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Dark Matter as an active gravitational agent in cloud complexes

Article  in  The Astrophysical Journal · January 2012

DOI: 10.1088/0004-637X/748/2/101 · Source: arXiv

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The Astrophysical Journal, 748:101 (11pp), 2012 April 1 doi:10.1088/0004-637X/748/2/101C© 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

DARK MATTER AS AN ACTIVE GRAVITATIONAL AGENT IN CLOUD COMPLEXES

Andres Suarez-Madrigal, Javier Ballesteros-Paredes, Pedro Colın, and Paola D’AlessioCentro de Radioastronomıa y Astrofısica, Universidad Nacional Autonoma de Mexico, Apdo. Postal 72-3 (Xangari),

Morelia, Michocan, Mexico C.P. 58089, Mexico; a.suarez@crya.unam.mxReceived 2011 October 29; accepted 2012 January 18; published 2012 March 13

ABSTRACT

We study the effect that the dark matter background (DMB) has on the gravitational energy content and, in general,on the star formation efficiency (SFE) of a molecular cloud (MC). We first analyze the effect that a dark matter halo,described by the Navarro–Frenk–White density profile, has on the energy budget of a spherical, homogeneous cloudlocated at different distances from the halo center. We found that MCs located in the innermost regions of a massivegalaxy can feel a contraction force greater than their self-gravity due to the incorporation of the potential of thegalaxy’s dark matter halo. We also calculated analytically the gravitational perturbation that an MC produces over auniform DMB (uniform at the scales of an MC) and how this perturbation will affect the evolution of the MC itself.The study shows that the star formation in an MC will be considerably enhanced if the cloud is located in a denseand low velocity dark matter environment. We confirm our results by measuring the SFE in numerical simulationsof the formation and evolution of MCs within different DMBs. Our study indicates that there are situations wherethe dark matter’s gravitational contribution to the evolution of the MCs should not be neglected.

Key words: dark matter – evolution – ISM: clouds

1. INTRODUCTION

A great effort from the astronomical community is currentlydirected toward the detailed understanding of star formation, inparticular to the precise mechanisms involved in the conversionof some part of a molecular cloud’s (MC’s) mass into stars. Theevolution of MCs and the formation of stars within them arethought to be regulated by actors as diverse as gravity, large-scaleflows, turbulence, magnetic fields, and stellar feedback, amongothers (see reviews by Shu et al. 1987; Vazquez-Semadeni2010).

While evaluating the dynamical state of an MC due toall relevant forces acting on it, the Virial Theorem is fre-quently invoked to define an equilibrium condition by equal-ing the gravitational energy W of the cloud to twice its kineticenergy K:

2K = −W. (1)

Traditionally, this relationship has been interpreted in termsof clouds being in a state of quasi-equilibrium and long-livedentities, and deviations from this condition are assumed tomean that the cloud is collapsing or expanding, depending onwhich term is dominant (e.g., Myers & Goodman 1988a, 1988b;McKee & Zweibel 1992). However, it has been demonstratedthat a cloud fulfilling the so-called equilibrium condition willnot necessarily remain dynamically stable (Ballesteros-Paredes2006). In fact, numerical simulations show that clouds col-lapsing in a chaotic and hierarchical way will develop sucha relationship (e.g., Vazquez-Semadeni et al. 2007; Ballesteros-Paredes et al. 2011a, 2011b).

Furthermore, when calculating the Virial Theorem for MCs, itis a common practice to consider that the gravitational term is thegravitational energy, assuming that the cloud is, in practice, anisolated entity. However, different recent works have shown thatthe media surrounding the dense structures can importantly alterthe evolution of such entities (e.g., Burkert & Hartmann 2004;Gomez et al. 2007; Vazquez-Semadeni et al. 2007; Heitsch &Hartmann 2008; Ballesteros-Paredes et al. 2009b, 2009a). Inthese works, the influence of baryonic matter outside the regionof interest is studied.

On an apparently completely different topic, evidence froma wide range of observations has long led astronomers to arguein favor of the existence of much more matter than is actuallyseen: the elusive dark matter. Although the very nature of thismass component has not yet been discovered, it is presumedto permeate smoothly the majority of galaxies and in manycases to greatly surpass their visible extension, while beingtheir dominant mass component. Compared to the sizes of giantGalactic MCs, dark matter halos are huge and, for most practicalpurposes, homogeneous. Thus, one would expect that such adistribution will not cause a substantial gravitational effect onthe cloud.

In the present contribution, the idea that dark matter halosmay have an effect on the energy content of the gas in galaxiesis investigated. As a first approach, we focus our study in thecenter of massive galaxies, where the effect is expected to be thegreatest. A cusped dark matter profile such as the one proposedby Navarro et al. (1996, hereafter NFW) would provide thecentral region of a halo with a potential well which couldimportantly contribute to the gravitational energy of an MClocated around there.

In addition, we explore the idea that a baryonic mass con-centration is able to introduce a perturbation in an otherwisehomogeneous dark matter distribution. As such, the mere pres-ence of an MC would modify the dark matter halo in which itis embedded. This perturbation would in turn place the cloudin a local external potential well, which would play as an ex-tra agent in the struggle that determines its dynamical state. Inthis case, the external medium to the MC will contribute to itscollapse. When compared to a situation identical to this one butwithout the dark mater background (DMB), one would expectthat an MC in this scenario would display an enhanced capacityto form stars, since there is an extra factor working in the samedirection as the cloud’s internal gravitational energy. As such,measuring the star formation efficiency (SFE) of clouds em-bedded in different DMBs would hint toward the importance ofthe environment in their evolution in each case. We analyticallystudy the situation and give light as to what physical conditionswould be required for this to be an important effect.

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The Astrophysical Journal, 748:101 (11pp), 2012 April 1 Suarez-Madrigal et al.

Furthermore, we present results from numerical simula-tions of MC formation and evolution from two convergentmonoatomic flows, placed in different dark matter contexts.Although the DMBs explored are not very realistic, they aredesigned to test the analytical predictions and illustrate the im-portance of accounting for the external gravitational potential inMC analysis.

The structure of the article is as follows: in Section 2, weintroduce the tidal energy term that is part of the total gravita-tional energy of a matter distribution. In Section 3, we describe asemi-analytical procedure to evaluate the tidal contribution thatthe complete halo of a galaxy can have on an MC and applyit to a few scenarios. In Section 4.1, we analytically derive anexpression to assess the tidal effect of the DMB perturbationcaused by the MC itself, while Section 4.2 evaluates this ef-fect in different DMB environments. In Section 5, we presentnumerical simulations of cloud formation in order to test the an-alytical predictions in a more realistic scenario, and in Section 6we show the results obtained. Finally, Section 7 gives generalconclusions for the two studied effects.

2. FULL GRAVITATIONAL CONTENT OF AMOLECULAR CLOUD IN A GALAXY

MCs, like any other physical system, are bound to follow allthe internal and external forces acting on them and determiningtheir behavior over time. In the present article, the total gravi-tational content for MCs situated in different environments willbe evaluated, in order to look for the effect that their surround-ings can have on their evolution. Hence, this section derives anexpression for the tidal energy component, which will allow foran evaluation of its gravitational effect on the cloud.

With the intention of determining an MC’s dynamical con-dition, an energy balance is commonly performed to it by in-voking the Virial Theorem. In this theorem, the gravitationalenergy term W considered for a mass distribution of volume Vand density ρ embedded in a gravitational potential Φ is givenby

W = −∫

V

ρ xi

∂Φ∂xi

d3x (2)

(see, e.g., Shu et al. 1987). When applied to MCs, this term istraditionally assumed to be equal to the gravitational energy ofthe cloud:

Egrav = 1

2

∫V

ρcl Φcl d3x, (3)

where ρcl represents the density of the cloud and Φcl the potentialthat it generates. This assumption considers that the relevantgravitational potential is due only to the mass of the cloud,which is a valid approximation for an isolated mass distribution(Ballesteros-Paredes 2006). However, in a given galaxy, thereis a considerable amount of material (both baryonic and darkmatter) coexisting with an MC. Thus, following Ballesteros-Paredes et al. (2009b), one may write the gravitational potentialΦ of an MC in a galaxy as the potential due to the mass ofthe cloud itself, Φcl, plus the contribution of any other externalsources, Φext,

Φ = Φcl + Φext, (4)

such that the gravitational term entering the Virial Theorembecomes

W = Egrav + Wext, (5)

0.001

0.01

0.1

1

10

0.1 1 10

δ

η

δNFW = η-1(1+η)-2

Figure 1. Dimensionless NFW density profile (δNFW) as a function of radius(η). The density has a cusp in the center of the matter distribution and diverges atη = 0. This profile appropriately describes dark matter halos found in numericalsimulations.

where

Wext ≡ −∫

V

ρcl xi

∂φext

∂xi

dV (6)

is the tidal energy, i.e., the gravitational energy contained in thecloud, but due to any mass distribution external to the cloud.As explained in detail in Ballesteros-Paredes et al. (2009b), theexternal gravitational energy can either contribute to the collapseof the cloud or to its disruption, depending on the concavity ofthe external gravitational potential where it is located.

In the following sections, the tidal energy felt by a given MCwill be compared to its internal gravitational energy; its relativeimportance will determine the role that any material surroundingan MC will play in its evolution and future star formation.

3. TIDAL ENERGY IN THE POTENTIAL WELL OFA DARK MATTER HALO

To evaluate the effect that dark matter might have on starformation, we first consider a toy model consisting of a sphericalMC embedded in a spherical dark matter halo, with the cloudlocated a distance s from the halo’s center. The halo densityprofile is given by the simulation-inspired NFW density profile(Navarro et al. 1996):

ρNFW(r) = ρs

(r/rs)(1 + r/rs)2, (7)

where rs is a scale radius, the radius where the logarithmicderivative of the density is equal to −2, and ρs is a characteristicdensity, the density at r = rs . The profile can be expressed in adimensionless form by

δNFW(η) = η−1(1 + η)−2, (8)

where δNFW(η) ≡ ρNFW(r)/ρs and η ≡ r/rs . As seen in eitherFigure 1 or Equations (7) and (8), the NFW profile has a cuspat the center. It goes to infinity as η−1 for η → 0.

The NFW gravitational potential can be expressed as

φNFW(r) = −4πGρsr2s

ln (1 + r/rs)

r, (9)

where G is the gravitational constant. A dimensionless versionof it would be

ψNFW(η) = ln (1 + η)

η, (10)

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The Astrophysical Journal, 748:101 (11pp), 2012 April 1 Suarez-Madrigal et al.

where ψNFW ≡ −φNFW/4πGρsr2s and η ≡ r/rs . It is note-

worthy that, although the density profile is divergent, the po-tential is finite everywhere due to the mild divergence of thedensity; the limit of the NFW potential as the radius goes tozero is

limr→0

φNFW = −4πGρsr2s . (11)

We assume, for simplicity, an MC modeled as a spherewith constant density ρcl and radius R. When calculating Wext(Equation (6)), it is important to note that the integration shouldbe done over the volume of the MC, while the expressionfor the NFW potential is centered on the dark matter halo.A coordinate transformation is then necessary to match bothsystems of reference (separated by the distance s), which makesthe expression significantly more elaborate for any configurationwhere the cloud’s center does not coincide with the center of thehalo (i.e., s �= 0). For this reason, a numerical integrator needsto be used to obtain the tidal energy.

To get a quantitative idea of the importance of the contributionof the dark matter halo, Wext is compared to the internalgravitational energy content of the cloud. For a spherical andhomogeneous distribution, this amounts to

Eg = −16

15π2Gρ2

clR5. (12)

The energy ratio Wext/Eg measures the relative impor-tance of the gravitational energy contribution of the darkmatter compared to the gravitational energy of the clouditself.

We have, in total, five parameters available to explore: twoNFW halo parameters (characteristic density and radius of thedensity profile), two MC properties (density and extent of thecloud), and the distance of the cloud from the center ofthe halo (s). We choose to fix first the halo characteristicsand the distance s and calculate Wext/Eg values for a rangeof the cloud’s parameters. We present energy ratio maps fortypical MC radii and densities, one per halo type and cloudseparation.

The gravitational effect of a Milky Way type dark matter haloon an MC is first presented. As NFW parameters, we choosevalues predicted by the ΛCDM cosmology with Ωm = 0.3and ΩΛ = 0.7, for a halo of Mvir = 1012 M� (the estimatedmass for our Galaxy; Battaglia et al. 2005). For this mass,Bullock et al. (2001) predict a concentration c = 7.9, wherec ≡ Rvir/rs . The virial radius Rvir is defined as the radius wherethe mean density of the halo is δ times the average density ofthe universe. For the cosmology considered here δ = 337 atz = 0. Moreover, Mvir is simply the mass inside Rvir. The modelby Bullock et al. (2001) gives the median concentration for agiven mass, redshift, and cosmology, based on statistical resultsfrom cosmological simulations. Once we have Mvir and c, theNFW profile is completely specified; for example, using thedefinition of Rvir and the value of c one can obtain rs, while ρs

can simply be computed by integrating the NFW profile from0 to Rvir and using the values of rs and Mvir. Thus, for the darkmatter halo expected to permeate a Milky Way type galaxy,ρs = 3.6 × 10−3 M� pc−3 and rs = 25.8 kpc.

Figure 2 shows the ratio log(Wext/Eg) in gray scale as afunction of the cloud’s density (y-axis) and radius (x-axis)for different values of the separation s between the cloud andhalo’s center. On the top left panel, we show the case wherethe test cloud is centered at the halo’s center (s = 0 pc). Forthis particular setup, the tidal energy is quite important relative

Figure 2. Wext/Eg map for constant density clouds of radius Rcl and density ρcl,located inside a Galactic-type dark matter halo (M = 1012 M�) and centeredat a distance s from the halo center. When the cloud center is located at theorigin of the halo, its tidal energy can have a comparable effect to that of thegravitational energy. In general, the smaller and less dense clouds feel a greatergravitational contribution from the external medium.

to the gravitational energy of the cloud (the energy ratio isbigger than 0.2) for a wide range of cloud’s densities and radii:ρcl � 103 cm−3 and Rcl � 50 pc. The top right panel showsa cloud placed 1 kpc away from the center of the halo. Theimportance of the tidal energy has already decreased for highdensity clouds of the same size. As the cloud moves away fromthe center of the halo (s = 2, 3 kpc, lower panels), the tidalcontribution rapidly becomes smaller, as can be seen in thebottom panels. Here, the tidal contribution is negligible exceptfor very low density clouds.

Only clouds located very near the center of the halo feela noticeable influence from it: the values for the energy ratiodecay rapidly as the separation between the cloud and the haloincreases. This is due to the central cuspiness of the NFW profile,shown in Figure 1, since Wext is dependent on the potentialgradient, which grows steeper near the origin. It is also evidentthat values of the tidal energy are more important for smallerand less dense clouds. Since the external gravity is “pulling”the particles in the cloud, it can be understood that a cloud ofsmaller diameter and lower density will be less tightly boundand easier to influence.

As a second example, we present the results from an identicalestimate but for a dark matter halo of mass 1010 M�. In this case,the NFW parameters employed are ρs = 8.5 × 10−3 M� pc−3

and rs = 3.8 kpc. As in Figure 2, in Figure 3 we presentthe log (Wext/Eg) maps for this halo’s characteristics. We notethat for a halo with these parameter values the effect is evenweaker, and fades away faster as the MC moves away from thecenter of the halo. This smaller contribution can be explainedas a result of the shallower potential well of this less massivehalo.

It is important to mention that the ratios presented here onlyshow the importance of the dark matter tidal energy comparedto the gravitational energy of a cloud. We do not make anycomparison with the gravitational influence of other possiblematter components, such as a galaxy’s central black hole orstar population. With the results shown, we conclude that adark matter halo is only capable of influencing the evolutionof small, diffuse clouds located near the center of massivegalaxies.

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The Astrophysical Journal, 748:101 (11pp), 2012 April 1 Suarez-Madrigal et al.

Figure 3. Same as Figure 2 but for a galaxy with a halo mass of 1010 M�. Asimilar effect is noticed, but scaled down due to the potential well of such a halobeing shallower.

4. GRAVITATIONAL FEEDBACK FROMA PERTURBED HALO

4.1. Perturbation Analysis

To continue with another perspective of the dark matterinfluence in an MC’s evolution, this and subsequent sectionsstudy the gravitational feedback effect that a perturbation on thelocal DMB, produced by a baryonic cloud, has on the energybudget of the cloud itself.

Dark matter interacts with baryonic matter only gravitation-ally; as such, dark matter particles behave as a non-collisionalfluid that feels a gravitational attraction to a mass distributionlike an MC. Because galaxies are immersed in dark halos, anMC residing in it will coexist with a DMB. Following a darkhalo distribution like that of NFW, this DMB would have an al-most constant density at scales of the MC, because the variationscale of the halo’s density profile is much larger than the extentof even a giant molecular cloud (GMC). However, the merepresence of the baryonic cloud perturbs the otherwise homoge-neous DMB and promotes a density enhancement in it, whichgenerates a potential well that will influence the MC in return.We analyze how important this perturbation can be to alter theevolution of an MC in different dark matter environments.

Although MCs are highly irregular and structured (Combes1991; Ballesteros-Paredes et al. 2007 and references therein),we model them as Plummer spheres in order to follow upan analytical expression that may be easy to use. The massdistribution of such a sphere follows a density profile given by

ρcl = 3Mcl

4πa3

(1 +

r2

a2

)−5/2

, (13)

where Mcl is the total mass of the cloud, r is the radial distancefrom the center of the cloud, and a is the scale-radius ofthe Plummer potential, which gives an idea of the size of thecore of the distribution. The associated gravitational potentialcan be written as

Φcl(r) = −GMcl

a

1√1 + r2/a2

, (14)

where G is the gravitational constant. A cloud with these profilesis then placed within a DMB with constant density ρDM and avelocity dispersion σDM.

Following Hernandez & Lee (2008), the dark matter particlesare described by a Maxwell–Boltzmann velocity distributionfunction f0(v) ∝ exp(−v2/2σ 2

DM). For the purpose of this study,the dark matter particles will have an average velocity of 0, thatis, the DMB as a bulk is considered at rest with respect tothe baryonic matter. Although the gravitational potential of thedark matter halo Φ0 is constant at the scales of the MC, the MCinduces a perturbation Φ1 so that the total potential Φ is theaddition of both contributions, i.e.,

Φ(r) = Φ0(r) + εΦ1(r), (15)

and the original density ρ0 is enhanced by a perturbation ρ1 andshould become

ρ(r) = ρ0 + ερ1(r). (16)

Similarly, the original Maxwell–Boltzmann distribution func-tion f0(v) of the local dark matter halo gets perturbed by anamount f1(r, v) and the new distribution function will be givenby

f (r, v) = f0(v) + εf1(r, v). (17)

In order to know how the dark matter potential will be affectedby the presence of the MC, we must solve the Boltzmannequation for non-collisional systems

∂f

∂t+ v · ∇f − ∇Φ · ∇vf = 0, (18)

in which ∇ represents the gradient operator with respect tospatial coordinates and ∇v the gradient operator with respect tovelocity coordinates. The quantities involved in Equation (18)are the ones taking into account the perturbation caused by thebaryonic matter, i.e., the enhanced potential (Equation (15)) anddistribution function (Equation (17)). For simplicity, we lookfor a first-order stationary solution to the Boltzmann equation(i.e., ∂f/∂t = 0). By using Jean’s swindle (Φ0 = 0), the factthat f0 does not explicitly depend on the spatial coordinates(∂f0/∂xi = 0), and considering only first-order terms, the radialsolution can be obtained as

v · ∇f = ∇Φ · ∇vf,

= ∇(Φ0 + Φ1) · ∇v(f0 + f1),

= ∇Φ1 · ∇vf0 + ∇Φ1 · ∇vf1,

≈ ∇Φ1 · ∇vf0. (19)

We can calculate the gradient of the gravitational potentialperturbation Φ1, which is actually the cloud potential Φcl, as

∂Φ1

∂r= GMcl

a3

r

(1 + r2/a2)3/2. (20)

Using the explicit form of the Maxwell–Boltzmann velocitydistribution function to get the velocity gradient,

∂f0

∂v= − v

σ 2DM

f0(v), (21)

and after inserting the last two expressions into Equation (19),it becomes

v∂f1

∂r= rGMcl

a3(1 + r2/a2)3/2

(− v

σ 2DM

)f0(v), (22)

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The Astrophysical Journal, 748:101 (11pp), 2012 April 1 Suarez-Madrigal et al.

which furthermore can be integrated over velocity space as

∂ρ1

∂r= −GMcl

r

a3(1 + r2/a2)3/2

(1

σ 2DM

)ρ0(v). (23)

Remembering that the original DMB density has a constantvalue of ρDM, this equation can be solved to yield

ρ1 = GMcl

a σ 2DM

1√1 + r2/a2

ρDM, (24)

which is the density perturbation in the dark matter distributionproduced by the MC. Such a density perturbation in the darkmatter halo will produce its own non-constant gravitationalfield, which may, in principle, influence the gravitational energybudget of the MC. Our interest lies, thus, in the gravitationalfeedback of this enhancement over the cloud itself.

In order to calculate the tidal energy produced by theperturbed dark matter halo over the cloud, it is necessaryto calculate the gradient of the gravitational potential φDMassociated with the perturbation ρ1 given by Equation (24).The gravitational potential as a function of the radius r that isproduced by the perturbation ρ1 will be given by

φDM(r) = −4πG

{1

r

∫ r

0ρ1(r ′)r ′2dr ′ +

∫ RDM

r

ρ1(r ′)r ′dr ′},

(25)

integrated over the radial coordinate r ′ and where the secondintegral will be evaluated up to some radius RDM, such that theperturbation is fully contained within this radius, i.e., RDM istaken such that ρ1(r = RDM) = ρ0. Introducing Equation (24)and integrating, we get an expression for the perturbed darkmatter potential:

φDM(r) = −2πG2Mcla

σ 2DM

ρDM{(r/a)−1 ln (r/a +√

1 + (r/a)2)

−√

1 + (r/a)2 + 2√

1 + (RDM/a)2}, (26)

whose gradient is given by

∂ΦDM

∂r= −4πG2Mcla

σ 2DM

ρDM

{a ln (r/a +

√(r/a)2 + 1)

2r2

−√

1 + (r/a)2

2r

}. (27)

Introducing this equation and the Plummer density distribution(13) into Equation (6), we obtain

Wext = −2πG2aM2clρDM

σ 2DM

{2R/a

1 + (R/a)2− arctan (R/a)

− ln (R/a +√

1 + (R/a)2)

(1 + (R/a)2)3/2

}, (28)

where R is the truncation radius of the Plummer profile for theMC. This expression quantifies the tidal energy of the perturbeddark matter distribution over the cloud; in order to evaluateits importance in the process, it should be compared to thegravitational energy of the cloud itself, given by Equation (3).

Figure 4. Ratio f = fext/fg for a Plummer cloud of truncation size R andcharacteristic scale a = 10 pc. Note that the ratio is always positive, thusimplying that the tidal energy always contributes to the collapse, according toEquation (30).

Using Equations (14) and (13) to evaluate this expression, thegravitational energy of the cloud becomes

Egrav = −3GM2cl

2a

{R/a

8(1 + (R/a)2)− R/a

4(1 + (R/a)2)2

+1

8arctan (R/a)

}. (29)

Thus, the ratio of the gravitational to the tidal energy is

Wext

Eg

=(

a

10pc

)2 (ρDM

M�pc−3

)( σDM

km s−1

)−2f (R/a), (30)

where we have defined

f (r/a) = −1.8025fext(R/a)

fg(R/a), (31)

with

fext(R/a) ={

2R/a

1 + (R/a)2− arctan (R/a)

− ln (R/a +√

1 + (R/a)2)

(1 + (R/a)2)3/2

}, (32)

and

fg(R/a) ={

R/a

8(1 + (R/a)2)− R/a

4(1 + (R/a)2)2

+1

8arctan (R/a)

}. (33)

Equation (30) shows how important is a given dark matterenvironment on the gravitational energy budget of an MC.Several points must be stressed from this equation. First of all,the ratio of tidal to gravitational energy does not depend on themass of the cloud, neither on its density. Furthermore, the tidalenergy on the cloud produced by the perturbed halo will be moreimportant for larger clouds and larger background halo densities.On the other hand, for a halo’s larger velocity dispersion σDM,the perturbation must be smaller and, as a consequence, the tidalenergy will also be smaller, when compared to the gravitationalenergy, as showed by the σ−2

DM dependence in Equation (30).In Figure 4 we plot the ratio f versus R/a, as given by

Equations (31)–(33). Since the function is always positive, wenote that the ratio of tidal to gravitational energy, Equation (30),

5

The Astrophysical Journal, 748:101 (11pp), 2012 April 1 Suarez-Madrigal et al.

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

1 10

Wex

t/Eg

R/a

a=10pc; M=107 Mo

Figure 5. Wext/Eg ratio for a cloud of Plummer radius a = 10 pc and differenttruncation radii (R), embedded in a dark matter halo typical of dwarf galaxies(M ∼ 107 M�), at a radius r = 0.1rs . In this case, even small molecular cloudswould feel a non-zero tidal effect from their environment.

will also be positive for any physically consistent combinationof parameters R/a, ρDM, σDM. This means that the tidal energycaused by such a DMB will always contribute to the collapseof the MC perturbing it. This effect, when relevant, is expectedto be reflected on the evolution of the cloud. In a molecularregion forming stars, a rise in the compressional forces actingon it would show an increased SFE, as more stars will formfaster when there is relatively less support against collapse. Inthe following section, we explore the parameter space relevantto this scenario (R/a, ρDM, σDM).

4.2. Semi-analytical Results

In order to get an idea of the magnitude of the ratio Wext/Eg indifferent dark matter environments, we need to provide DMBsto the MCs under study. As in Section 3, we make use of theNFW model to estimate the structural parameters of the darkmatter halos, where the MCs will be embedded. We need todetermine the halo’s density ρDM and velocity dispersion σDM,given its total mass Mvir and the radius r at which these propertiesare evaluated, at a specified redshift z. We only consider hererelatively nearby galaxies; that is, we assume redshift z = 0.

To find ρDM and σDM we thus proceed as follows: for agiven halo mass Mvir, we obtain the concentration parameter cfollowing Bullock et al. (2001). With these values, we calculateits density profile parameters as specified in Section 3. Oncespecified, we can evaluate the profile at the desired radius rto know the local density ρDM. On the other hand, in order toobtain σDM, we adopt the relation given by Lokas & Mamon(2001, see their Equation (14)) for the case where velocities ofdark matter particles are considered isotropic. Such expressionis dependent only on cvir, Mvir, and Rvir and so the velocitydispersion obtained for a given halo is a constant. These recipesare used in our analysis to get the DMB parameters needed indifferent cases.

To begin with, we first choose parameters consistent with aMilky Way type dark matter halo evaluated at a radius of 8 kpc,adequate to the solar neighborhood. However, in this case wepreferred to use a more realistic concentration parameter c = 12,closer to the observationally estimated value by, e.g., Battagliaet al. (2005). With these conditions, the procedure just describedpredicts a DMB with density ρDM ∼ 0.01 M� pc−3 and avelocity dispersion σDM = 165 km s−1. With these parameters,Equation (30) predicts that in order to get a non-negligible tidal

Figure 6. Wext/Eg map for Plummer clouds with characteristic radius a = 10 pcand R/a = 10, embedded in a homogeneous dark matter background ofparameters ρDM and σDM. The tidal energy can contribute to the collapse ofclouds immersed in dwarf galaxies. As the energy ratio increases as a2, massivegalaxies can also show this effect for large enough MCs.

Figure 7. Same as in Figure 6, but for R/a = 5. The differences from Figure 6are not significant because Plummer spheres contain most of their mass withinfive characteristic radii.

effect (Wext/Eg ∼ 0.15) the cloud’s scale radius a has to behuge (a = 1900 pc). A cloud of this size greatly surpasses theextent of local MCs. On the other hand, since the energy ratioWext/Egrav varies as a2, we note that for more typical clouds,a ∼ 10 pc and R/a ∼ 10, the contribution of the tidal energy isnegligible.

Figure 5 presents a more interesting case in which the energyratio is non-negligible for a range of cloud sizes R/a. The cloudhas a Plummer radius of a = 10 pc and it is embedded in a darkmatter halo with mass M = 107 M� at a distance of r = 0.1rs .

We can see, in this case, that even small clouds of the orderof 10 pc will feel a non-zero tidal effect from the dark matterenvironment.

Figure 6 shows, in gray scale, the energy ratios Wext/Eg

evaluated using an NFW dark matter density profile with arange of velocity dispersions σDM and densities ρDM. In thisfigure, the parameters of the Plummer cloud are a = 10 pc andR/a = 10. Similar maps are presented in Figures 7 and 8, butfor clouds with R/a = 5 and 1, respectively. As a reference, in

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Figure 8. Same as in Figure 6, but for R/a = 1. In this case, the tidal effectis diminished when compared to Figure 6 because we are considering a moreconcentrated sphere in which the tidal forces have more difficulty to act.

these images we also indicate the typical densities and velocitydispersions of dark matter halos of galaxies with total massesranging from 107 to 1011 M�. Density profiles are computed atr = 0.1rs .

Two points can be highlighted from Figure 6. First, the tidalenergy of the dark matter halo becomes important for largeclouds (R ∼ 100 pc) in galaxies with masses ranging from107 to 108 M�. Second and as noticed previously, since the ratioWext/Egrav scales as a2, this contribution also becomes importantin larger galaxies for large enough clouds. For instance, as can beinferred from Figure 7, ∼10% of the total gravitational energycontent of a giant cloud complex (a ∼ 100 pc, R ∼ 0.5 kpc)embedded in a 1010 M� dark matter halo will be due to the tidalcompression energy produced by the potential well induced bythe cloud itself.

The high similarity between Figures 6 and 7 is likely due tothe mass of a Plummer sphere being concentrated in the innerfive characteristic radii (93% of the mass of the distributionlies at r < 4a). On the other hand, Figure 8, which showsthe case R/a = 1, does show smaller contributions from theenvironment to the energy budget of the cloud than those shownin previous figures. The reason can be traced again to the inherentconcentration of a Plummer sphere: as we truncate the sphereat r = a, we are only considering the inner core of the cloud,which has a higher gravitational energy and in which the tidalcontribution has more trouble to act.

To further study the “big cloud” scenario, a different explo-ration of parameters is shown in Figure 9, in which the densityand velocity dispersion of the DMB are set according to the the-oretical values of a Milky Way type halo, at a radius of 50 kpc(ρDM = 0.0002 M� pc−3, σDM = 150 km s−1), while the cloudcharacteristic radius a and truncation radius R are varied. This al-lows us to identify the characteristics of a mass distribution thatwould suffer a relevant contraction by the tidal forces: the figureshows that, for a radius larger than ∼8000 pc (a = 8000 pc,R/a = 1, or a = 6000 pc, R/a = 1.3) and located some 50 kpcfrom the center of the Milky Way, it will produce a perturbationin the dark matter halo which will in turn have a small contri-bution to the gravitational energy it contains (around 1%). Thislocation and dimensions are similar to that of a satellite galaxylike the Large Magellanic Cloud. The effect is indeed low, but

Figure 9. Wext/Eg map for Plummer clouds of characteristic radius a andtruncation radius R, embedded in a homogeneous dark matter background withdensity and velocity dispersion corresponding to a Milky Way dark matter halo50 kpc from its center (ρDM = 0.0002 M� pc−3, σDM = 150 km s−1). Abig mass distribution (like a satellite galaxy) would feel a small tidal energycontribution.

Figure 10. Same as Figure 9, but for a Milky Way dark matter halo 2 kpc fromits center (ρDM = 0.04 M� pc−3, σDM = 150 km s−1). In this case, a largecloud complex would suffer a tidal effect from the dark matter, compared to theprevious case.

the theoretical implications it suggests are interesting, as wecould say that at least some contribution to the star formationseen in orbiting galaxies or in galaxy mergers could come fromthe tidal energy caused by a dark matter perturbation.

Still, one more scenario to consider is that of a cloud closerto the center of a Milky Way type galaxy. In Figure 10, thedark matter parameters are fixed to fit the Milky Way’s darkmatter halo at 2 kpc from its center (ρDM = 0.04 M� pc−3,σDM = 150 km s−1), while the cloud dimensions are explored.Note that in this case, a large cloud complex of a ∼ 1 kpcand R ∼ 2 pc will have a tidal energy (∼6%) that willslightly contribute to its collapse. This possibility suggeststhat a complex like the molecular ring in the Milky Waymay be affected somehow by the tidal energy produced bythe perturbation in the halo that the cloud complex imprintson it.

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5. NUMERICAL SIMULATIONS OF CLOUD FORMATIONWITHIN A DM HALO

Different studies support the idea that MCs in the Milky Wayare transient entities produced by compressions from large-scale streams in the interstellar medium (Ballesteros-Paredeset al. 1999a, 1999b; Hartmann et al. 2001; Vazquez-Semadeniet al. 2007; Heitsch & Hartmann 2008; Vazquez-Semadeni et al.2010). On the other hand, in Sections 4.1 and 4.2, we derivedsemi-analytical calculations that suggest that a dark matterwell produced by the perturbation of a baryonic distributionof mass can significantly add to its gravitational budget, andthus contribute to its collapse, if the right conditions are given.In order to test this scenario within a more realistic setting,we have performed a series of numerical experiments in whicha dense cloud is formed by the collision of two streams ofdiffuse gas. The colliding gas cools down, becomes denser,and collapses. While doing so, the collapsing cloud perturbs thelocally homogeneous dark matter halo1 in which it is embedded,producing a potential well. This will give an extra tidal energycontribution to the original cloud, which is expected to translateinto a larger star formation activity. Given the differencesbetween the semi-analytical and the numerical circumstances(initial conditions and treatment), we do not expect semi-analytical results to faithfully reflect the behavior of the gas anddark matter in the simulations; still, the gravitational feedbackeffect must appear in the simulations. This effect can be thenestimated in the numerical experiments through the comparisonof the cloud’s SFE with and without dark matter.

To perform the simulations, we use the Adaptive RefinementTree (Kravtsov et al. 1997; Kravtsov 2003) code (ART), whichemploys an N-body algorithm to solve the gravitational interac-tions of the system. The chosen version of the code additionallysolves gas hydrodynamics and can handle radiative cooling andstar formation, as presented by Vazquez-Semadeni et al. (2010).

The initial conditions for the gas are taken from previousstar formation simulations by Vazquez-Semadeni et al. (2010).We consider a 256 pc per side box filled with tenuous gas(n0 = 1 cm−3) at a temperature of T0 = 5000 K, whichrepresents a warm neutral hydrogen medium (molecular weightμ0 = 1.27). The gas has an initial turbulent velocity fluctuationdistribution of magnitude vrms = 0.1 km s−1 everywhere. Ontop of that two gas streams moving toward each other alongthe x-axis are created by adding an initial velocity componentof 7.5 km s−1, to the material contained in two 112 pc longcylinders with radii of 64 pc, lying on each side of the box.Since the sound speed for the gas amounts to cs = 7.4 km s−1,the flows are transonic. The gas that comprises the streamsconstitutes less than 20% of the total mass in the box. The cloudevolution is followed during 40 Myr.

The local DMB is represented by particles that interact onlygravitationally, both with other dark matter particles and with thegas. Table 1 shows in the first and second columns the values ofthe density ρDM and the velocity dispersion σDM, respectively,used in each simulation. These were calculated following theschemes described in Section 4. In the third column, we showthe typical mass of a halo (galaxy) associated with those ρDMand σDM values, given at the radius shown in the fourth column.

To decouple the effect of the DMB on the behavior ofthe gas from other effects, we need to study its evolution

1 As the scale radii of the halos considered in these simulations are muchgreater than the size of the clouds, this is a good approximation.

Table 1Variable Parameters in the Simulations

ρDM σDM log (Mvir(M�)) r Linestyle in Figure 12(M� pc−3) (km s−1) (rs )

0 0 (. . .)a (. . .)a Solid0.1 37 10 0.15 Dashed0.1 10 8 0.26 Long-dashed0.1 2 (. . .)b (. . .)b Dash-dotted0.1 1 (. . .)b (. . .)b Dotted0.17 3.16 7 0.13 Dot-long dashed0.24 3.16 7 0.1 Short-log dashed

Notes. Dark matter density (ρDM) and velocity dispersion (σDM) are shown, aswell as the mass of a galaxy (Mvir) whose characteristic NFW halo representsthese values at the radius (r) given, when applicable.a Reference run.b There are no realistic NFW dark matter halos with this characteristics.

without a DMB. This reference run was taken from (Vazquez-Semadeni et al. 2010, SAF0, which stands for Small AmplitudeFluctuations without stellar feedback) and it has the same valuesof the gas parameters described above. These authors report theformation of a “central cloud” of irregular shape with a radiusof the order of 10 pc, where star formation is concentrated.Although the physical configuration of the simulation differsfrom the one of the analytical case we studied previously, webelieve it is a good test case for investigating the general effect:the promotion of a higher star formation rate.

In order to quantify the differences in the evolution of thedifferent runs, we calculated the SFE as defined by, for example,Vazquez-Semadeni et al. (2010):

SFE = M∗M∗ + Mgas

, (34)

where M∗ represents the mass contained in stars and Mgasaccounts for the mass in the dense gas (n � 102 cm−3) atany given moment during the evolution of the system. We alsocompute the relative difference between the SFE of any run withdark matter and the reference run (without it), at any given time,Δ, that is,

Δ = SFEref − SFErun

SFEref. (35)

In this context, the first point to clarify is whether thesevariables allow us to distinguish between the evolution of runswith different physics (e.g., no dark matter included versus aparticular configurations of dark matter halos), and the evolutionof runs with the same physics, but different initial fluctuations.Thus, in Figure 11, we show the time evolution of the SFE(upper panel) for three runs with identical physics (with darkmatter properties ρDM = 0.24 M� pc−3, σDM = 3.16 km s−1),but with different phases for the initial density and velocityfluctuations. However, in this case and unlike explained before,what is drawn as Δ in the lower panel of Figure 11 is the relativedifference between the SFE of the run plotted with a solid line2

and the runs plotted with a dashed or a dotted line (which differonly in the phase of the initial conditions). As can be seen inthis figure, the differences among the three runs are not large,so the different initial conditions do not constitute a significantsource for variations in the SFE. Thus, if any of our performedruns with dark matter produces substantially larger differences,

2 The seed used to run this model is the same used to run the models withdark matter shown in Table 1.

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SAF0

Figure 11. SFE and differential values (with respect to the solid line run) for threedifferent runs with different random phases of the initial density and velocityfield, but otherwise identical physical conditions: ρDM = 0.24 M� pc−3,σDM = 3.16 km s−1. As can be noticed, the differences are not large. Thus, forruns with a dark matter halo, any difference substantially larger than those shownin this figure will be considered representative of the gravitational influence ofthe dark matter halo.

it must be due mainly to the gravitational influence of the darkmatter.

6. NUMERICAL RESULTS

As commented above, the collision of the streams in the simu-lations develops instabilities in the atomic gas which changes itsphase and becomes molecular material with lower temperaturesand higher densities; this new phase forms condensations thatgrow with time. As the shocked region increases its mass, thecloud starts contracting. The star formation starts at ∼25 Myrafter the beginning of the simulation and has a rapid growth soonafter. Around 10 Myr later, most of the gas in the simulation boxhas been transformed into stars and hence the SFE approachesits maximum possible value of 1 (see Figure 12).

In Figure 12 we show the time evolution of the SFE (upperpanels) and Δ (lower panels) for the different runs outlinedin Table 1, as compared to the reference run, which is shownas a solid line in all panels. On the left side we present theruns with density ρDM = 0.1 M� pc−3 and velocity dispersionsσDM = 37 km s−1 (dashed line) and σDM = 10 km s−1 (long-dashed line). Aside from the initial fluctuations, these runsshow no significant difference in the evolution of the SFE,as compared to the reference run (solid line). This behavioris consistent with the results predicted for Wext/Egrav by oursemi-analytical approach (see Figures 6–8). As far as the SFE isconcerned, these simulations proceeded as if the dark matterwas not present. The central panel depicts two more runswith the same density (ρDM = 0.1 M� pc−3) but with lowervelocity dispersions: σDM = 2 km s−1 (dash-dotted line) and

Figure 12. SFE and Δ values for the simulation runs described in Table 1. The reference run is displayed as the solid line in all three panels. The left panel compareshigh velocity dispersion DMBs (σDM � 10 km s−1) with the reference run. Both runs in this panel have a density of ρDM = 0.1 M� pc−3: the dashed line shows therun with σDM = 37 km s−1, while the long-dashed line depicts the one with σDM = 10 km s−1. No significant difference is found from the reference run, as expectedfrom the analytical predictions. The central panel shows two more runs with the same density (ρDM = 0.1 M� pc−3) but smaller velocity dispersions: σDM = 2 km s−1

(dash-dotted line) and σDM = 1 km s−1 (dotted line). These cases show a systematically higher SFE than the reference run, also as expected. Two more runs withhigher densities are displayed on the right panel: ρDM = 0.17 M� pc−3, σDM = 3.16 km s−1 (dot-long dashed line) and ρDM = 0.24 M� pc−3, σDM = 3.16 km s−1

(short-long dashed line). Again, the star formation seems a lot more active than the reference run.

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σDM = 1 km s−1 (dotted line). In these cases, there is amarked difference in the overall evolution of the SFE. TheσDM = 2 km s−1 run distinguishes from the previous ones in thatits SFE is consistently higher than the one of the reference; thereis a period of time in which it has an SFE a factor of two higherthan the reference value. The dotted line in the same panel hasan SFE that is systematically higher than all previous runs. Itcan clearly be seen that runs with lower dark matter velocitydispersions (density remaining constant) produce higher SFEs.As can be deduced from Figure 7, an important contributionis expected from the external gravitational potential toward thecollapse of the cloud: a ratio Wext/Eg in the range 0.3–0.8 forthese cases. This translates into a significant enhancement ofthe star formation rate. The right panel, in the same Figure 12,shows two other cases. These runs have both a higher densitythan those runs considered in the left and middle panels. Onerun has ρ = 0.17 M� pc−3 (dot-long dashed line) and theother one ρ = 0.24 M� pc−3 (short-long dashed line). They aresimulated with the same velocity dispersion σDM = 3.16 km s−1.These models also show an enhanced SFE in comparison tothe reference run, but here the contribution due to the DMBseems somewhat larger than the expected from the analyticalpredictions (see Figures 6–8). Such a deviation is, however, notsurprising: the predicted external contribution comes from anideal model that considers a spherical MC at rest, while thesimulation contains more realistic conditions.

7. DISCUSSION AND CONCLUSIONS

We investigated the effect that the gravitational contributionfrom dark matter, in the form of tidal energy, has on the en-ergy balance of an MC and its evolution. On considering thelarge-scale dark matter halos, where galaxies are supposed tobe immersed, we noticed that there is an important gravitationalcontribution only for clouds very near the center of these ha-los, especially for low-density, small MCs. The effect of thiscollapse-promoting force disappears very rapidly as the cloudlocation is moved away from the center. To our knowledge,this effect has not been studied before, perhaps in part be-cause clouds were not expected to be affected by the pres-ence of the dark matter. In light of these results, these predic-tions can be applied to structures encountered in the centralregion of galaxies. In fact, Genzel et al. (1990) report on a se-ries of observations regarding the rich gas structure in the inner10 pc of the Milky Way, comprised of a couple of GMCs, a cir-cumnuclear disk and an H ii region. These entities lie at the rightposition, are small and clumpy (averaging ∼3 pc in diameter),and span a large range of densities (Vollmer & Duschl 2002).Such gas distributions can, according to our results, be feelinga great gravitational compression from the external medium,relative to their self-gravity. These results also suggest that theintense star formation activity found in the innermost regionsof large galaxies (like ours) may have some contribution fromtidal compression caused by the dark matter halo of the galaxy.

In this particular case, the presence of the central black hole(with an estimated mass MBH = 4 × 106 M�; Schoedel et al.2002) is only relevant when the MC structure reaches distancessmaller than ∼1 pc, in accordance with the radius of influenceof a central black hole rh = GMBH/v0 (where v0 is the velocitydispersion of stars at the center; Peebles 1972; for our Galaxy,rh ≈ 0.5 pc). Following the same analysis presented hereto set an example, we estimate that the gravitational energycontribution (WBH) that the Milky Way’s central black holeexerts on a 3 pc diameter cloud with density n = 103 cm−3,

located 5pc away from the center of the Galaxy, is given byWBH/Eg < 0.01, which is not significant in comparison to thedark matter contribution.

On the other hand, the local DMB permeating an MC is shownto be able to contribute to its star formation if the appropriateconditions are fulfilled. The high dark matter densities and lowvelocity dispersions, found in dwarf spheroidal galaxies, neededfor the tidal energy to be important for the evolution of MCs arenot expected in galaxies where there is a significant amount ofmolecular gas, namely, in spiral and irregular galaxies. Yet,small galaxies dominated by dark matter with a significantfraction of gas are not ruled out. Moreover, although the solarneighborhood does not meet the dark matter conditions wherestar formation would be greatly enhanced by its presence, wepredict a small contribution for big clouds.

This work has shown that despite the huge difference betweenthe length scales of MCs and dark matter halos, the lattercould influence the evolution of a star forming region. Weshow that this effect is negligible in our local environment,but that it can have possible applications in other environments,other locations, or other galaxies. In summary, our study firmlysuggests that the external dark matter gravitational contributionshould be considered, when analyzing the dynamical evolutionof an MC, if the right conditions are fulfilled.

We are grateful to A. Kravtsov for the numerical codeused in our study. This work has received partial supportfrom grants UNAM/DGAPA IN110409 to JBP. A.S.-M. ac-knowledges CONACyT master’s degree grant. This workmakes extensive use of the NASA-ADS database system.We thank an anonymous referee for a careful reading of themanuscript.

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