.6 eV), k H ph isnot sensitive to T ∗ , whereas k H − ph , which reflects the radiation field above 0.755 eV, dependssignificantly on the adopted T ∗ . H formation proceeds via a two step process (H − channel),H + e → H − + γ, (25)and H − + H → H + e. (26)If H − is photodissociated, it can not activate the second step (26). This is the reason why alower T ∗ radiation field leads to a less efficient H formation rate and to a lower H fractionalabundance than a higher T ∗ radiation field.Cooling via H − f-b emission occurs through the radiative association reaction (25).Subsequently, H − is collisionally dissociated through the reactionH − + H →
2H + e, (27)and the whole process results in a net cooling by the emitted photon. Thus, as long as thecollisional dissociation rate exceeds the photodissociation rate, the H − f-b emission is hardlyquenched, even in the presence of a strong FUV radiation field. Due to the small opacity,H − f-b cooling becomes important only at high densities ( & cm − ) and temperatures ( ∼ several 10 K), where the above condition is always satisfied. Then, H − f-b cooling is notaffected by photodissociation.As shown in Figure 3, radiative cooling and compressional heating rates almost balancesfor a wide range of densities. Namely, in equation (10), the two terms in the right hand sideare almost cancels and the left hand side is much smaller than those terms. Suppose thatthe radiative cooling rate per unit volume depends on the density and temperature asΛ rad /ρ gas ∝ ρ α − T β . (28) 13 –Since the compressional heating rate − p ddt ρ gas = pρ gas t col ∝ ρ / T, (29)the thermal balance of those two terms results in the following temperature evolution: T ∝ ρ gas / − αβ − . (30)Both the H line and H − f-b emissions are very sensitive to temperature, and thus β >
1. Forthose collisional processes, α = 2 for fixed chemical abundances. With chemical evolution,it deviates from 2, but remains > /
2. Therefore, the exponent in equation (30) is negativefor the atomic-cooling track as long as the cloud is optically thin: the temperature decreaseswith density as observed in Figures 1 and 2. On the other hand, on the molecular-coolingtrack, α = 1 for densities higher than the critical value for the LTE. Thus, the temperatureincreases with density for n H & cm − .The existence of a threshold UV background and the discontinuity of thermal evolutionat this value are due to the presence of non-local thermodynamic equilibrium (non-LTE) toLTE transition of H ro–vibrational level population at ∼ cm − . When the gas density ishigher than this value, the cooling rate saturates and more H is needed to compensate forcompressional heating. In addition, after the LTE is reached, collisional dissociation rate isenhanced owing to a large H level population in the excited levels. Thus, if a strong FUVradiation delays H formation and cooling until the critical density for LTE is reached, afraction of the remaining H is collisionally dissociated. Thus the gas cloud is no longer ableto cool by H even at a later phase of the evolution. On the other hand, if the UV backgroundis slightly smaller than the threshold, the cloud begins to cool by H and the temperaturebegins to fall before the collisional dissociation effect becomes significant (see Figure 5b inOmukai 2001 for cooling rates by each process in such a case). The lower temperature allowsfurther H formation and resultant cooling. The cooling proceeds in this accerelated fashionand the temperature eventually reaches the molecular cooling track. This is the origin of thedichotomy between the atomic and molecular cooling tracks. To summarize, the main effectof the FUV radiation is to photodissociate H directly and to decrease the H formationrate through photodissociating H − . If these two processes inhibit H formation and coolinguntil the critical density for LTE is reached, the gas remains warm ( & several thousands K)and H is collisionally dissociated at higher densities. Thus the high density evolution is notaffected by the presence of the FUV field and depends only on the temperature at the H critical density. 14 – In this section, we show the effects induced by the presence of metals and dust grains onthe thermal evolution of gas clouds irradiated by a FUV field with a mean intensity largerthan J , thr . In what follows, the total metallicity is expressed relative to the solar value,as [M / H] ≡ log( Z/Z ⊙ ). Unless specified otherwise, the fractions of metals in the gas phaseand in dust grains are assumed to be the same as in the interstellar medium (ISM) of theGalaxy. Specifically, the number fractions of C and O nuclei in the gas phase with respect toH nuclei are y C , gas = 0 . × − Z/Z ⊙ and y O , gas = 3 . × − Z/Z ⊙ . The mass fractionof dust grains relative to the mass in gas is 0 . × − Z/Z ⊙ below the ice-vaporizationtemperature ( T gr .
100 K).In Figure 5 we present the thermal evolution of clouds with metallicity in the range − ≤ [M/H] ≤ − T ∗ = 10 K, J = 10 and (b) T ∗ = 10 K, J = 3 × ,respectively. Under these conditions, the clouds would collapse only via atomic coolingin the absence of metals or dust grains (see Figs. 1 and 2). For a metallicity as low as[M/H] . −
6, the predicted thermal evolution follows the metal–free track. In both panelsof Figure 5, deviations from the metal–free tracks start to appear at a density ∼ cm − when the metallicity is [M/H] ≃ − .
3. For the sake of comparison, thin lines show theexpected evolution in the absence of radiation for the same initial values of metallicity.At metallicity [M/H]= − .
3, although the temperature drops and eventually reaches themolecular-cooling track at ∼ cm − , this arrival is after the minimum in the molecularcooling at ∼ cm − . With a slightly higer metallicity of [M/H]= −
5, this arrival takesplace at ∼ − cm − , and the temperature subsequently decreases to the minimum in theno-radiation case. For higher metallicities, the temperature drop occurs at lower density andthe temperature minima becomes lower. In Figure 6 we show the cooling and heating ratescontributed by each process during the evolution of the cloud with T ∗ = 10 K, J = 10 and [M/H]=-5. Up to 10 cm − , cooling is dominated by the H line emission (denoted as“H” in the Figure; . cm − ) and H − f-b emission (“H − f-b”; & cm − ), and the cloudcollapses along the atomic cooling track (see Fig.5 a). However, at a density ∼ cm − ,cooling by the dust grain (“grain”) becomes dominant and causes the sudden temperaturedrop. Now the temperature is lower than that in the atomic cooling track, the H collisionaldissociation rate is also reduced, which causes a high equilibrium value of the H fraction.As a result of H cooling, the temperature decreases further, although this effect is almostcompletely balanced by heating due to H formation (“H form”). Note that fine-structureline cooling (“CII, OI”) is not important at such low metallicities (see the discussion below).Eventually, the thermal evolutionary tracks reach those of the corresponding metallicity inthe no–radiation case (shown as thin curves in Fig.5) and evolve along them thereafter. 15 –We also run models with an external UV field with parameters T ∗ = 10 K, J = 10 ,which is 10 times stronger than that considered in Fig.5 (a) and we found that the criticalvalue of the metallicity at which deviations from the metal–free evolution appear, Z cr ∼ × − Z ⊙ does not depend on the intensity of the FUV radiation as long as J > J , thr .Furthermore, for the same value of the metallicity, the evolutionary tracks at high densities( & cm − ) are independent of the type of external radiation, as can be seen comparing theresults in panels (a) and (b) of Figure 5. In fact, once the evolution has reached the densityat which molecular and atomic cooling tracks bifurcate ( & cm − ), collisional processesrather than radiative ones dominate the energy balance (see also § § / H] = − ∼ cm − is due to complete vaporization of grains, whichoccurs at ≃ ≃
130 K (at ∼ cm − for [M / H] = − ∼ cm − for [M / H] = −
4) are due to vaporization of water ice. This figureindeed shows that despite the high gas temperature, the dust temperature remains low at afew 10K, which allows survival of grains until very high densities.As discussed above, OI and CII line emission contributes negligibly to gas cooling inthe metallicity and density range where the effects of dust grains start to become relevant( Z cr ∼ × − Z ⊙ , n H ∼ cm − ). Metal–line cooling causes a deviation from the metal–free atomic track only when the metallicity reaches [M / H] ∼ −
3. In this case, since thetemperature track converges to the J = 0 track before the dust–cooling phase, two tem-perature minima appear at 10 cm − and 10 cm − (see Fig.5 a, b). In the absence of dustgrains, a higher fraction of metals is required to cool the gas at a rate such that the thermalevolution deviates from the atomic cooling metal–free tracks. To demonstrate this, we haveperformed a numerical experiment where we have suppressed the contribution of dust grainsto the energy balance of the collapsing clouds. The results for models with radiation fieldparameters of ( T ∗ = 10 K, J = 10 ) are shown in Figure 8. When the metallicity is 16 –below [M / H] ≃ − .
5, the temperature evolution is exactly the same as the metal-free one(shown by the 2 × − Z ⊙ track in the Figure). For higher metallicity, fine-structure linecooling becomes dominant when n H . cm − and the temperature drops abruptly by morethan two orders of magnitude. Therefore we find that the critical metallicity [M / H] ≃ − . ≃ × − Z ⊙ ) required to modify the thermal evolution is almost two orders of magnitudehigher than in models with dust. This level of metallicity is approximately the same as thevalue at which metal cooling rate exceeds the H cooling rate in clouds which are already col-lapsing by molecular cooling (Bromm & Loeb 2003b, Santoro & Shull 2006; Frebel, Johnson& Bromm 2007). The thermal properties of star–forming clouds have an important influence on how theyfragment into stars (Larson 2005). There is observational evidence that proto–stellar coreshave a mass spectrum which resemble the stellar initial mass function (IMF), indicating thatcloud fragmentation must be responsible for setting some fundamental properties of the starformation process (e.g., Motte, Andre, & Neri 1998; Lada et al. 2007; for the recent reviews,see Bonnell, Larson & Zinnecker 2006 and Elmegreen 2008).Roughly speaking, fragmentation occurs efficiently when the effective adiabatic index γ ≡ ∂ ln p/∂ ln ρ .
1, i.e., during the temperature drops, and almost stops when isothermalitybreaks ( γ &
1) as also shown by the simulations of Li, Klessen & Mac Low (2003). Thus,consistent with Schneider et al. (2002, 2003, 2006) we can adopt the density at which theequation of state first becomes softer than γ ≈ M frag = M J ( n frag , T frag ) ∝ T / n − / . (31)In the absence of an external FUV radiation field, the temperature of metal–free cloudsdecreases with density in the range 1cm − . n H . cm − and increases at higher densities,after the major coolant H has reached the LTE. Dense cores form around this density withtypical masses of 10 M ⊙ , which is close to the Bonnor–Ebert mass at this thermal state(Bromm, Coppi, & Larson 1999, 2002; Abel, Bryan, & Norman 2002). As the metallicityincreases to Z cr = 10 − Z ⊙ , dust–induced fragmentation leads to solar or sub–solar fragments(Schneider et al. 2006), making a fundamental transition in the characteristic mass scales ofproto–stellar cores. 17 –It is important to stress that the presence of a temperature dip in the thermal evolution,and the softening of the equation of state γ <
1, imply only the possibility of fragmentation.For example, fragmentation depends also on the initial conditions, and requires the existenceof sufficiently large initial density perturbations. In the first cosmological objects, which arebarely able to cool and collapse, fragmentation can be less efficient. Still, self–gravitatingcores of mass comparable with that predicted by the above criterion are observed to form inhigh–resolution 3D simulations (Abel, Bryan, & Norman 2002, Yoshida et al. 2006). Evenfor turbulent molecular clouds of solar metallicity, 3D simulations show that fragmentationis efficient when γ ≈ . γ increases to ≈ . γ , is presented in Figure 9 forclouds with initial metallicities [M/H]= −∞ , -6, -5.3, -5, -4, and -3, irradiated by a field withparameters T ∗ = 10 K and J ∗ = 10 , whose temperature evolution is shown in Fig. 5 a. Theapplication of the above arguments to predict the typical fragment mass from the thermalevolution of the clouds is not straightforward because, along the metal–free atomic coolingtracks and over a broad density range 10 − cm − , the effective adiabatic index remains γ ≃
1, although slightly below unity (0.95 - 1; see Fig. 9, top panel). If we adopt γ frag = 1 as thethreshold value of the effective adiabatic index for fragmentation, in this case fragmentationwould be expected to occur up to densities of ∼ cm − , leading to solar–mass fragments,as discussed by O2001 and Omukai & Yoshii (2003). In contrast, the numerical simulationsby BL03 show that down to the highest density reached by the simulations ( . cm − )fragmentation is very inefficient. Even with some degree of rotation, the cloud fragmentsat most into two pieces, resulting in a binary system. Although fragmentation might occurat higher densities, in BL03’s calculations neither efficient fragmentation leading to theformation of a star cluster, nor the growth of elongation of the clouds is observed. Wespeculate that this result is due to the following reasons. The objects considered by BL03are those only marginally able to collapse by atomic cooling, and thus are initially close tothe hydrostatic equilibrium. During this initial epoch, the Jeans mass is large, and densityand velocity perturbations are erased by pressure forces. In addition to this little initial seedperturbation, since γ is only slightly below unity, the growth of perturbation would be veryslow. Thus the perturbation might not grow enough to cause fragmentation.Note however that, although we find that along the atomic cooling tracks H − coolingis the dominant cooling agent at high densities, & cm − , this process is not consideredin the simulation of BL03, which implements only H Ly α cooling. To check whether thisomission might cause the lack of fragmentation, we have followed the evolution of a metal–free cloud under the influence of an external FUV radiation field with T ∗ = 10 K and J = 10 but turning off the H − cooling by hand. The result is shown in Figure 10. Withno H − cooling, the cloud follows a slightly higher temperature track when the density is 18 – & cm − . However, below ∼ cm − , the difference is small and it has a weak effect onthe cloud dynamics. Therefore the inclusion of H − cooling would not affect the results ofBL03’s simulation, which is limited to densities < cm − .On the basis of these considerations, we assume that for metal–free clouds irradiated bya strong FUV background, fragmentation does not occur during the atomic–cooling phase,where γ ≃
1, and it occurs only when the temperature drops more rapidly, where γ < γ frag <
1, by molecular cooling, that is when J ν < J ν, thr .In the metal–enriched, irradiated clouds we studied, the temperature dip due to dustcooling occurs at very high densities, 10 cm − , deep in the interior of the collapsing clouds,where pre–existing density perturbations might also be erased by pressure forces. However,in this regime we find γ . . / H] & −
5, when the thermal evolutionary tracks shownin Figure 5 suddenly deviate from the atomic–cooling track, in other words, when γ fallssufficiently below unity (Figure 9), the clouds begin a vigorous fragmentation, which thenlasts until the temperature increases again. The value of γ frag to cause fragmentation isuncertain as discussed above, but likely to be slightly below unity. In the following, for thesake of definiteness, adopt γ frag = 0 . γ = 0 . γ soon falls below . . γ frag , say, by ∼ .
1, leads to fragmentation densities whose differences are within anorder of magnitude. For example, when [M/H]= − n H (cm − ) =7 . ,
8, and 8.3 for γ frag = 0 . , .
8, and 0 .
9, respectively. We assume that fragmentation stopswhen γ exceeds unity again. For [M/H]= −
4, this occurs at n H ∼ cm − .The mass scale of the final fragments is given by the Jeans mass at the temperatureminimum, i.e., when γ exceeds unity. When the initial metallicity is [M / H] ≃ −
5, the tem-perature minimum corresponds to 300 K at n H = 10 cm − , and thus the typical fragmentmass is 0 . M ⊙ . As the metallicity increases, both the density and temperature at the frag-mentation scale decrease, being (10 cm − , 150K) for [M / H] ≃ −
4, and (10 cm − , 30K) for[M / H] ≃ −
3. However, the corresponding fragment mass scale remains ∼ . M ⊙ , becausethe variations of density and temperature almost cancel out (see eq. 31). In some cases, e.g.[M / H] ∼ − J > J , thr field, two fragmentation epochs (log n H = 3 . − . . − . / H] = −
3) appear, which corresponds to two dips in the temperature (or 19 – γ ) evolutionary track. The outcome of this kind of track is not clear without any numericalwork studying their effect. Here, we speculate that the first dip produces clumps as a resultof the fragmentation of clouds. Then the clumps fragments again into cores owing to thesecond dip.In the absence of dust, the temperature minimum appears at a lower density, n H ∼ cm − , and higher metallicity [M / H] ≃ − . − M ⊙ and the formation of sub–solar mass fragmentsis not possible in this case. This property of pre–stellar clouds enriched only by gas–phasemetals has been already proven to hold in the absence of external FUV fields (Schneider etal. 2006).To summarize, our results show that in the presence of a sufficiently strong FUV radia-tion field the collapse of metal–free clouds by molecular cooling is inhibited and it can proceedonly via atomic cooling. Under these conditions, cloud fragmentation is highly inefficient,leading at most to the formation of a binary system. The typical mass of pre–stellar cloudsis therefore 10 − M ⊙ and the formation of a super massive star, seed of a super massiveblack hole, is the likely outcome of the evolution (BL03). However, this scenario is alteredas soon as trace amounts of metals and dust grains are present in the collapsing clouds: dustcooling leads to fragmentation of the clouds into sub–clumps with mass as low as ∼ . M ⊙ already at a floor metallicity of Z cr ∼ × − Z ⊙ . This conclusion holds independently ofthe intensity and spectrum of the FUV radiation field. In the absence of dust, an enrichmentlevel of Z cr ∼ × − Z ⊙ is required for OI and CII line cooling to fragment the cloud; thefragments in this case are predicted to be relatively more massive, ∼ − M ⊙ . Since dust–induced fragmentation takes place at high densities, a dense proto–stellarcluster is expected to form (Omukai et al. 2005, Schneider et al. 2006, Clark et al. 2008). Asan example, when the initial metallicity of the collapsing cloud is [M / H] = −
5, the suddentemperature drop, where γ < γ frag = 0 .
8, begins at T drop ∼ n drop ≃ . cm − .At this stage, the size and mass of the cooling region, or proto–cluster, are given by thecorresponding Jeans length, λ J ≃ × − pc, and mass M cl ≃ M ⊙ . When a differentthreshold value γ frag is adopted, these quantities change, e.g., to λ J ≃ × − pc, and M cl ≃ M ⊙ for γ frag = 0 . λ J ∼ × − pc and M cl ∼ M ⊙ as typical values. After virialization, the proto–stellar cluster has a size half of this. Since each ultimate fragment has a typical mass of M frag ∼ . M ⊙ , which is set by the Jeans mass at the end of the fragmentation process 20 –( n H ∼ cm − ), we expect that up to N ∗ ∼ M cl /M frag ∼ ∼ .
01 pc. The difference between the formationepochs of each protostar is of the order of the free–fall time of the proto–cluster gas. Sincethe cluster begins to form in a dense cloud with density ∼ cm − , protostar formation issynchronized on a timescale of ∼
300 yrs.The fate of dense, compact star clusters has been discussed extensively in the literature(see, e.g. Rasio et al. 2004 for a recent review, focusing on the possibility of intermediate BH,IMBH, formation through a runaway collapse that is relevant in our case). It is importantto stress that, even assuming a star formation efficiency of order unity (which seems likelywhen the density exceeds & cm − ; Alves, Lombardi & Lada 2007), the stellar IMF willbe strongly affected by gravitational interactions, collisions and mergers. In fact, observedproperties of present–day star forming regions, as well as numerical simulations, suggestthat gravitational fragmentation is probably responsible for setting a characteristic stellarmass but the full mass–spectrum and the Salpeter–like slope of the IMF are most likelyformed through continued accretion and dynamical interactions in a clustered environment(see Bonnell et al. 2006 and references therein). Furthermore, in young and compact starclusters supermassive stars may form through repeated collisions (e.g. Portegies Zwart et al.1999, 2004; Ebisuzaki et al. 2001).We can therefore ask, what is the expected fate of the dense star cluster forming in ourclouds? The evolution of a star cluster with half–mass radius R cl and mass M cl proceeds onthe dynamical friction timescale (Binney & Tremaine 1987), t fric ≃ . × lnΛ yr (cid:18) r . (cid:19) (cid:18) R cl . (cid:19) − / (cid:18) M cl M ⊙ (cid:19) / (cid:18) m ∗ M ⊙ (cid:19) − , (32)which is the time required for a star with mass m ∗ , which is on the massive side of thespectrum, to sink from the radius r to the cluster center by gravitational interactions withbackground, less massive stars. In the following, the Coulomb logarithm lnΛ is taken to be ≃
7, a value typical for open clusters. In the above equation, the density profile is assumedto be isothermal, ρ ∝ /r . The dynamical friction timescale was originally derived for afixed background. What actually occurs for a cluster on this timescale is the equipartitionof kinetic energy among the member stars. Heavy stars move slowly and then drop deeperin the potential well, leading to mass segregation in the cluster. The stellar merger rate isgreatly enhanced if higher–mass stars reach the cluster center within the lifetime of a verymassive star, i.e. if t fric is less than a few Myr.Using the estimated size and mass of the proto–cluster at the onset of dust–inducedfragmentation for the [M/H] = -5 track, the dynamical friction timescale for a star initially 21 –at radius r can be rewritten more generically as t fric ≃ . × yr (cid:18) M r M ⊙ (cid:19) (cid:16) µ . (cid:17) / (cid:18) T drop (cid:19) − / (cid:18) m ∗ M ⊙ (cid:19) − , (33)where M r is the mass enclosed inside radius r and we have expressed M cl and R cl in termsof n drop and T drop (note that n drop drops out of the equation). Following Portegies Zwart etal. (2004), we assume that a very massive star can be formed by stellar mergers if, t fric < . (34)From equation (33), we infer that the inner region of mass M fric = 5 × M ⊙ (cid:16) µ . (cid:17) − / (cid:18) T drop (cid:19) / (cid:18) m ∗ M ⊙ (cid:19) / (35)satisfies the condition (34). The mass fraction of sinking stars relative to the cluster back-ground stars, f sink , is uncertain and probably depends on the stellar mass spectrum, butcan be safely assumed to be less than a half. In addition, not all the stars that sink tothe center are incorporated into the runaway merging object. Since merging events usuallyproceed via three– or more body interactions, where the runaway object and a star coalesceby kicking the lightest star (Portegies Zwart et al. 1999), the merging efficiency, f merg , withthe runaway object among stars fallen to the cluster center can be assumed to be about ahalf. Thus, the mass of the central object resulting from this process can be estimated as, M cen = f sink f merg M fric (36) ≈ . × M ⊙ (cid:18) f sink . (cid:19) (cid:18) f merg . (cid:19) (cid:18) T drop (cid:19) / (cid:18) m ∗ M ⊙ (cid:19) / . (37)As can be seen in Figures 5 and 9, dust–induced fragmentation in clouds with a metal-licity [M / H] . −
4, starts at T drop ∼ n drop & cm − and the correspondingproto–cluster mass is M cl < M ⊙ . That is, M cl < M fric and the entire cluster satisfies thecondition (34). In this case, the mass of the central object is limited by the mass of the clusterrather than by M fric and its final mass can be estimated as, M cen = f sink f merg M cl . M ⊙ .On the other hand, dust–induced fragmentation of clouds with metallicity, − . [M / H] . −
3, leads to proto–stellar clusters with masses M cl & M fric and the mass of the centralobject is given by equation (37). Thus, in this case a very massive star can form with M cen . M ⊙ . In either case, we note that the metal–poor, massive star ultimately form-ing at the center of the halo is likely to leave behind a seed BH remnant – except in a narrowrange of metallicity, where they produce a pair instability supernova – either by direct col-lapse or by fallback (Heger et al. 2003). For higher metallicity, there are two episodes of 22 –fragmentation; a metal–induced one at low–density and a dust–induced one at high–density.In this case, the mass scale of the star cluster is relatively low and a massive BH seed is notformed.As pointed out above, the critical metallicity levels we find in the case of strong FUVirradiation, at which the cloud behavior is modified from the metal–free case, is very similarto the critical metallicity found in earlier work for J = 0. It is therefore important to askwhether the presence of the flux will, in fact, make any difference to the ultimate fate ofthe cloud. At metallicities above [M / H] ≃ −
3, the flux has essentially no impact on theevolutionary track of clouds at high densities. For example, comparing the thin and thicksolid curves of Figure 5 (a) it is clear that when [M / H] = −
3, the flux has no effect at n H ∼ > cm − . At lower densities, we expect that the first fragmentation phase will occuras the Jeans mass drops from M J ∼ M ⊙ to ∼ M ⊙ at n H ∼ cm − in the J = 0case, and to ∼ M ⊙ at 10 cm − in the strong UV background case. Therefore, the size ofthe molecular clumps that form is larger in the J = 0 case. However, the thermal evolutionthereafter is the same: the molecular clumps will experience a second phase of fragmentationwhen the Jeans mass falls further, from M J ∼
10 M ⊙ to ∼ . ⊙ . Thus, for protostellar gasclouds with [M / H] ≃ −
3, the presence of an external UV field determines only the amountof gas in the envelope in which the ∼
10 M ⊙ star cluster is embedded.In contrast, the presence of a FUV background significantly affects the evolution ofprotostellar clouds with lower values of metallicity, [M / H] < −
3. In these models, theatomic gas experiences only the second phase of fragmentation induced by dust cooling, andtherefore more massive star clusters (10 − M ⊙ , increasing with metallicity) are formedcompared to those in the J = 0 limit (a few - 10 M ⊙ , depending weakly on metallicity). Inaddition to the size of the star clusters, the mass and physical conditions of the correspondingenvelopes are different: when J = 0, the star cluster is embedded in a molecular envelopeof 10 − M ⊙ with temperature of a few 100 K, while, in the presence of an external UVfield, the surrounding envelope is more massive (10 − M ⊙ ) and it is made by atomic gas atseveral 1000 K. As a consequence, the presence of a strong UV background case favors theformation of a more massive central star by stellar merger (larger stellar cluster mass andhigher T drop in eq. 37).Finally, our results and the discussion above suggest that the direct formation of asupermassive star or SMBH as massive as ∼ − M ⊙ , as envisioned in the metal–free case,is not possible when the metallicity is above a critical value, and the gas fragments intosmaller pieces. However, we note that if fragmentation of the inner regions of the collapsingprotogalaxy is not fully efficient, and if radiative and mechanical feedback from the starsdoes not expel the leftover gas from the nucleus, then the star cluster at the center of 23 –the halo, and its coalesced massive remnant star, can be embedded within a thick residualgaseous envelope with temperature T env ∼ several thousand K. Since this gas envelope isself–gravitating, with a temperature below the virial temperature, it can undergo dynamicalcollapse, and may still produce a SMBH either directly or by accretion onto the centralstellar–mass BH at the Bondi rate (see, e.g. Begelman et al. 2006 and discussion therein).In particular, in the UV irradiated case, the accretion rate onto the star cluster, and possiblyto the central coalesced star, is high – this is because of the high temperature in the envelope,and since the accretion rate of the self–gravitating gas is given by (Shu 1977):˙ M ≃ c G (38)= 4 × − M ⊙ / yr (cid:18) T env (cid:19) / . (39)In conclusion, higher initial mass, higher accretion rate, and larger amount of reservoir gasare more favorable for BH growth in the strong UV case than the case without radiation.
4. Summary and Conclusions
In this paper, we have investigated the thermal evolution and fate of proto–stellar gasclouds in T vir & K halos irradiated by a strong FUV background. Under these conditions,which may apply to some dwarf galaxies collapsing close to the epoch of reionization, wefind that: • The effect of an external UV background is to photodissociate H directly and todecrease the H formation rate through photodissociation of H − . When the UV back-ground reaches a critical threshold value, J , thr , these two processes inhibit H forma-tion and cooling until the critical density for LTE is reached. Thereafter, the gas cloudcan cool only via atomic hydrogen transitions. • For gas clouds of primordial composition, an external UV background with intensity J < J , thr only delays the onset of H formation and H cooling becomes importantat some (higher) density: fragmentation occurs at densities 10 cm − − cm − leadingto average fragment masses in the range 10 M ⊙ − M ⊙ , similarly to the case with J = 0. • For J > J , thr , not enough H is formed to activate cooling and the evolution ofprimordial clouds is controlled by atomic (H and H − ) cooling. The clouds collapsenearly isothermally with a temperature of several thousands K (“atomic track”) up 24 –to very high densities ∼ cm − . According to previous numerical calculation byBL03, the clouds collapse directly into a single 10 − M ⊙ object, leading to supermassive star and SMBH formation. A core–envelope structure inevitably developsunder these circumstances, and the star grows by accretion onto an initially smallinner core. During the accretion phase, radiative and mechanical feedback effects mightbecome important and eventually halt the accretion at some phase (e.g., Omukai &Palla 2003; McKee & Tan 2007). If so, the mass of the central object can remain farbelow 10 − M ⊙ . • Independently of the values and properties of the external FUV field (as long as it is J > J , thr ), deviations from the metal–free ”atomic track” start to appear when thegas is enriched by even trace amounts of metals and dust. When Z > Z cr ≃ × − Z ⊙ ,dust cooling induces fragmentation at n H ∼ cm − and a proto–stellar cluster isexpected to form with average proto–stellar (fragment) mass of ∼ . M ⊙ . If onlygas–phase metals are present, a two orders of magnitude larger value of metallicity isneeded, Z cr ∼ × − Z ⊙ , before CII and OI line–cooling induce a deviation fromthe ”atomic track”, leading to fragmentation at n H ∼ cm − and to proto–stellarclusters with average proto–stellar (fragment) mass of 10 − M ⊙ . • The physical processes responsible for the origin of a critical metallicity and of itsnumerical value are the same as those found in the absence of an external FUV field.However, due to the higher gas temperature, the final outcome of the proto–stellarcloud collapse can be significantly affected. Namely, if we assume that the size of theproto–stellar cluster formed by dust–induced fragmentation is set by the Jeans massat the onset of the rapid temperature drop, it depends on the intensity of the FUVbackground field: when J < J , thr , relatively small clusters are formed (a few - 10 M ⊙ )whereas when J > J , thr , very dense star clusters with masses 100 − M ⊙ areformed, at the center of which stellar coalescences are expected to occur. The centralmerger object might grow to a very massive star of a few 100 M ⊙ . • In addition, the presence of an external FUV background affects the physical conditionsof the envelope surrounding the proto-stellar clusters: when J < J , thr the envelopeis fully molecular, with a mass of 10 − M ⊙ and a temperature of a few 100 K.Conversely, when J > J , thr the envelope is made of atomic gas and reaches a massof 10 − M ⊙ and temperature of several 1000 K. The higher temperature and largergas reservoir favors BH growth by accretion, which can be as high as 10 − M ⊙ yr − .According to the above, the conditions that would allow the formation of the directformation of a SMBH are (i) to be hosted within a T vir ∼ K halo, which is ii) irradiated 25 –by a strong UV field with J > J , thr , and (iii) still metal and dust free, with Z < Z cr . Themain new result of the present paper is that if the metalicity is too high, so that condition(iii) does not hold, then instead of a SMBH, a dense cluster of low–mass star forms at thehalo nucleus. The stars in such a cluster may still rapidly coalesce into a single massivestar, which may produce an intermediate–mass BH remnant, but with a smaller mass of M ∼ < − M ⊙ .While the above conclusion that even trace amounts of dust enable cooling and frag-mentation of the proto–stellar clouds appear to be robust, the exact value of the thresholdmetallicity is vulnerable to the uncertain nature of dust in early protogalaxies (see alsoSchneider et al. 2006). However, it is interesting to note that when Z ≥ Z cr the characteris-tic fragment mass – which is related to the characteristic stellar mass – is highly insensitiveto environmental conditions, such as the presence of an external FUV radiation field, as alsorecently discussed by Elmegreen, Klessen & Wilson (2008).We warn the reader that our discussion on the nature and evolution of the resultingproto–stellar cluster is still speculative as it is based on a few numerical experiments whichapply to dense stellar systems in present–day star forming regions (see the discussion andreferences in section 3.4). For example, we assume that the size of the cluster is set at theonset of the efficient cooling phase, which appears plausible but has not yet been confirmed.In particular, in models with J > J , thr and Z > − Z ⊙ (when both dust grains andgas-phase metals are present), the thermal evolution curves appear to have two separatetemperature minima, which correspond to metal– and dust– induced cooling and fragmen-tation. The fate of these collapsing clouds is at present unknown, and dedicated numericalsimulations would be highly desirable. If proto-stellar clusters are indeed formed under theconditions that we suggest, the formation and the nature of a central object by repeatedcollisions and accretion is highly uncertain. For example, the fraction of stars on the mas-sive side of the spectrum which falls to the cluster center (i.e., f sink in eq. 37) by dynamicalfriction is unknown and may vary significantly depending on the IMF.Despite the above uncertainties, our results suggest that even trace amount of metalspreclude the rapid formation of SMBHs as massive as M ≈ − M ⊙ in protogalactic halos.While such promptly appearing SMBHs would help solving the puzzle of the M ∼ > M ⊙ quasar black holes at z ∼ >
6, our results suggest that the low–metallicity halos may insteadproduce dense stellar clusters – the cluster may coalesce to produce an IMBH, but still witha much lower mass of M ≈ − M ⊙ .This study is supported in part by the Grants-in-Aid by the Ministry of Education,Science and Culture of Japan (16204012, 18740117, 18026008, 19047004:KO), by NASA 26 –through grant NNG04GI88G (to ZH) and by the Pol´anyi Program of the Hungarian NationalOffice for Research and Technology (NKTH). 27 – REFERENCES
Abel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93Alves, J., Lombardi, M., & Lada, C. J. 2007, A&A, 462, L17Bakes, E. L. O. & Tielens, A. G. G. M. 1994, ApJ, 427, 822Begelman, M., Volonteri, M., & Rees, M. J. 2006, MNRAS, 370, 289Binney, J. & Tremaine, S. 1987, Galactic Dynamics (Princeton Univ. Press)Bianchi, S. & Schneider, R. 2007, MNRAS, 378, 973Birnboim, Y. & Dekel, A. 2003, MNRAS, 345, 349Bonnell, I. A., Larson, R. B. & Zinnecker, H. 2007, Protostars and Planets V, B. Reipurth,D. Jewitt, and K. Keil eds., University of Arizona Press, Tucson, p.149-164Bromm, V., Coppi, P. S., & Larson, R. B. 1999, ApJ, 527, L5Bromm, V., Coppi, P. S., & Larson, R. B. 2002, ApJ, 564, 23Bromm, V. & Loeb, A. 2003a, ApJ, 596, 34 (BL03)Bromm, V. & Loeb, A. 2003b, Nature, 425, 812Chandrasekhar, S. & Fermi, E. 1953, ApJ, 118, 116Clark, P. C., Glover, S. C. O., & Klessen, R. S. 2008, ApJ, 672, 757Comerford, J., Haiman, Z., & Schaye, J. 2002, ApJ, 580, 36Dijkstra, M., Haiman, Z., Rees, M. J., & Weinberg, D. H. 2004, ApJ, 601, 666Dijkstra, M., et al. 2008, in preparationDunkley, J., et al. 2008, ApJ, submitted, arXiv.org:0803.0586Ebisuzaki, T. et al. 2001, ApJ, 562, L19Elmegreen, B. G. 2008, The Evolving ISM in the Milky Way and Nearby Galaxies: Recyclingin the Nearby Universe, arXiv e-print (arXiv:0803.3154)Elmegreen, B. G., Klessen, R. S., Wilson C. D. 2008, ApJ, in press, arXiv e-print(arXiv:0803.4411) 28 –Fan, X. 2006, New Astronomy Reviews, 50, 665Frebel, A., Johnson, J. L., & Bromm, V. 2007, MNRAS, 380, L40Glover, S. C. O. 2008, in ”First Stars III”, eds. B. O’Shea, A. Heger & T. Abel, AmericanInstitute of Physics Press, 25Haiman, Z 2004, ApJ, 613, 36Haiman, Z., & Loeb, A. 2001, ApJ, 552, 459Haiman, Z., Rees, M. J., & Loeb, A. 1997, ApJ, 484, 985Heger, A., et al. 2003, ApJ, 591, 288Hollenbach, D. & McKee, C. F. 1979, ApJS, 41, 555Inoue, A. K., Iwata, I., Deharveng, J.-M., Buat, V., Burgarella, D. 2005, A&A, 435, 471Inutsuka, S. & Miyama, S. M. 1997, ApJ, 480, 681Jappsen, A. K., Klessen, R. S., Larson, R. B., Li, Y., & Mac Low, M. M., 2005, A&A, 435,611Keeton, C., Kuhlen, M., & Haiman, Z. 2005, ApJ, 621, 559Kereˇs, D., Katz, N., Weinberg, D. H., & Dav´e, R. 2005, MNRAS, 363, 2Kitayama, T., Yoshida, N., Susa, H., & Umemura, M. 2004, ApJ, 613, 631Lada, C. J., Muench, A. A., Rathborne, J. M., Alves, J. F., & Lombardi, M. 2007, ApJ, inpress, arXiv.org:0709.1164Larson, R. B., 1985, MNRAS, 214, 379Larson, R. B., 2005, MNRAS, 359, 211Leitherer, C., Ferguson, H. C., Heckman, T. M., Lowenthal, J. D. 1995, ApJ, 454, L19Li, Y., Klessen, R. S.& Mac Low, M. M., 2003, ApJ, 592, 975Lodato, G., & Natarajan, P. 2006, MNRAS, 371, 1813McKee, C., F. & Tan, J. C., 2007, preprint (astro-ph:2007arXiv0711.4116)Mathis, J. S., Rumpl, W. & Nordsieck, K. H., 1977, ApJ, 217, 425 29 –Maiolino, R., Schneider, R., Oliva, E., Bianchi, S., Ferrara, A., Mannucci, F., Pedani, M. &Roca Sogorb, M., 2004, Nature, 431, 533Mesinger, A., Bryan, G. L., & Haiman, Z. 2006, ApJ, 648, 835Miyama, S. M., Narita, S., & Hayashi, C. 1987, Prog. Theor. Phys., 78, 1273Motte, F., Andre, P., & Neri, R. 1998, A&A, 336, 150Nozawa, T. et al. 2007, ApJ, 666, 955Nagasawa, M. 1987, Prog. Theor, Phys. 77, 635Oh, S. P., & Haiman, Z. 2002, ApJ, 569, 558Omukai, K. 2000, ApJ, 534, 809Omukai, K. 2001, ApJ, 546, 635 (O2001)Omukai, K. & Palla, F. 2003, ApJ, 589, 677Omukai, K. & Yoshii, Y. 2003, ApJ, 599, 746Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626, 627Pollack, J. B., Hollenbach, D., Bechwith, S., Simonelli, D. P., Roush, T., & Fong, W. 1994,ApJ, 421, 615Portegies Zwart, S. F., Makino, J., McMillan, S. L. W. & Hut, P. 1999, A&A, 348, 117Portegies Zwart, S. F., Baumgardt, H., Hut, P., Makino, J. & McMillan, S. L. W. 2004,Nature, 428, 724Rasio, F. A., Freitag, M., & G¨urkan, M. A. 2004, in Carnegie Observatories AstrophysicsSeries, Vol. 1: Coevolution of Black Holes and Galaxies, ed. L. C. Ho (Cambridge:Cambridge Univ. Press), p. 138Richards, G., Strauss, M. A., Pindor, B., Haiman, Z., Fan, X., Eisenstein, D., Schneider, D.P., Bahcall, N. A., Brinkmann, J., & Brunner, R. 2004, AJ, 127, 1305Santoro, F., & Shull, J. M. 2006, ApJ, 643, 26Schneider, R., Ferrara, A., Natarajan, P., & Omukai, K. 2002, ApJ, 571, 30Schneider, R., Ferrara, A., Salvaterra, R., Omukai, K., Bromm, V. 2003, Nature, 422, 869 30 –Schneider, R., Omukai, K. Inoue, A. K., & Ferrara, A. 2006, MNRAS, 369, 1437Semenov, D., Henning, Th., Helling, Ch., Ilgner, M., & Sedlmayr, E. 2003, A&A, 410, 611Shapiro, S. L. 2005, ApJ, 620, 59Shapiro, P. R., & Kang, H. 1987, ApJ, 318, 32Shu, F. H. 1977, ApJ, 214, 488Spaans, M., & Silk, J. 2006, ApJ, 652, 902Stratta, G., Maiolino, R., Fiore, F. & D’Elia, V. 2007, ApJ, 661, L9Susa, H., Uehara, H., Nishi, R., Yamada, M. 1998, PThPh, 100, 63Tielens, A. G. G. M. & Hollenbach, D. J. 1985, ApJ, 291, 722Tohline, J. E. 1980, ApJ, 239, 417Volonteri, M., & Rees, M. J. 2005, ApJ, 633, 624Volonteri, M., & Rees, M. J. 2006, ApJ, 650, 669Yoo, J., & Miralda-Escud´e, J. 2004, ApJ, 614, 25Yoshida, N., Omukai, K., Hernquist, L., Abel, T. 2006, ApJ, 652, 6Whalen, D., Abel, T., & Norman, M.L. 2004, ApJ, 610, 14Willott, C. J., McLure, R. J., & Jarvis, M. J. 2003, ApJ, 587, L15
This preprint was prepared with the AAS L A TEX macros v5.2.
31 –Fig. 1.— Temperature evolution of metal-free clouds irradiated by a UV flux. The spec-tral shape is that of a black–body spectrum with T ∗ = 10 K. Models are shown withFUV intensities at the Lyman limit of J = 0 , , ,
100 and 10 , in the usual units of10 − erg cm − sr − s − Hz − (solid and dashed curvesfrom bottom to top; see the legendin the panel). Diagonal dotted lines correspond to different constant Jeans mass. Modelswith higher initial temperature (210 K in contrast to 21 K in the fiducial models) are alsoshown by dotted lines. For those with J = 10 ,
100 and 10 , the temperature evolution at 32 – n H & . cm − completely overlaps with that predicted by the fiducial models.Fig. 2.— The same as Figure 1, except assuming a harder spectrum, with T ∗ = 10 K andintensities J = 0 , , , , , , and 3 × (solid and dashed curves from bottomto top). 33 –Fig. 3.— Contributions of various processes to the total cooling rate, as a function ofthe number density, for a metal-free cloud irradiated by an extremely intense FUV field( T ∗ = 10 K, J = 10 ; with the temperature evolution shown in Figure 1). The meaning ofthe symbols is as follows: “compr” indicates compressional heating; “H − f-b” cooling by H − free-bound emission; “H” cooling by H line emission; “H ” cooling by H line emission. 34 –Fig. 4.— H and H − photodissociation rate coefficients as a function of radiation temperaturefor J = 1. Also shown for reference are the Habing parameter G and dilution factor W . Fora fixed intensity J = 1 at 13.6 eV, the H photodissociation coefficient is almost constantwith T ∗ while the H − coefficient is about four orders of magnitude higher for T ∗ = 10 K, andthus reduces H formation by a large factor (see text). 35 – 36 –Fig. 5.— The temperature evolution of clouds with initial metallicity [M/H]= -6 (solid),-5.3 (dotted), -5 (short-dashed), -4 (long-dashed), and -3 (dash-dotted) irradiated by a FUVfield with (a) T ∗ = 10 K, and J = 10 , and (b) T ∗ = 10 K, and J = 3 × . Thin curvesshow the results obtained without an external FUV field for initial metallicities [M/H]= -5.3,-5, -4, and -3 (the same line types as the irradiated cases with the same metallicity). Dueto photoelectric heating, when [M/H]=-3, the temperature at the lowest densities is higherthan in the other models. 37 –Fig. 6.— Cooling and heating rates contributed by each process during the collapse of a cloudwith a metallicity of [M/H]=-5 and external FUV radiation with ( T ∗ = 10 K, J = 10 ).The corresponding temperature drop at n H ∼ cm − (shown in Figure 5) is caused bydust cooling. The meaning of the symbols is as follows: “compr” indicates compressionalheating; “grain” cooling by dust thermal emission; “H − f-b” cooling by the H − free-boundemission; “CII, OI” cooling by the CII and OI fine-structure line emission; “H” cooling bythe H line emission; “H ” cooling by the H line emission; “H form” heating by the H formation. 38 –Fig. 7.— The dust temperature of clouds (solid curves) with metallicity [M/H]=-6, -5,and -4 and with an external radiation of ( T ∗ = 10 K, J = 10 ) as a function of density.For the same models the dotted lines indicate the corresponding gas temperatures. Thedust temperature curve disappears at about n H ∼ cm − for [M/H]=-6, reflecting thevaporization of the grains.Fig. 8.— Temperature evolution of clouds when all metals are assumed to be in the gasphase (i.e. no dust). The radiation parameters are T ∗ = 10 K, J = 10 . Models with 39 –initial metallicities Z = 2 × − Z ⊙ (solid), 3 × − Z ⊙ (dashed), and 10 − Z ⊙ (solid) areshown (from top to bottom).Fig. 9.— Evolution of effective adiabatic indices γ = ∂ ln p/∂ ln ρ for the models shown inFigure 5(a), i.e., clouds irradiated by a field with parameters T ∗ = 10 K and J ∗ = 10 ,with metallicities [M/H]= −∞ , -6, -5.3, -5, -4, and -3 (from top to bottom). The casesof [M/H]= −∞ and -6 are identical and shown in the same panel (top). The horizontallines indicate γ = 0 . γ becomes & − free-bound cooling on the temperature evolution. The evolutionof a metal-free gas irradiated by a FUV field with ( T ∗ = 10 K, J = 10 ) is shown with andwithout the H −−