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Draftversion January31,2012 PreprinttypesetusingLATEXstyleemulateapjv.5/2/11 DARK MATTER AS AN ACTIVE GRAVITATIONAL AGENT IN CLOUD COMPLEXES Andrés Sua´rez-Madrigal1, Javier Ballesteros-Paredes, Pedro Col´ın, and Paola D’Alessio CentrodeRadioastronom´ıayAstrof´ısica,UniversidadNacionalAutónomadeMéxico, Apdo. Postal72-3(Xangari),Morelia,Michocań,México. C.P.58089 Draft version January 31, 2012 ABSTRACT 2 Westudytheeffectthatthedarkmatterbackground(DMB)hasonthegravitationalenergycontent 1 and,ingeneral,onthestarformationefficiencyofamolecularcloud(MC).We firstanalyzetheeffect 0 that a dark matter halo, described by the Navarro et al. (1996) density profile, has on the energy 2 budget of a spherical, homogeneous, cloud located at different distances from the halo center. We n found that MCs located in the innermost regions of a massive galaxy can feel a contraction force a greater than their self-gravity due to the incorporation of the potential of the galaxy’s dark matter J halo. WealsocalculatedanalyticallythegravitationalperturbationthataMCproducesoverauniform 7 DMB (uniform at the scales of a MC) and how this perturbation will affect the evolution of the MC 2 itself. The study shows that the star formation in a MC will be considerably enhanced if the cloud is located in a dense and low velocity dark matter environment. We confirm our results by measuring ] the star formation efficiency in numerical simulations of the formation and evolution of MCs within R different DMBs. Our study indicates that there are situations where the dark matter’s gravitational S contribution to the evolution of the molecular clouds should not be neglected. . h Subject headings: ISM: clouds — evolution — dark matter p - o 1. INTRODUCTION itational term is the gravitationalenergy, assuming that r t Agreateffortfromtheastronomicalcommunityiscur- the cloud is, in practice, an isolated entity. However, s different recent works have showed that the media sur- a rently directed towards the detailed understanding of [ star formation, in particular to the precise mechanisms rounding the dense structures can importantly alter the evolution of such entities (e.g., Burkert & Hartmann involved in the conversion of some part of a molecu- 2 2004; Gómez et al. 2007; Va´zquez-Semadeni et al. 2007; lar cloud’s mass into stars. The evolution of molecular v Heitsch & Hartmann 2008; Ballesteros-Paredeset al. clouds(MCs)andtheformationofstarswithinthemare 3 2009b,a). Intheseworks,theinfluenceofbarionicmatter thought to be regulated by actors as diverse as gravity, 9 outside the region of interest is studied. 1 large-scaleflows, turbulence, magnetic fields, and stellar On an apparently completely different topic, evidence 4 feedback, among others (see reviews by Shu et al. 1987; from a wide range of observations has long led as- . Va´zquez-Semadeni 2010). 1 tronomerstoargueinfavoroftheexistenceofmuchmore While evaluating the dynamical state of a molecular 0 matterthanisactuallyseen: theelusivedarkmatter. Al- cloud due to all relevant forces acting on it, the Virial 2 though the very nature of this mass component has not Theorem is frequently invoked to define an equilibrium 1 been discovered yet, it is presumed to smoothly perme- condition by equaling the gravitational energy W of the : atethe majorityofgalaxiesandinmanycasestogreatly v cloud to twice its kinetic energy K: surpass their visible extension, while being their domi- i X nant mass component. Compared to the sizes of giant 2K =−W. (1) r Galactic molecular clouds, dark matter halos are huge a Traditionally, this relationship has been interpreted and, for most practical purposes, homogeneous. Thus, in terms of clouds being in a state of quasi-equilibrium one would expect that such a distribution will not cause and long-lived entities, and deviations from this condi- a substantial gravitationaleffect on the cloud. tion are assumed to mean that the cloud is collapsing or In the present contribution, the idea that dark matter expanding, depending on which term is dominant (e.g., halosmayhaveaneffectontheenergycontentofthegas Myers & Goodman 1988a,b; McKee & Zweibel 1992). in galaxies is investigated. As a first approach, we focus However, it has been demonstrated that a cloud full- ourstudyinthe centerofmassivegalaxies,wherethe ef- filing the so-called equilibrium condition will not nec- fectisexpectedtobethegreatest. Acuspeddarkmatter essarily remain dynamically stable (Ballesteros-Paredes profilesuchasthe one proposedbyNavarro et al.(1996, 2006). In fact, numerical simulations show that clouds hereafter NFW) would provide the central region of a collapsing in a chaotic and hierarchicalway will develop halo with a potential well which could importantly con- sucha relationship(e.g., Va´zquez-Semadeni et al.2007; tribute to the gravitational energy of a molecular cloud Ballesteros-Paredeset al. 2011a,b). located around there. Furthermore, when calculating the Virial Theorem for In addition, we explore the idea that a barionic mass MCs, it is a common practice to consider that the grav- concentration is able to introduce a perturbation in an otherwise homogeneous dark matter distribution. As [email protected] such,themerepresenceofamolecularcloudwouldmod- 2 Sua´rez-Madrigalet al. ify the dark matter halo in which it is embedded. This termis traditionally assumedto be equalto the gravita- perturbationwouldinturn place the cloudin a localex- tional energy of the cloud: ternalpotentialwell,whichwouldplayasanextraagent in the struggle that determines its dynamical state. In E = 1 ρ Φ d3x, (3) thiscase,theexternalmediumtotheMCwillcontribute grav 2 cl cl ZV toitscollapse. Whencomparedtoasituationidenticalto where ρ represents the density of the cloud and Φ thisonebutwithoutthedarkmaterbackground(DMB), cl cl the potential energy that it generates. This assump- one would expect that a MC in this scenario would dis- tion considers that the relevant gravitational poten- playanenhancedcapacitytoformstars,sincethereisan tial is due only to the mass of the cloud, which is a extrafactor workingin the same directionas the cloud’s valid approximation for an isolated mass distribution internal gravitational energy. As such, measuring the (Ballesteros-Paredes 2006). However, in a given galaxy, star formation efficiency (SFE) of clouds embedded in there is a considerable amount of material (both bari- different DMBs would hint towards the importance of onic and dark matter) coexisting with a MC. Thus, fol- the environment in their evolution in each case. We an- lowingBallesteros-Paredeset al. (2009b), one may write alytically study the situation and give light as to what the gravitationalpotential Φ of a MC in a galaxy as the physical conditions would be required for this to be an potentialduetothemassoftheclouditself,Φ ,plusthe important effect. cl contribution of any other external sources, Φ , Furthermore, we present results from numerical simu- ext lations of molecular cloud formation and evolution from Φ=Φ +Φ , (4) two convergent monoatomic flows, placed in different cl ext dark matter contexts. Although the DMBs explored are such that the gravitationalterm entering the Virial the- notveryrealistic,theyaredesignedtotesttheanalytical orem becomes predictions and illustrate the importance of accounting fortheexternalgravitationalpotentialinmolecularcloud W =E +W . (5) grav ext analysis. where The structure of the article is as follows: in section §2 we introduce the tidal energy term that is part of ∂φ the total gravitational energy of a matter distribution. W ≡− ρ x ext dV, (6) ext cl i ∂x In section §3 we describe a semi-analytical procedure to ZV i evaluate the tidal contributionthat the complete haloof is the tidal energy, i.e., the gravitational energy con- a galaxy can have on a molecular cloud and apply it to tained in the cloud, but due to any mass distribution a few scenarios. In section §4.1 we analytically derive external to the cloud. As explained with detail in an expressionto asess the tidal effect of the dark matter Ballesteros-Paredeset al. (2009b), the external gravita- background perturbation caused by the molecular cloud tionalenergy caneither contribute to the collapse of the itself, while section §4.2 evaluates this effect in different cloudor to its disruption, depending onthe concavityof dark matter background environments. In section §5 we the external gravitationalpotential where it is located. present numerical simulations of cloud formation, in or- Inthefollowingsections,thetidalenergyfeltbyagiven der to test the analytical predictions in a more realistic MCwillbecomparedtoitsinternalgravitationalenergy; scenario,andin§6weshowtheresultsobtained. Finally, its relative importance will determine the role that any §7 gives general conclusions for the two studied effects. materialsurroundingaMCwillplayinits evolutionand future star formation. 2. FULL GRAVITATIONAL CONTENT OF A MOLECULAR CLOUD IN A GALAXY 3. TIDAL ENERGY IN THE POTENTIAL WELL OF A DARK MATTER HALO Molecular clouds, like any other physical system, are bound to follow all the internal and external forces act- To evaluate the effect that dark matter might have on ing on them and determining their behavior over time. starformation,wefirstconsideratoymodelconsistentof Along the presentarticle, the totalgravitationalcontent asphericalmolecularcloudembeddedinasphericaldark forMCssituatedindifferentenvironmentswillbe evalu- matter halo, with the cloud located at a distance s from ated, in order to look for the effect that their surround- thehalo’scenter. Thehalodensityprofileisgivenbythe ings can have on their evolution. Hence, this section de- simulation-inspired NFW density profile (Navarro et al. rivesanexpressionforthetidalenergycomponent,which 1996): will allow for an evaluation of its gravitational effect on ρ s the cloud. ρNFW(r)= (r/r )(1+r/r )2, (7) With the intention of determining a MC’s dynamical s s condition, an energy balance is commonly performed to where r is a scale radius,the radius where the logarith- s it by invoking the Virial Theorem. In this theorem, the mic derivative of the density is equal to −2, and ρ is a s gravitationalenergytermW consideredforamassdistri- characteristic density, the density at r =r . The profile s bution of volume V and density ρ, embedded in a gravi- can be expressed in a dimensionless form by tational potential Φ is given by δ (η)=η−1(1+η)−2, (8) NFW ∂Φ W =− ρ xi ∂x d3x (2) where δNFW(η) ≡ρNFW(r)/ρs and η ≡r/rs. As seen in ZV i Fig. 1 or eqs. (7) and (8), the NFW profile has a cusp (see, e.g., Shu et al. 1987). When applied to MCs, this at the center. It goes to infinity as η−1 for η →0. Dark matter in cloud complexes 3 10 We have,intotal,fiveparametersavailabletoexplore: δNFW = η-1(1+η)-2 two NFW halo parameters (characteristic density and radius of the density profile), two molecular cloud prop- 1 erties(densityandextentofthecloud),andthedistance of the cloud from the center of the halo (s). We choose tofixfirstthehalocharacteristicsandthedistancesand δ 0.1 calculate W /E values for a range of cloud’s parame- ext g ters. We presentenergyratiomaps fortypicalmolecular cloud radii and densities, one per halo type and cloud 0.01 separation. ThegravitationaleffectofaMilkyWay-typedarkmat- terhaloonaMCisfirstpresented. AsNFWparameters, 0.001 0.1 1 10 wechoosevaluespredictedbytheΛCDMcosmologywith η Ωm = 0.3 and ΩΛ = 0.7, for a halo of Mvir = 1012 M⊙ (the estimated mass for our Galaxy; Battaglia et al. Fig. 1.—DimensionlessNFWdensityprofile(δNFW)as afunc- 2005). Forthismass,Bullock et al.(2001)predictacon- tion of radius (η). The density has a cusp in the center of the matter distribution and diverges at η = 0. This profile appropri- centration c = 7.9, where c ≡ Rvir/rs. The virial radius atelydescribesdarkmatterhalosfoundinnumericalsimulations. R is defined as the radius where the mean density of vir the halo is δ times the average density of the universe. For the cosmology considered here δ = 337 at z = 0. The NFW gravitationalpotential can be expressed as Moreover,M issimplythemassinsideR . Themodel vir vir by Bullock et al. (2001) gives the median concentration ln(1+r/r ) φ (r)=−4πGρ r2 s , (9) for a given mass, redshift and cosmology, based in sta- NFW s s r tistical results from cosmological simulations. Once we have M and c, the NFW profile is completely spec- where G is the gravitational constant. A dimensionless vir ified; for example, using the definition of R and the version of it would be vir valueofconecanobtainr ,whileρ cansimplybecom- s s ln(1+η) putedbyintegratingtheNFWprofilefrom0toRvir and ψNFW(η)= η , (10) using the values of rs and Mvir. Thus, for the dark matter halo expected to permeate a Milky Way-typegalaxy, nwohteerweoψrtNhFyWth≡at,−aφltNhFoWug/h4πtGheρsdrs2enasnitdy ηpr≡ofilre/riss.diIvteris- ρsF=ig3u.r6e×2s1h0o−w3sMt⊙heprca−ti3oalongd(Wrse=xt/2E5.g8)kinpcg.rayscale,as gent, the potential is finite everywhere due to the mild a function of the cloud’s density (y-axis) and radius (x- divergenceofthedensity;thelimitoftheNFWpotential axis) for different values of the separations between the as the radius goes to zero is cloud and halo’s center. On the top left panel, we show thecasewherethetestcloudiscenteredatthehalo’scen- limφ =−4πGρ r2. (11) ter(s=0pc). Forthisparticularsetup,thetidalenergy r→0 NFW s s is quite important relative to the gravitationalenergy of the cloud(the energy ratiois biggerthan 0.2)for a wide We assume, for simplicity, a molecular cloud modeled range of cloud’s densities and radii: ρ <∼ 103cm−3 and asaspherewithconstantdensityρ andradiusR. When cl cl < R ∼ 50 pc. The top right panel shows a cloud placed calculatingW (eq. 6), it is importantto note that the cl ext 1 kpc awayfrom the center of the halo. The importance integrationshouldbedoneoverthevolumeofthemolec- of the tidal energy has already decreased for high den- ular cloud, while the expression for the NFW potential sity clouds of the same size. As the cloud moves away is centeredonthe darkmatterhalo. A coordinatetrans- from the center of the halo (s = 2,3 kpc, lower panels), formation is then necessary to match both systems of thetidalcontributionrapidlybecomessmaller,ascanbe reference(separatedbythe distances),whichmakesthe seen in the bottom panels. Here, the tidal contribution expression significantly more elaborate for any configu- is negligible except for very low density clouds. rationwherethecloud’scenterdoesn’tcoincidewiththe Only clouds located very near the center of the halo center of the halo (i.e. s 6= 0). For this reason, a nu- feel a noticeable influence from it: the values for the merical integrator needs to be used to obtain the tidal energyratiodecayrapidlyasthe separationbetweenthe energy. cloud and the halo increases. This is due to the central To get a quantitative idea of the importance of the cuspiness of the NFW profile, shown in figure 1, since contribution of the dark matter halo, W is com- ext W isdependentonthepotentialgradientwhichgrows pared to the internal gravitationalenergy content of the ext steeper near the origin. It is also evident that values of cloud. For a spherical and homogeneous distribution, the tidal energy are more important for smaller and less this amounts to dense clouds. Since the external gravity is “pulling” the 16 particles in the cloud, it can be understood that a cloud Eg =−15π2Gρ2clR5. (12) ofsmaller diameter andlowerdensity will be less tightly bound and easier to influence. The energy ratio Wext/Eg measures the relative impor- As a second example, we present the results from an tanceofthegravitationalenergycontributionofthedark identical estimate but for a dark matter halo of mass mattercomparedtothegravitationalenergyofthecloud 1010M⊙. In this case, the NFW parameters employed itself. 4 Sua´rez-Madrigalet al. 4. GRAVITATIONAL FEEDBACK FROM A PERTURBED HALO 4.1. Perturbation Analysis Tocontinue withanotherperspective ofthe darkmatter influence in a molecular cloud’s evolution, this and subsequent sections study the gravitational feedback effect that a perturbation on the local dark matter background (DMB), produced by a barionic cloud, has upon the energy budget of the cloud itself. Dark matter interacts with barionic matter only gravitationally; as such, dark matter particles behave as a non-colissional fluid that feels a gravitational attraction to a mass distribution like a molecular cloud. Because galaxies are immersed in dark halos, a MC residing in it willcoexistwithaDMB.Followingadarkhalodistribu- tion like that of NFW, this DMB would have an almost Fig.2.—Wext/Eg mapforconstantdensitycloudsofradiusRcl constant density at scales of the MC, because the varia- and density ρcl, located inside a Galactic-type dark matter halo tionscaleofthehalo’sdensityprofileismuchlargerthan (M =1012M⊙)andcenteredatadistancesfromthehalocenter. theextentofevenagiantmolecularcloud(GMC). How- Whenthecloudcenterislocatedattheoriginofthehalo,itstidal ever, the mere presence of the barionic cloud perturbs energy can have a comparable effect to that of the gravitational energy. Ingeneral,thesmallerandlessdensecloudsfeelagreater the otherwise homogeneous DMB and promotes a den- gravitationalcontributionfromtheexternalmedium. sity enhancement in it, which generates a potential well thatwillinfluencetheMCinreturn. Weanalyzehowim- portantthisperturbationcanbetoaltertheevolutionof amolecularcloudindifferentdarkmatterenvironments. Although molecular clouds are highly irregular and structured (Combes 1991; Ballesteros-Paredeset al. 2007, and references therein), we model them as Plum- merspheresinordertofollowupananalyticalexpression whichmaybe easytouse. Themassdistributionofsuch a sphere follows a density profile given by −5/2 3M r2 cl ρ = 1+ , (13) cl 4πa3 a2 ! where M is the total mass of the cloud, r is the radial cl distance from the center of the cloud, and a is the scale- radius of the Plummer potential, which gives an idea of the size of the core of the distribution. The associated Fig.3.— As in Fig. 2 but for a galaxy with a halo mass of gravitationalpotential can be written as 1010M⊙. A similar effect is noticed, but scaled down due to the potential wellofsuchahalobeingshallower. GM 1 Φ (r)=− cl , (14) cl a 1+r2/a2 are ρs = 8.5×10−3M⊙ pc−3 and rs = 3.8kpc. As in whereGisthegravitationalconpstant. Acloudwiththese Fig. 2, in Fig. 3 we present the log(W /E ) maps profiles is then placedwithin a DMB with constantden- ext g for this halo’s characteristics. We notice that for a halo sity ρ and a velocity dispersion σ . DM DM withtheseparametervaluestheeffectisevenweaker,and Following Hernandez & Lee (2008), the dark matter fadesawayfasterasthemolecularcloudmovesawayfrom particlesaredescribedbyaMaxwell-Boltzmannvelocity the center of the halo. This smaller contribution can be distribution function f (v) ∝ exp(−v2/2σ2 ). For the 0 DM explained as a result of the shallower potential well of purposeofthisstudy,thedarkmatterparticleswillhave this less massive halo. an average velocity of 0, that is, the DMB as a bulk is It is important to mention that the ratios presented considered at rest with respect to the barionic matter. here only show the importance of the dark matter tidal Although the gravitational potential of the dark matter energy compared to the gravitational energy of a cloud. halo Φ is constant at the scales of the molecular cloud, 0 We do not make any comparison with the gravitational the MC induces a perturbation Φ , so that the total po- 1 influence of other possible matter components, such as tential Φ is the addition of both contributions, i.e. a galaxy’s central black hole or star population. With the results shown, we conclude that a dark matter halo Φ(r)=Φ (r)+ǫΦ (r), (15) is only capable of influencing the evolution of small, dif- 0 1 fuse clouds which are located near the center of massive andtheoriginaldensityρ isenhancedbyaperturbation 0 galaxies. ρ and should become 1 Dark matter in cloud complexes 5 GM 1 ρ(r)=ρ +ǫρ (r). (16) cl 0 1 ρ = ρ , (24) 1 a σ2 1+r2/a2 DM Similarily, the original Maxwell-Boltzmann distribu- DM tion function f0(v) of the local dark matter halo gets whichisthedensityperturpbationinthedarkmatterdis- perturbed by an amount f (r,v) and the new distribu- tribution produced by the molecular cloud. Such a den- 1 tion function will be given by sity perturbation in the dark matter halo will produce its own non-constant gravitational field, which may, in f(r,v)=f (v)+ǫf (r,v). (17) principle, influence the gravitational energy budget of 0 1 the molecular cloud. Our interest lies, thus, in the grav- Inordertoknowhowthedarkmatterpotentialwillbe itational feedback of this enhancement over the cloud affectedby the presence ofthe molecularcloud,we must itself. solvetheBoltzmannequationfornon-collisionalsystems In order to calculate the tidal energy produced by the perturbeddarkmatterhalooverthecloud,itisnecessary ∂f +v·∇f −∇Φ·∇ f =0, (18) to calculate the gradient of the gravitational potential v ∂t φ associatedtotheperturbationρ givenbyeq. (24). DM 1 inwhich∇representsthegradientoperatorwithrespect The gravitationalpotential as a function of the radius r tospatialcoordinatesand∇v thegradientoperatorwith that is producedby the perturbationρ1 will be givenby respect to velocity coordinates. The quantities involved in eq. (18) are the ones taking into account the pertur- 1 r bation caused by the barionic matter, i.e., the enhanced φ (r)=−4πG ρ (r′)r′2dr′ DM 1 r potential(eq. 15)anddistributionfunction(eq. 17). For (cid:26) Z0 simplicity,welookforafirstorderstationarysolutionto RDM the Boltzmann equation (i.e. ∂f/∂t = 0). By using the + ρ1(r′)r′dr′ , (25) Jean’sswindle(Φ0 =0),thefactthatf0 doesnotexplic- Zr (cid:27) itly depend on the spatial coordinates (∂f0/∂xi = 0), integrated over the radial coordinate r′ and where the and considering only first order terms, the radial solu- secondintegralwillbeevaluateduptosomeradiusR , DM tion can be obtained as such that the perturbation is fully contained within this radius, i.e. R is taken such that ρ (r = R ) = ρ . DM 1 DM 0 Introducing (24) and integrating, we get an expression v·∇f=∇Φ·∇ f, v for the perturbed dark matter potential: =∇(Φ +Φ )·∇ (f +f ), 0 1 v 0 1 =∇Φ1·∇vf0+∇Φ1·∇vf1, 2πG2M a ≈∇Φ1·∇vf0. (19) φDM(r)=− σ2 cl ρDM (r/a)−1ln r/a+ 1+(r/a)2 We can calculate the gradient of the gravitational po- DM (cid:26) (cid:16) p (cid:17) − 1+(r/a)2+2 1+(R /a)2 , (26) tential perturbation Φ , which is actually the cloud po- DM 1 tential Φ , as (cid:27) cl p p whose gradient is given by ∂Φ GM r 1 cl = . (20) ∂r a3 (1+r2/a2)3/2 ∂Φ 4πG2M a aln r/a+ (r/a)2+1 DM =− cl ρ UsingtheexplicitformoftheMaxwell-Boltzmannveloc- ∂r σD2M DM(cid:26) (cid:16) 2pr2 (cid:17) ity distribution function to get the velocity gradient, 1+(r/a)2 − . (27) ∂f0 =− v f (v), (21) p 2r (cid:27) ∂v σ2 0 Introducing this equation and the Plummer density dis- DM tribution (13) into eq. (6), we obtain and after inserting the last two expressions in eq. (19), it becomes 2πG2aM2ρ 2R/a W =− cl DM −arctan R/a ext σ2 1+(R/a)2 ∂f rGM v DM (cid:26) v 1 = cl − f (v), (22) (cid:0) (cid:1) ∂r a3(1+r2/a2)3/2 σD2M! 0 −ln R/a+ 1+(R/a)2 , (28) (cid:16) (1+(Rp/a)2)3/2 (cid:17) whichfurthermore,canbe integratedovervelocityspace (cid:27) as where R is the truncation radius of the Plummer profile for the molecular cloud. This expression quantifies the tidal energy of the perturbed dark matter distribution ∂ρ1 =−GM r 1 ρ (v). (23) over the cloud; in order to evaluate its importance in ∂r cla3(1+r2/a2)3/2 σD2M! 0 the process, it should be compared to the gravitational energy of the cloud itself, given by equation (3). Us- Remembering that the original DMB density has a con- ing eqs. (14) and (13) to evaluate this expression, the stant value of ρ , this equation can be solved to yield gravitationalenergy of the cloud becomes: DM 6 Sua´rez-Madrigalet al. 3GM2 R/a R/a E =− cl − grav 2a 8 1+(R/a)2 4 1+(R/a)2 2 (cid:26) 1(cid:0) (cid:1) (cid:0) (cid:1) + arctan R/a . (29) 8 (cid:27) (cid:0) (cid:1) Thus, ratio of the gravitationalto the tidal energy is W a 2 ρ σ −2 ext DM DM = f(R/a), Eg (cid:18)10pc(cid:19) (cid:18)M⊙pc−3(cid:19)(cid:18)kms−1(cid:19) (30) where we have defined f(r/a)=−1.8025 fext(R/a), (31) Fig.4.— Ratio f =fext/fg for a Plummer cloud of truncation f (R/a) size R and characteristic scale a = 10 pc. Note that the ratio is g always positive, implying thus that the tidal energy always con- tributestothecollapse,accordingtoeq. (30.) with 2R/a 4.2. Semi-analytical Results f (R/a)= −arctan R/a ext 1+(R/a)2 In order to get an idea of the magnitude of the ra- (cid:26) (cid:0) (cid:1) tio Wext/Eg in different dark matter environments, we ln R/a+ 1+(R/a)2 need to provide dark matter backgrounds to the molec- − , (32) ular clouds under study. As in section §3, we make use (cid:16) (1+(Rp/a)2)3/2 (cid:17) (cid:27) oftheNFWmodeltoestimatethestructuralparameters and of the dark matter halos,where the MCs will be embedded. We need to determine the halo’s density ρ and DM velocity dispersion σ , given its total mass M and DM vir R/a R/a the radius r at which these properties are evaluated, at f (R/a)= − g 8 1+(R/a)2 4 1+(R/a)2 2 a specified redshift z. We only consider here relatively nearby galaxies; that is, we assume redshift z =0. (cid:0) (cid:1)+1a(cid:0)rctan R/a (cid:1). (33) To find ρDM and σDM we thus proceed as follows: for 8 a given halo mass M , we obtain the concentration pa- vir (cid:27) (cid:0) (cid:1) rametercfollowingBullock et al.(2001). Withtheseval- Equation(30)tellshowimportantisagivendarkmat- ues, we calculate its density profile parameters as spec- ter environment on the gravitationalenergy budget of a ified in section §3. Once specified, we can evaluate the molecular cloud. Several points must be stressed from profile at the desired radius r to know the local density this equation. First of all, the ratio of tidal to gravita- ρ . On the other hand, in order to obtain σ , we DM DM tional energy does not depend on the mass of the cloud, adopt the relation given by Lokas & Mamon (2001, see neither on its density. Furthermore, the tidal energy on their eq. 14) for the case where velocities of dark mat- the cloud produced by the perturbed halo will be more ter particles are considered isotropic. Such expression important for larger clouds, and larger background halo is dependent only on c , M and R and so the ve- vir vir vir densities. On the other hand, for a halo’s larger velocity locity dispersion obtained for a given halo is a constant. dispersion σDM, the perturbation must be smaller and, These recipes are used in our analysis to get the DMB as a consequence, the tidal energy will also be smaller, parameters needed in different cases. when compared to the gravitational energy, as showed To begin with, we first choose parameters consistent by the σ−2 dependence in eq. (30). with a Milky-Way type dark matter halo evaluated at DM In Fig. 4 we plot the ratio f vs R/a, as given by eqs. a radius of 8 kpc, adequate to the solar neighborhood. (31–33). Since the function is always positive, we note However,inthiscasewepreferredtouseamorerealistic that the ratio of tidal to gravitational energy, eq. (30), concentration parameter c = 12, closer to the observa- will also be positive for any physically consistent com- tionally estimated value by, e.g., Battaglia et al. (2005). bination of parameters R/a, ρ , σ . This means With these conditions, the procedurejust described pre- DM DM that the tidal energy caused by such a DMB will al- dicts a DMB with density ρDM ∼ 0.01M⊙pc−3 and a ways contribute to the collapse of the MC perturbing velocity dispersion σ = 165 kms−1. With these pa- DM it. This effect, when relevant, is expected to be reflected rameters, equation (30) predicts that in order to get a ontheevolutionofthecloud. Inamolecularregionform- non-negligible tidal effect (W /E ∼ 0.15) the cloud’s ext g ing stars, a rise in the compressional forces acting on it scale radius a has to be huge (a = 1900pc). A cloud wouldshowanincreasedstarformationefficiency(SFE), of this size greatly surpasses the extent of local molec- asmorestarswillformfasterwhenthereisrelativelyless ular clouds. On the other hand, since the energy ratio support againstcollapse. In the following section we ex- W /E varies as a2, we note that for more typical ext grav ploretheparameterspacerelevanttothisscenario(R/a, clouds, a∼10 pc and R/a∼10, the contribution of the ρ , σ ). tidal energy is negligible. DM DM Dark matter in cloud complexes 7 0.22 7 a=10pc; M=10 M 0.2 o 0.18 0.16 g 0.14 E /xt 0.12 e W 0.1 0.08 0.06 0.04 0.02 1 10 R/a Fig. 5.—Wext/Eg ratioforacloudofPlummerradiusa=10pc anddifferenttruncationradii(R),embeddedinadarkmatterhalo typical of dwarfgalaxies (M ∼107M⊙), at aradius r=0.1rs. In Fig.6.— Wext/Eg mapfor Plummerclouds withcharacteristic thiscase,evensmallmolecularcloudswouldfeelaanon-zerotidal radius a=10 pc and R/a=10, embedded ina homogeneous dark effectfromtheirenvironment. matterbackgroundofparametersρDMandσDM. Thetidalenergy cancontributetothecollapseofcloudsimmersedindwarfgalaxies. Astheenergyratioincreasesasa2,alsomassivegalaxiescanshow Figure 5 presents a more interesting case in which the thiseffectforlargeenoughMCs. energy ratio is non-negligible for a range of cloud sizes R/a. The cloud has a Plummer radius of a=10 pc and it is embedded in a dark matter halo with mass M = 107M⊙ at a distance of r = 0.1rs. We can see, in this case,thatevensmallcloudsofthe orderof10pcwillfeel anon-zerotidaleffectfromthedarkmatterenvironment. Figure 6 shows, in grey scale, the energy ratios W /E evaluated using a NFW dark matter density ext g profilewitharangeofvelocitydispersionsσ andden- DM sitiesρ . Inthisfigure,theparametersofthePlummer DM cloudarea=10pc,andR/a=10. Similarmapsarepre- sentedinFigs.7and8, butfor cloudswith R/a=5and 1, respectively. As a reference, in these images we also indicate the typical densities and velocity dispersions of dark matter halos of galaxies with total masses ranging from 107 to 1011 M⊙. Density profiles are computed at r =0.1r . s Two points can be highlighted from Fig. 6. First, the tidal energy of the dark matter halo becomes important for large clouds (R ∼ 100 pc) in galaxies with masses ranging from 107 to 108 M⊙. Secondly and as noticed Fig.7.—SameasinFig.6,butforR/a=5. Thedifferenceswith previously, since the ratio W /E scales as a2, this Fig. 6 are not significant because Plummer spheres contain most ext grav oftheirmasswithin5characteristicradii. contributionbecomeimportantalsoinlargergalaxiesfor large enough clouds. For instance, as can be inferred fromFig.7,∼10%ofthetotalgravitationalenergycon- contribution has more trouble to act. tent of a giant cloud complex (a∼100 pc, R∼ 0.5 kpc) To further study the “big cloud” scenario, a different embedded in a 1010 M⊙ dark matter halo will be due to exploration of parameters is shown in Figure 9, in the tidal compression energy produced by the potential which the density and velocity dispersion of the dark well induced by the cloud itself. matter background are set according to the theoretical The highsimilaritybetweenFigs. 6and7is likelydue values of a Milky Way-type halo, at a radius of 50 kpc to the mass of a Plummer sphere being concentrated in (ρDM = 0.0002 M⊙pc−3, σDM = 150 kms−1), while the the inner 5 characteristic radii (93% of the mass of the cloud characteristic radius a and truncation radius R distribution lies at r < 4a). On the other hand, Figure are varied. This allows us to identify the characteristics 8, which shows the case R/a = 1, does show smaller of a mass distribution that would suffer a relevant contributionsfromtheenvironmenttotheenergybudget contractionbythetidalforces: thefigureshowsthat,for of the cloud than those shown in previous figures. The a radius larger than ∼ 8000 pc (a = 8000 pc, R/a = 1, reasoncanbe tracedagaintothe inherentconcentration or a = 6000 pc, R/a = 1.3) and located at some 50 kpc ofaPlummersphere: aswetruncatethesphereatr =a, from the center of the Milky Way, it will produce a weareonlyconsideringtheinnercoreofthecloud,which perturbation in the dark matter halo which will in turn has a higher gravitational energy and in which the tidal have a small contribution to the gravitational energy 8 Sua´rez-Madrigalet al. Fig.8.— Same as in Fig. 6, but for R/a=1. In this case, the Fig.10.— Same as in Fig. 9, but for a Milky Way dark tidaleffectisdiminishedwhencomparedtoFig.6because weare matter halo at 2 kpc from its center (ρDM = 0.04 M⊙pc−3, considering a more concentrated sphere in which the tidal forces σDM = 150 kms−1). In this case, a large cloud complex would havemoredifficultiestoact. suffer a tidal effect from the dark matter, compared to the previ- ouscase. largecloudcomplexofa∼1kpc andR∼2 pc willhave a tidal energy (∼6%) that will slightly contribute to its collapse. Thispossibilitysuggeststhatacomplexlikethe molecularringintheMilyWaymaybeaffectedsomehow by the tidal energy produced by the perturbation in the halo that the cloud complex imprints on it. 5. NUMERICAL SIMULATIONS OF CLOUD FORMATION WITHIN A DM HALO Different studies support the idea that molecular clouds in the Milky Way are transient entities produced by compressions from large scale streams in the in- terstellar medium (Ballesteros-Paredeset al. 1999a,b; Hartmann et al. 2001; Va´zquez-Semadeni et al. 2007; Heitsch & Hartmann 2008; Va´zquez-Semadeni et al. 2010). On the other hand, in sections 4.1 and 4.2 we derived semi-analytical calculations which suggest that a dark matter well produced by the perturbation of Fig.9.— Wext/Eg map for Plummer clouds of characteristic a barionic distribution of mass can significantly add radius a and truncation radius R, embedded in a homogeneous darkmatter background withdensity andvelocity dispersioncor- to its gravitational budget, and thus contribute to its responding to a Milky Way dark matter halo at 50 kpc from its collapse, if the right conditions are given. In order to center (ρDM =0.0002 M⊙pc−3, σDM =150 kms−1). A bigmass testthisscenariowithinamorerealisticsetting,wehave distribution(likeasatellitegalaxy)wouldfeelasmalltidalenergy performed a series of numerical experiments in which contribution. a dense cloud is formed by the collision of two streams of diffuse gas. The colliding gas cools down, becomes it contains (around 1%). This location and dimensions denser and collapses. While doing so, the collapsing are similar to that of a satellite galaxy like the Large cloud perturbs the locally homogeneous dark matter Magellanic Cloud. The effect is indeed low, but the halo2 in which it is embedded, producing a potential theoretical implications it suggests are interesting, as well. This will give an extra tidal energy contribution we could say that at least some contribution to the star to the originalcloud, which is expected to translate into formation seen in orbiting galaxies or in galaxy mergers a larger star formation activity. Given the differences could come from the tidal energy caused by a dark between the semi-analytical and the numerical circum- matter perturbation. stances (initial conditions and treatment), we do not expect semi-analytical results to faithfully reflect the Still, one more scenario to consider is that of a cloud behavior of the gas and dark matter in the simulations; closer to the center of a galaxy Milky-Way type. In Fig- still, the gravitational feedback effect must appear in ure 10, the dark matter parameters are fixed to fit the Milky Way’s dark matter halo at 2 kpc from its center 2 Asthe scaleradiiofthe halosconsideredinthesesimulations (ρDM = 0.04 M⊙pc−3, σDM = 150 kms−1), while the aremuchgreaterthanthesizeoftheclouds,thisisagoodaprox- cloud dimensions are explored. Note that in this case, a imation. Dark matter in cloud complexes 9 and it has the same values of the gas parameters de- TABLE 1 scribed above. These authors report the formation of a Variable parametersin the simulations “central cloud” of irregular shape with a radius of the orderof10pc, where starformationis concentrated. Al- ρDM σDM log(Mvir r/rs Linestyle (M⊙pc−3) (kms−1) [M⊙]) inFig. 12 though the physical configuration of the simulation dif- fers from the one of the analytical case we studied pre- 0 0 (...)a (...)a solid viously,we believe it is a goodtest case forinvestigating 0.1 37 10 0.15 dashed 0.1 10 8 0.26 long-dashed the generaleffect: the promotionof a higher star forma- 0.1 2 (...)b (...)b dash-dotted tion rate. 0.1 1 (...)b (...)b dotted In order to quantify the differencees in the evolu- 0.17 3.16 7 0.13 dot-longdashed tion of the different runs, we calculated the Star For- 0.24 3.16 7 0.1 short-logdashed mation Efficiency (SFE) as defined by, for example, Va´zquez-Semadeni et al. (2010): Note. — Dark matter density (ρDM) and vel. dispersion (σDM)areshown,aswellasthemassofagalaxy(Mvir)whose M∗ characteristic NFW halo represents these values at the radius SFE= , (34) (r)given,whenapplicable. M∗+Mgas a Referencerun b TherearenorealisticNFWdarkmatterhaloswiththischar- where M∗ represents the mass contained in stars and acteristics Mgas accounts for the mass in the dense gas (n >∼ 102 cm−3), at any given moment during the evolution the simulations. This effect can be then estimated in of the system. We also compute the relative difference the numerical experiments through the comparison of between the SFE of any run with dark matter and the the cloud’s star formation efficiency with and without reference run (without it), at any given time, ∆; that is, dark matter. SFE −SFE To perform the simulations, we use the Adaptive Re- ref run ∆= . (35) finementTree(Kravtsov et al.1997;Kravtsov2003)code SFE ref (ART), which employs an N−body algorithm to solve In this context, the first point to clarify is whether the gravitationalinteractions of the system. The chosen these variables allow us to distinguish between the evo- versionofthecodeadditionalysolvesgashydrodynamics lutionofrunswithdifferentphysics(e.g.,nodarkmatter and can handle radiative cooling and star formation, as included vs. a particular configurations of dark matter presented by Va´zquez-Semadeni et al. (2010). halos), and the evolution of runs with the same physics, The initial conditions for the gas are taken but different initial fluctuations. Thus, in Fig. 11, we from previous star formation simulations by show the time evolution of the SFE (upper panel) for Va´zquez-Semadeni et al. (2010). We consider a 256pc- threerunswithidenticalphysics(withdarkmatterprop- apter-asidteembpoexratfiullreedowfitTh t=enu5o0u0s0Kga,sw(hni0ch=rep1rcemse−n3t)s erties ρDM = 0.24M⊙pc−3, σDM = 3.16kms−1), but 0 with different phases for the initial density and velocity a warm neutral hydrogen medium (molecular weight fluctuations. However, in this case and unlike explained µ = 1.27). The gas has an inital turbulent velocity 0 before,whatisdrawnas∆inthelowerpanelofFig.11is fluctuation distribution of magnitude vrms = 0.1kms−1 therelativedifferencebetweentheSFEoftherunplotted everywhere. On top of that, two gas streams moving with a solid line3 and the runs plotted with a dashed or towards each other along the x−axis are created by adottedline (whichdifferonlyinthephaseofthe initial adding an initial velocity component of 7.5kms−1, to conditions). As canbe seeninthis figure,the differences the material contained in two 112pc long cylinders with amongthethreerunsarenotlarge,sothedifferentinitial radii of 64pc, lying on each side of the box. Since the conditionsdonotconstituteasignificantsourceforvaria- sound speed for the gas amounts to cs = 7.4kms−1, tionsintheSFE.Thus,ifanyofourperformedrunswith the flows are trans-sonic. The gas that comprises the dark matter produces substantially larger differences, it streams constitutes less than 20% of the total mass in must be due mainly to the gravitationalinfluence of the the box. The cloud evolution is followed during 40 Myr. dark matter. The local dark matter background is represented by particles that interact only gravitationally, both with 6. NUMERICAL RESULTS other dark matter particles and with the gas. Table 1 Ascommentedabove,thecollisionofthestreamsinthe shows in the first and second column the values of the simulationsdevelopsinstabilitiesintheatomicgaswhich density ρ and the velocity dispersion σ , respec- DM DM changes its phase and becomes molecular material with tively, used in each simulation. These were calculated lower temperatures and higher densities; this new phase followingtheschemesdescribedinsection6. Inthethird formscondensationsthatgrowwithtime. Astheshocked column, we show the typical mass of a halo (galaxy) as- region increases its mass, the cloud starts contracting. sociated with those ρDM and σDM values, given at the Thestarformationstartsat∼25Myrafterthebegining radius shown in the fourth column. of the simulation, and has a rapid growth soon after. To decouple the effect of the dark matter back- Around 10 Myr later, most of the gas in the simulation ground on the behavior of the gas from other effects, box has been transformed into stars and hence the SFE we need to study its evolution without a dark mat- approachesitsmaximumpossiblevalueof1(seeFig. 12). ter background. This reference run was taken from Va´zquez-Semadeni et al.(2010)(SAF0,whichstandsfor 3 The seed used to run this model is the same used to run the Small Amplitude Fluctuations without stellar feedback) modelswithdarkmattershowninTable1 10 Sua´rez-Madrigalet al. mationrate. Therightpanel,inthesameFig. 12,shows two other cases. These runs have both a higher density than those runs considered in the left and middle panels. Onerunhasρ=0.17M⊙pc−3 (dot-longdashedline) and the other one ρ = 0.24M⊙pc−3 (short-long dashed line). They are simulated with the same velocity dispersion σ = 3.16 kms−1. These models also show an DM enhanced SFE in comparison to the reference run, but here the contribution due to the DMB seems somewhat larger than the expected from the analytical predictions (see Figs. 6, 7 and 8). Such a deviation is, however, notsurprising: thepredictedexternalcontributioncomes fromanidealmodelthatconsidersasphericalmolecular cloudatrest,whilethesimulationcontainsmorerealistic conditions. 7. DISCUSSION AND CONCLUSIONS Weinvestigatedtheeffectthatthegravitationalcontri- butionfromdarkmatter,intheformoftidalenergy,has on the energy balance of a molecular cloud and its evolution. Onconsideringthelarge-scaledarkmatterhalos, where galaxies are supposed to be immersed, we noticed thatthereisanimportantgravitationalcontributiononly for clouds very near to the center of these halos, spe- Fig. 11.— SFEanddifferentialvalues(withrespecttothesolid cially for low-density,small molecular clouds. The effect linerun)forthreedifferentrunswithdifferentrandomphasesofthe of this collapse-promoting force dissapears very rapidly initial density and velocity field, but otherwise identical physical conditions: ρDM = 0.24M⊙pc−3,σDM = 3.16kms−1. As can be as the cloud location is movedawayfrom the center. To noticed, the differences are not large. Thus, for runs with dark our knowledge, this effect has not been studied before, matterhalo,anydifferencesubstantiallylargerthantheseshownin perhaps in part because clouds were not expected to be thisfigurewillbeconsideredasrepresentativeofthegravitational affected by the presence of the dark matter. In light of influenceofthedarkmatterhalo. these results, these predictions can be applied to structures encountered in the central region of galaxies. In In Figure 12 we show the time evolution of the SFE fact, Genzel et al. (1990) report on a series of observa- (upperpanels)and∆(lowerpanels)forthedifferentruns tions regarding the rich gas structure in the inner 10 pc outlined in Table 1, as compared to the reference run, ofthe Milky Way, comprisedofa couple giantmolecular which is shown as a solid line in all panels. On the left clouds, a circumnuclear disk and an HII region. These sidewepresenttherunswithdensityρDM =0.1M⊙pc−3 entities lie at the right position, are small and clumpy and velocity dispersions σDM = 37kms−1 (dashed line) (averaging ∼ 3 pc in diameter) and span a large range andσDM =10kms−1 (long-dashedline). Asidefromthe of densities (Vollmer, B. & Duschl, W. J. 2002). Such initial fluctuations, these runs show no significantdiffer- gas distributions can, according to our results, be feel- enceintheevolutionofthe SFE,ascomparedtothe ref- ing a great gravitational compression from the external erence run (solid line). This behavior is consistent with medium, relative to their self-gravity. These results also theresultspredictedforWext/Egravbyoursemi-anlytical suggestthat the intense star formation activity found in approach(seeFigs. 6,7and8). AsfarastheSFEiscon- the innermost regions of large galaxies (like ours) may cerned, these simulations proceeded as if the dark mat- have some contribution from tidal compression caused ter was not present. The centralpanel depicts two more by the dark matter halo of the galaxy. runswiththesamedensity(ρDM =0.1M⊙pc−3)butwith In this particular case, the presence of the central lowervelocity dispersions: σDM =2kms−1 (dash-dotted black hole (with an estimated mass MBH = 4×106M⊙; line) and σ = 1kms−1 (dotted line). In this cases, Schoedel et al. 2002) is only relevant when the molecu- DM there is a marked difference in the overall evolution of larcloudstructurereachesdistancessmallerthan∼1pc, the SFE. The σ = 2kms−1 run distinguishes from in accordance with the radius of influence of a central DM the previous ones in that its SFE is consistenly higher black hole r = GM /v (where v is the velocity h BH 0 0 than the one of the reference; there is a period of time dispersion of stars at the center; Peebles 1972, for our in which it has an SFE a factor of two higher than the Galaxy, r ≈ 0.5pc). Following the same analysis pre- h reference value. The dotted line in the same panel has sentedheretosetanexample,weestimatethatthegravi- an SFE that is systematicaly higher than all previous tationalenergycontribution(W )thattheMilkyWay’s BH runs. It can clearly be seen that runs with lower dark central black hole excerts on a 3pc diameter cloud with matter velocitydispersions(density remaining constant) density n=103cm−3, located 5pc away from the center produce higher SFEs. As can be deducedfrom Figure 7, of the Galaxy is given by W /E < 0.01, which is not BH g an important contribution is expected from the external significant in comparison to the dark matter contribu- gravitationalpotentialtowardsthe collapseofthe cloud: tion. aratioW /E intherange 0.3-0.8forthesecases. This On the other hand, the local dark matter background ext g translates into a significant enhancement of the star for- permeating a molecular cloud is shown to be able to