Gravitational Collapse and Filament Formation: Comparison with the Pipe Nebula
aa r X i v : . [ a s t r o - ph . GA ] S e p Draft version November 2, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
GRAVITATIONAL COLLAPSE AND FILAMENT FORMATION: COMPARISON WITH THE PIPE NEBULA
Fabian Heitsch , Javier Ballesteros-Paredes , and Lee Hartmann Draft version November 2, 2018
ABSTRACTRecent models of molecular cloud formation and evolution suggest that such clouds are dynamic andgenerally exhibit gravitational collapse. We present a simple analytic model of global collapse onto afilament and compare this with our numerical simulations of the flow-driven formation of an isolatedmolecular cloud to illustrate the supersonic motions and infall ram pressures expected in models ofgravity-driven cloud evolution. We apply our results to observations of the Pipe Nebula, an especiallysuitable object for our purposes as its low star formation activity implies insignifcant perturbationsfrom stellar feedback. We show that our collapsing cloud model can explain the magnitude of thevelocity dispersions seen in the CO filamentary structure by Onishi et al. and the ram pressuresrequired by Lada et al . to confine the lower-mass cores in the Pipe nebula. We further conjecture thathigher-resolution simulations will show small velocity dispersions in the densest core gas, as observed,but which are infall motions and not supporting turbulence. Our results point out the inevitability ofram pressures as boundary conditions for molecular cloud filaments, and the possibility that especiallylower-mass cores still can be accreting mass at significant rates, as suggested by observations.
Subject headings: turbulence — methods:numerical — ISM:clouds — ISM:kinematics and dynamics—stars:formation INTRODUCTION
The supersonic, “turbulent” motions observed inmolecular clouds must play an important role in starformation. Early numerical models of molecular cloudsoften imposed supersonic velocities as either a continu-ous forcing term (Mac Low et al. 1998; Stone et al. 1998;Padoan & Nordlund 1999) and/or as initial conditions(Bate et al. 2002, 2003). Yet without an understandingof how supersonic turbulence originates, it is difficult todevelop a predictive theory of the processes leading tothe formation of stars.The recognition that molecular clouds might of-ten if not generally result from accumulation of gasby large scale flows (Ballesteros-Paredes et al. 1999a,b;Hartmann et al. 2001) and in particular from atomicflows (Koyama & Inutsuka 2002; Audit & Hennebelle2005; Heitsch et al. 2006b) has made it plausible thatthe turbulence is arising as a consequence of the cloud’sformation. V´azquez-Semadeni et al. (2007) showed thatturbulence could develop in clouds formed by variable-velocity flows. By imposing a fixed spatial variation inthe supersonic inflow velocities, they in effect identifiedthe driving mechanism as variations in the inflow speeds(also Hennebelle et al. 2008). In contrast, Heitsch et al.(2006b, 2008b) showed that even uniform inflows can pro-duce turbulent substructure if the shock interface is notplanar and/or precisely perpendicular to the flows.It is clear from the studies described in the previousparagraph that molecular clouds swept up by supersonicflows in the interstellar medium will begin their existenceas both structured and turbulent. However, this sweep- Dept. of Physics & Astronomy, University of North Carolinaat Chapel Hill, CB 3255, Chapel Hill, NC 27599-3255, U.S.A Dept. of Astronomy, University of Michigan, 500 Church St.,Ann Arbor, MI 48109-1042, U.S.A Centro de Radioastronom´ıa y Astrof´ısica, UNAM, Apdo.Postal 72-3 (Xangari), Morelia, Michoac´an 58089, Mexico up by itself only results in cold clouds that are at mostmildly supersonic if not subsonic (Koyama & Inutsuka2002; Audit & Hennebelle 2005; Heitsch et al. 2006b,2008b). The dominant mechanism for producing super-sonic ”turbulent” motions in molecular clouds – espe-cially at column densities typical of star-forming clouds– is gravity (V´azquez-Semadeni et al. 2007, 2008, 2009;Heitsch et al. 2008b; Heitsch & Hartmann 2008). Thiswas already seen by Burkert & Hartmann (2004), whopointed out that clouds with many Jeans masses andnon-spherical geometry are generically susceptible togenerating large, spatially-variable gravitationally-drivenflows, as commonly seen in simulations with non-periodicgravity allowing global collapse (Bate et al. 2002, 2003;Heitsch et al. 2008b; V´azquez-Semadeni et al. 2007,2009). Hartmann & Burkert (2007) went further to ar-gue that large-scale gravitational collapse is a feature ofat least the Orion A molecular cloud (see also Tobin et al.2009).Relatively quiescent regions, undisturbed by energyinput from young stars, can provide good tests of thegravity-driven picture of cloud evolution. The Pipe Neb-ula is a prominent and well-studied example of a cloudof significant molecular gas mass with little perturb-ing star formation (Lombardi et al. 2006; Forbrich et al.2009). The densest regions of the Pipe generally lie alonga well-defined filamentary structure (Alves et al. 2007;Rathborne et al. 2007; Muench et al. 2007; Lada et al.2008; Rathborne et al. 2009) – a common feature ofstar-forming clouds (e.g., Schneider & Elmegreen 1979).While much attention has been paid by the above authorsto the mass function of the dense cloud cores found alongthe filaments, our interest is in the dynamical environ-ment and evolution of these cores.Lada et al. (2008) concluded that significant externalpressures were needed to confine many of Pipe cores,and attributed these pressures to ”the weight of the sur-rounding molecular cloud”. As the surrounding cloud Heitsch et al.is highly unlikely to be in hydrostatic equilibrium, this”weight” is most likely to be a dynamic pressure. In par-ticular, as the filament and cores represent a significantmass concentration, gravitational acceleration of exter-nal material should provide a confining pressure. A sig-nature of this confining infall should be supersonic linebroadening. While the internal non-thermal velocity dis-persions of the Pipe cores are subsonic, as detected indense gas tracers (see also Myers 1983; Andr´e et al. 2007;Kirk et al. 2007, for other regions), Onishi et al. (1999)found supersonic line widths in CO, qualitatively whatwould be expected from gravitationally-accelerated, in-falling material which enters the cores and filamentsthrough shocks at the boundaries (G´omez et al. 2007;Gong & Ostriker 2009). Our goal here is to make thisqualitative picture more quantitative.In this paper we attempt to develop an understandingof the gas flows driven by global gravitational collapseonto a dense filament. We start out with an analyti-cal model to estimate the expected velocities in the col-lapsing gas ( § §
3) of the flow-driven formationof an isolated low-mass cloud. These simulations showthat filament formation and global collapse are a naturalconsequence of the cloud formation process. We com-pare the analytical and numerical results, and then usethem as a guide to interpret recent observations of densegas filaments in the Pipe Nebula ( § THE INFINITE CYLINDER AND INFALL
Before examining the results from the numerical sim-ulations, it is useful to develop estimates from a simplemodel to serve as a benchmark. Consider a uniform, infi-nite, self-gravitating filament extended in the z direction,with R the radial distance in cylindrical coordinates. Thecritical line density m c (mass per unit length in the z direction), obtained by integrating momentum balancefrom R = 0 to R = ∞ is a function only of temperature(Ostriker 1964); m c = 2 c s /G , (1)where c s is the isothermal sound speed. Assuming amean molecular weight of 2 . m H , m c = 16 . T M ⊙ pc − , (2)where T is the gas temperature in (typical molecularcloud) units of 10 K. The filament has a density structureas a function of cylindrical radius of ρ = ρ (1 + R / (4 H )) − , (3) where the scale height H is given by H = c s / (2 G Σ ) = 0 . T A − V pc . (4)Here we have assumed that the relation between extinc-tion and molecular hydrogen column density is A V =1 . × /N ( H ).We may also relate the scale height to the central den-sity; H = c s πGρ . (5)For reference, the half-mass radius is at R = 2 H .Now consider a parcel of gas in free-fall toward thiscylinder, starting from rest at a cylindrical radial dis-tance R ◦ . The gravitational acceleration is a G = − Gm/R , (6)and thus the infall velocity v at R is v = 2( Gm ln( R ◦ /R )) / . (7)If we put this in terms of a static filament, v = 2 / c s ( m/m c ) / )(ln( R ◦ /R )) / (8)= 0 .
53 ln( R ◦ /R )) / ( m/m c ) / T / km s − . The infall velocity is therefore relatively insensitive tothe initial position.In a realistic situation, infalling material will shock atthe filament boundary as it adds mass to the filament.The subsequent velocity dispersion in the post-shock, fil-amentary gas will be subsonic but non-zero (G´omez et al.2007; Gong & Ostriker 2009), with a precise value de-pending upon the rate of radiative cooling.This model ignores any global motions which are gener-ally present in finite clouds. Burkert & Hartmann (2004)showed that the collapse of a circular, uniform sheet re-sults in a strong pile-up of material at the infalling edge(”gravitational edge focusing”) . Interior to this edge,the collapse timescale t c of a region of extent δr is rel-atively independent of radial position r , and is approxi-mately t c ∼ (cid:18) RπG Σ (cid:19) / , (9)where R is the inital radius of the sheet and Σ is the sur-face density. This is also approximately the time takenfor the edge of the sheet to reach the center. Equation(9) implies that the velocity difference across a region δr is roughly δv ∼ (cid:18) πG Σ R (cid:19) / δr . (10)The relative importance of the global collapse to thefilament-induced velocities will then depend upon rela-tive magnitudes of the surface density of the externalregion and the mass line density of the filament. We should point out that this effect has also been noted byLi (2001), who used it to explain clustered star formation, albeitin the context of magnetically dominated finite sheets subject toambipolar drift. hysics of the Pipe Nebula 3 NUMERICAL RESULTS
With the analytical estimates of the flow dynamicsaround filaments in hand, we apply them now to a lessidealized geometry, using a simulation of cloud growthdriven by large-scale gas flows. The goal is to testwhether the analytical estimate can reproduce the qual-itative behavior of gas collapsing onto a dense filamentin a more complex environment. We briefly summarizethe main properties of the simulation (for further de-tails, see Heitsch et al. 2008b). The simulation modelsthe flow-driven formation of an isolated molecular cloud,including the appropriate heating and cooling processesas well as self-gravity. Two gas flows colliding head-on ata shocked interface lead to compression, strong coolingand rapid fragmentation due to a combination of dynam-ical and thermal effects. The isolated cloud forming outof this collision is mildly turbulent, and with increas-ing mass it submits to global gravitational collapse per-pendicularly to the inflows. The finite cloud geometryleads to a sweep-up of material due to global gravita-tional edge focusing (see Burkert & Hartmann 2004), re-sulting in the formation of a dense filament at the cloud edge (see Fig. 1 of Heitsch & Hartmann 2008). As wewill show below, the gas infall onto the filament itselfis roughly cylindrical. It is the infall of gas onto thisfilament which we are interested in.This formation mechanism is able to reproduce someof the salient properties of molecular clouds, namelytheir internal turbulence, the predominantly filamentarystructure of their dense gas, and the observed rapid on-set of ”star” formation in the clouds. The setup is rathergeneric and is physically equivalent to e.g. the collisionof two supernova shells, the sweep-up of gas by an ex-panding shell (e.g., Patel et al. 1998), or gas swept up inspiral arms of galaxies (Elmegreen 1979, 2007; Kim et al.2003; Dobbs & Bonnell 2007).With a box size of 22 × pc and a resolution of256 × cells, our simulation does not have the spa-tial resolution to directly model the small cores seen inthe Pipe, since it does not reach the required densitiesand temperatures. Yet, we can use it to check the ap-proximate validity of the simplistic model of infall ontoa filament described in the previous section under morerealistic conditions, including the cloud’s evolution andglobal gravitational collapse.To compare the gas infall in our simulation to thephysically appropriate tracer, we estimate the CO lineemission from the model cloud. For this estimate, wemake a simple approximation, motivated by the notionthat CO formation requires shielding by dust grains (seeHeitsch & Hartmann 2008 for more details). We decidewhether CO is “present” in a particular grid cell by de-termining the attenuation of the ambient radiation fieldintegrated over solid angles. If the effective extinction isequivalent to that of an angle-averaged A V = 1 and thelocal temperature is T <
50 K, we assume CO is presentin high abundance. Because CO is rapidly dissociatedat lower extinctions (van Dishoeck & Black 1988), we donot advect CO for simplicity. The radiation field at eachgrid point is calculated by measuring the incident radi-ation for a given number of rays and averaging over theresulting sky. The ray number is determined such thatat a radius corresponding to n c = 256 cells (i.e. half the size of the larger box dimensions), each resolutionelement of the cartesian model grid is hit by one ray,i.e. n ray = 4 πn c . Thus, fine structures and strong den-sity variations are resolved (see also Heitsch et al. 2006a).With the distribution of CO thus determined, we ap-ply a three-dimensional version of the radiative transferMontecarlo code by Bernes (1979), as implemented byMardones et al. (in preparation). The code determinesthe level populations by assuming statistical equilibriumbetween collisions and radiation. The collision rates aretaken from the Leiden Atomic and Molecular Database .To lower the computational needs, we reduce the resolu-tion of the original data cube by a factor of 2. After fewtens of iterations, the level populations converge, and theline profiles in an arbitrary direction can be obtained.Figure 1 shows the velocity-integrated emission(grayscale) of the CO-map for a model cloud assembledby large-scale colliding flows (model Gf2 of Heitsch et al.2008b, at t = 14 . x )direction, and the frames measure 12 pc across. Over-plotted are the spectra for regions with T b ≥ . − − .Two features in the line profiles are noteworthy: they arefrequently asymetric, and some cases they exhibit morethan a single peak.Table 1 lists the first moments of the line profiles. Col-umn 1 shows the number of the region, as shown in Fig-ure 1, and column 2 shows the intensity in K km s − .Columns 3 & 4 list the centroid velocity and the veloc-ity dispersion, respectively, and finally, in column 5 wereport whether the line profile has multiple peaks or not.Figure 2 (left) shows the centroid velocity and the ve-locity dispersion (right), again in the y − z projection.Despite the fact that we view the cloud along the inflowdirection, the dense filament is coherent in velocity space(see also centroid velocities in center column of Table 1),except for the largish core on its right end, which is ap-proaching the observer at ≈ − . − . Thus, the ve-locity dispersion in the dense gas – the core-to-core veloc-ity dispersion – does not contain much information aboutthe velocities of the assembling flows ( ∼ . − ), infact, the motions in the (coherent) dense post-shock gasare subsonic.This also can be seen when considering the gas mo-tions perpendicular to the filament and in the plane ofthe sky, i.e. the infall of more diffuse gas onto the fila-ment (Fig. 3). The velocity profiles were calculated byidentifying the filament in the two-dimensional projec-tion (see Fig. 1). Comparing the velocity and the den-sity profiles (top and center panel), it is clear that gas athigher densities should show smaller velocity dispersions:at a distance of 2 pc from the axis of the filament, the in-fall velocity ranges between 0 . . − , i.e fromsubsonic to slightly supersonic. In other words, the densefilament is essentially a post-shock region (with subsonicinternal motions), confined by a dynamical pressure gen-erated by the (supersonic) infall of gas onto the filament.We note that the dense filament is not very well resolvedand that thus we decided not to model higher-densitytracers showing the subsonic motions more clearly –the line-profiles would not be reliable. Our choice of CO probes the immediate vicinity of dense cores, which ∼ moldata/ Heitsch et al.
Fig. 1.—
Total intensity map (in K km s − ) of CO emission from the model cloud, seen along the inflows. Line-of-sight velocity spectraare overplotted for regions with I ≥ . − . The numbers refer to Table 1. makes it an appropriate tracer for our purposes.The formation of subsonic cores in post-shock gas hasalso been discussed in the context of driven, supersonicturbulence assembling dense cores (Klessen et al. 2005;Ballesteros-Paredes et al. 2007). In that case, densitypeaks form at locations of maximum compression and ofminimum relative velocity difference. In our case (Fig. 3),the inflows are driven by the deepening gravitational po-tential of the assembling cloud, not relying on externallyforced turbulence.The bottom panel of Figure 3 compares the medianvalue of the pressures as an estimate of what processesdominate the gas dynamics in the environment of thefilament. The thick solid line corresponds to the rampressure obtained with the infall velocity along the z -direction (i.e. perpendicular to the filament in Figure 1),while the thin solid line shows the total ram pressureprofile. The gravitational pressure ρ Φ is shown by thedot-dashed line, while the thermal pressure is given bythe dashed line. Clearly, the region is more dynamicalthan just indicated by the mean infall velocities: as sug- gested by this figure, the infall (upper panel) is driven bythe gravitational potential (lower panel), while the ther-mal pressure responds only weakly to the gravitationalpressure.To get a clearer view of the dynamics, we show in Fig. 4the three projections of the datacube, with the line pro-files overplotted, as in Fig. 1. The x − y panel is on thetop of the cube, the x − z on the left, and the y − z onthe right. These projections reveal that the filament isactually tilted by approximately 45 degrees in the x − y plane, with the left side closer to the observer locatedat x → ∞ . From the three-dimensonal datacube, weknow that the infall velocities in the three directions areof similar magnitude. However, this does not necessar-ily translates to an infall, double-peaked line profile. Al-though the three projections exhibit some double-peakedprofiles, most of our CO lines exhibit only asymmetric,supersonic ( σ v ∼ . − ) profiles. Thus, the super-sonic line widths observed in molecular clouds that areusually attributed to random motions (or ”turbulence”),very likely contain a substantial component of ordered hysics of the Pipe Nebula 5 Fig. 2.—
Left:
Centroid velocity (in km s − ) of the model cloud, seen along the inflows. The filament is coherent in velocity space. Right:
Velocity dispersion (in km s − ) of the model cloud, as seen along the inflows. The densest regions have the smallest velocity dispersion. Fig. 3.—
Velocity, density and pressure profiles of gas falling ontothe filament, in the plane of sky, against distance to the filament.Negative distances are located below the filament. For the veloc-ities, the solid line denotes the centroid (mean) velocity, and thedashed lines indicate the 1 σ scatter. For the density, solid linesstand for median values, while dashed lines indicate the lower andupper quartiles. The pressure plot shows the ram, internal andgravitational pressure as indicated. The thin solid line stands forthe total ram pressure, i.e. all three components of the velocitycontribute. motions due to coordinated global collapse. A filamenttilted with respect to the line-of-sight – a very likely sce-nario –, only helps to suppress clear signatures of coor-dinated infall onto the filament, giving the impression of”turbulent” motions.Table 2 summarizes the measured properties of themodel filament, as identified in the two-dimensional pro-jection. Since the massive cores potentially could affectthe geometry, we show numbers for the full filament (in-cluding the cores), and the left half of the filament, whichdoes not contain massive cores.We note that the filament line density is approximatelytwice the expected limit for the isothermal case (for thehigher temperature of 28 K in the simulation). As men-tioned above, the filament is tilted, resulting in a shorterprojected length, and thus in a higher line density. Also,other effects such as limited resolution and the possibil-ity that the filament is not static but is still accretingmass on timescales of order 1 Myr, could play a role.Given that the model filament does not agree with thestatic approximation, we use equation (7) for the fullfilament to estimate v ∼ . R ◦ /R ) / km s − . Whatvalue of R ◦ to use is not clear, but if we take R ◦ = 2 pcand R = 0 . v ∼ . − , thisis about twice the average infall velocity at the filament(Figure 3).To summarize, Figure 3 demonstrates the overall col-lapse of the cloud in addition to acceleration onto thefilament. Obviously, equation (10) can describe the gasdynamics of the simulated cloud only in a very generalway, as (a) matter is being added to the cloud duringits collapse from the inflows, (b) the cloud (or filament)cross-section is not exactly circular, and as (c) there areshear components in the flows. Nevertheless, the overallcollapse motions are approximately cylindrical (see totalram pressure profile in Fig. 3), indicating that at thisstage of the simulation the massive filament dominates Heitsch et al. Fig. 4.—
The three intensity projections of the datacube. The flows are along the x -axis, and Figure 1 shows the ( y, z )-plane. the gravitational potential and thus the dynamics of thecloud, and justifying equation (10) as a rough descriptionof the actual dynamics. The total mass of the cloud, in-cluding the filament itself, at this epoch is ∼ ⊙ distributed over a region approximately 10 × ∼ × − g cm − or 1 . × cm − . Assuming the initial radius in the y -direction was ∼
10 pc, equation (10) then implies a ve-locity gradient across a region of 2 pc of δv ∼ . − ,which again is in reasonable agreement with the resultsin Figure 2. Considering the very crude nature of these estimates, agreement with the numerical simulation atthe factor of two level is adequate. COMPARISON WITH THE PIPE NEBULA
Motivated by the global, approximately cylindrical in-fall observed in the simulation ( §
3) under rather generalconditions, we first discuss in this section our analyticalestimates ( §
2) and then the simulation results in the con-text of the Pipe nebula. We summarize the relevant ob-servational findings first, and then use the approximateagreement between observations and models to interprethysics of the Pipe Nebula 7 number I [K km s − ] h v i [km s − ] σv [km s − ] mult1 3 . − .
65 0 .
82 yes2 9 . − .
48 0 .
67 yes3 3 . − .
78 0 .
72 yes4 10 . − .
67 0 .
51 yes5 2 . − .
79 0 .
86 yes6 4 .
66 0 .
01 0 .
50 no7 5 . − .
05 0 .
54 yes8 4 . − .
10 0 .
43 no9 3 .
04 0 .
35 0 .
64 no10 4 .
41 0 .
25 0 .
57 no11 7 .
69 0 .
12 0 .
54 yes12 8 . − .
30 0 .
50 yes13 8 . − .
41 0 .
53 yes14 5 . − .
52 0 .
53 no15 5 .
20 0 .
11 0 .
39 no16 25 . − .
11 0 .
29 no17 5 . − .
05 0 .
46 yes18 2 . − .
19 0 .
61 no19 14 . − .
45 0 .
48 no20 23 . − .
31 0 .
43 no21 8 . − .
56 0 .
53 yes22 3 .
77 0 .
13 0 .
48 no23 16 . − .
17 0 .
36 no24 7 .
53 0 .
27 0 .
56 no25 6 .
25 0 .
15 0 .
53 no26 3 .
60 0 .
05 0 .
51 no27 3 . − .
15 0 .
63 no28 6 . − .
34 0 .
61 no29 12 . − .
25 0 .
47 no30 5 . − .
85 0 .
58 yes31 2 . − .
81 0 .
73 no32 5 .
29 0 .
46 0 .
71 no33 7 .
63 0 .
25 0 .
55 no34 17 .
64 0 .
08 0 .
33 no35 7 .
99 0 .
22 0 .
52 yes36 4 .
26 0 .
31 0 .
68 yes37 21 .
40 0 .
12 0 .
52 yes38 3 .
53 0 .
53 0 .
80 no39 4 .
22 0 .
13 0 .
67 no40 11 .
92 0 .
25 0 .
55 no41 7 .
41 0 .
24 0 .
59 no42 2 .
99 0 .
27 0 .
65 no43 7 .
96 0 .
45 0 .
68 no44 12 .
80 0 .
19 0 .
41 no45 9 .
20 0 .
16 0 .
45 no46 5 . − .
29 0 .
53 no47 14 .
63 0 .
08 0 .
45 no48 6 .
65 0 .
28 0 .
60 yes49 25 .
58 0 .
14 0 .
43 no50 12 .
54 0 .
09 0 .
36 no51 6 .
06 0 .
15 0 .
52 no52 2 .
95 0 .
19 0 .
66 no53 3 .
48 0 .
14 0 .
58 no54 3 . − .
19 0 .
61 no55 4 . − .
06 0 .
63 no56 4 . − .
09 0 .
56 no57 6 .
97 0 .
18 0 .
50 no58 4 . − .
05 0 .
47 no59 4 . − .
05 0 .
46 no
TABLE 1Parameters of subfields in Figure 1. The last columnindicates the existence of an obvious second component.
M T L m c m c [M ⊙ ] [K] [pc] measured expectedfull filament 1 . ×
28 12 9 . . . ×
27 5 . . . TABLE 2Filament parameters for models, to be compared to thetoy model of §
2. The mass per length m c is given inM ⊙ pc − . The first line gives the numbers for the fullfilament (as used for Figure 3), while the second linerefers to the left half of the filament which does notcontain any cores. the physical conditions of the Pipe nebula in the contextof global infall ( § ∼ . ⊙ to about 25M ⊙ , with radii ranging from ∼ .
04 to ∼ . ) of ∼ cm − . The FWHM of the C O lines ranges fromabout 0.4 to about 1 km s − . The estimated visual ex-tinction rises slowly with mass until about M ∼ ⊙ , atwhich point A V rises rapidly from about ∼ − M ∼ . ⊙ , R ∼ .
07 pc,and FWHM ∼ .
36 km s − . Such a “median” core is typ-ically pressure-confined according to Lada et al. (2008).To simplify the comparison with observations, we assumea typical temperature of 10 K, though there is some ev-idence that some of the Pipe cores have slightly highertemperatures, ∼ −
15 K (Rathborne et al. 2007).To proceed further we need an estimate of the averageline density of the filament. The Pipe, though highlyelongated, is not a perfectly straight filament nor is ituniform. The sum of the core masses in Rathborne et al.(2009) is 228M ⊙ ; distributing this mass over a length ∼ ◦ ∼
20 pc results in an average line density ∼ ⊙ pc − . This estimate neglects the inter-core massbut includes the “bowl” region which is much more com-plex in structure. If we restrict attention to the por-tion of the pipe at negative galactic longitudes, the totalcore mass is 105M ⊙ over ∼
15 pc or m ∼ ⊙ pc − .As the Pipe does not exhibit considerable star forma-tion at the present epoch (Forbrich et al. (2009) esti-mate a star formation efficiency of 0 .
06% when compar-ing to the total cloud mass given by Onishi et al. 1999),it seems reasonable to assume an average line densityof m ∼ ⊙ pc − , to include mass outside the corebut not so much that the filament would be radiallygravitationally-unstable.To further fix ideas we take an average filament opticaldepth to be A V ∼
2. This is near the low end of thecore extinction values found in Rathborne et al. (2009).Then from equation (4), the scale height is H ∼ .
095 pc;this seems reasonable, seeing that the median core radiusgiven by Rathborne et al. (2009) is about 0.07 pc. Wemay also then derive a central density from equation (5)of ρ ∼ . × − g cm − or n ( H ) ∼ . × cm − .This density is a few times smaller than the average coredensity n ( H ) ∼ cm − , but we have not taken anyconfining pressure into account.We are now in a position to calculate an estimate ofthe infall velocity using equation (9). Taking R ◦ /R = 10, Heitsch et al.we find v ∼ .
77 km s − . The timescale for this motion issensitive to the choice of initial condition, i.e. the initialvelocity; for the above solution, the time taken to fall infrom 0 . . ∼ (4 /π ) × .
77 km s − . If we identify this with theFWHM or line width, our results are consistent withthe median CO line width of ∼ . − found byOnishi et al. (1999, see their Fig. 6b). It is larger thanthat observed in the C O cores by about a factor of 2 . Oobservations sample material both outside and inside thecores. As mentioned in §§ et al .state that, in the lower-mass cores, half or more of the to-tal line-of-sight column density arises from material out-side the core (also G´omez et al. 2007; Gong & Ostriker2009). Conversely, this means that for many cores, halfor less of the emitting material is outside the core. Thuswe would predict a core velocity width perhaps half thatabove, or around ∼ . − , in rough agreement withobservations (Muench et al. 2007). In addition, there isno specific reason to assume that the filament is orientedperpendicularly to the line of sight. Any (very likely) tiltof the filament with respect to the line of sight will resultin an even smaller projected velocity dispersion.Going back to the numerical simulations, we note thatthe velocity dispersion in the flow direction, calculatedas the second moment of the line profile (see Fig. 1 andTable 1), are or the order of ∼ . − . This impliesthat the FWHM, calculated as √ σ v , is of the orderof ∼ . − , a value close to the observed 1 . − by Onishi et al. (1999) for the Pipe. The discrepancywith the analytic result, however, is probably due in partto the more complex velocity field of the simulation, in-cluding the global collapse. In some cases the observedline width is inflated because of the superposition of sep-arate cores along the line of sight, which can be seen insystems with two peaks of emission (see also last columnin Table 1). In any case the simulations are in reasonableagreement with the Onishi et al . observations.While there are other possible mechanisms to gen-erate the observed velocity dispersions in moleculargas (one of them would be MHD turbulence, see e.g.McKee & Zweibel (1995) for a detailed discussion), wepoint out that in a cloud of many Jeans masses, of non-uniform density and of irregular geometry, it will be im-possible to avoid having significant gravitational accel-erations locally. These in turn will produce motions onthe order of 1 km s − (see Fig. 3 and § DISCUSSION & CONCLUSIONS
The results presented in this work support theidea that molecular clouds are in a general state ofglobal collapse, suggested more than 35 years ago byGoldreich & Kwan (1974). Although such global col-lapse of irregular structures develops internal turbulenceat some level, the large linewidths in MCs are causedmainly by the large-scale systematic inward motions inthis scenario. Given the irregularities, angular momen-tum conservation and the high Reynolds numbers, itwould actually be outright surprising if the collapse did not generate some ”turbulence”.Zuckerman & Evans (1974) suggested that such large-scale collapse would result in a star formation rate muchhigher than observed, mandating some mechanism ofcloud support to render star formation inefficient (seeMcKee 1999 for a summary). Yet, analytical and numer-ical studies (e.g., Burkert & Hartmann 2004; Field et al.2008; Heitsch et al. 2008b; Heitsch & Hartmann 2008;V´azquez-Semadeni et al. 2007, 2008, 2009) demonstratethat MCs, while globally collapsing, are highly suscepti-ble to strong fragmentation. While this fragmentation isto some extent a consequence of a combination of ther-mal and dynamical instabilities during the cloud forma-tion process (Heitsch et al. 2008a), it is mainly due to thefact that the cloud geometry is finite, leading to gravita-tional edge focusing, i.e. non-linear gravitational accel-erations as a function of position (Burkert & Hartmann2004). In other words, the existence of non-linear grav-itational accelerations as a function of position, leadingto a rapid piling up of material as well as local fragmen-tation, allows local collapse to proceed faster than globalcollapse. And it is precisely because small, high-densitystructures are rapidly developing, that the mass involvedin the densest regions has to be small, ensuring a smallstar formation efficiency. In other words, the observedlow star formation efficiency seems to be the testimonyof the importance of non-linear acceleration at particularplaces while the global collapse occurs. In this connec-tion it is worth pointing out that the major region of starformation within the Pipe (with 15 YSOs; Covey et al.2009), the B59 cloud, lies at one end of the filament,which is a preferred locus for gravitational edge focusing(Burkert & Hartmann 2004).While there are clearly a number of uncertainties inboth the analytical and numerical calculations and inthe precise observational quantities, a simple model ofgravitational infall toward the filament clearly can ac-count for the observed non-thermal velocity dispersionsin the Pipe. The gravity-driven model also does pre-dict small (core-to-core) velocity dispersions consistentwith observations (Hartmann 2002; Walsh et al. 2004,2007; Kirk et al. 2009). This agreement is possible be-cause the “turbulence” is not being continuously drivenby an ad hoc force but is the result of (global) gravita-tional acceleration. The simulation results also suggestthat while gas is falling onto the filament supersonically,the velocity dispersions in the dense post-shock gas aresubsonic, consistent with the more detailed treatment ofcore formation in post-shock gas by G´omez et al. (2007);Gong & Ostriker (2009). Yet the current simulation isnot sufficiently resolved to support more detailed state-hysics of the Pipe Nebula 9ments about the dense cores.Lada et al. (2008) suggested that the confining pres-sure for many Pipe cores was the result of the “weight”of the cloud. Here we refine this suggestion by point-ing out that a static pressure estimate is not strictly ap-propriate for a cloud in supersonic motion. The rampressure that we infer is, however, another aspect ofthe same physical mechanism - gravitational accelera-tion. It is also worth noting that, in this interpreta-tion, the non-thermal velocity dispersion in the cores andsurrounding regions are dominated by collapse motionsand thus do not provide pressure support against grav-ity (Ballesteros-Paredes et al. 1999b; Ballesteros-Paredes2006; V´azquez-Semadeni et al. 2008). Nor does in thispicture a driving source for turbulence within molecularclouds appear to be necessary (see also Field et al. 2008;Heitsch & Hartmann 2008).The numerical simulations, with an average infall ve-locity in the y -direction of ∼ . − and a densityof ∼ cm − at the filament, imply ram pressures oforder P/k ∼ × cm − K. Scaling this result from the ∼ ⊙ filament of the simulation to the ∼ ⊙ estimated for the Pipe region from CO (Onishi et al.1999) results in a predicted pressure 3 times larger, orabout 6 × cm − K. This is in reasonable agreementwith the confining pressures for Pipe cores estimated byLada et al. (2008) of
P/k ∼ × cm − K.The core mass function (CMF) inferred for the Pipe(Alves et al. 2007; Lada et al. 2008; Rathborne et al.2009) peaks at a mass well above the typically-estimatedpeak of the stellar initial mass function (IMF). In thegravity-driven picture, translating the CMF at an in-stant of time to an IMF is complicated because themodel of gravitationally-accelerated infall implies thatthe filament - and thus the cores - are accreting mass(see Clark et al. 2007 for a related argument). If we setthe ram pressure of infall equal to confining pressure in-ferred by Lada et al. (2008), and use v = 0 .
77 km s − from the analytic model, the implied density of infallingmaterial ρ ∼ . × − g cm − or N ( H ) ∼
480 cm − .The mass accumulation rate over a cylindrical regionof radius R = 0 . z = 0 . πRzρv ∼ . × − M ⊙ yr − . While this estimateis obviously sensitive to the adopted parameters, it isclear that the lower-mass cores can quite plausibly dou-ble their mass over timescales of less than 1 Myr (see also G´omez et al. 2007; Gong & Ostriker 2009). In fact, ob-servations of starless cores are starting to indicate thatquiescent, coherent dense starless cores do accrete ac-tively (e.g., Schnee et al. 2007). Alternatively, the nu-merical simulation shows a doubling of mass in cores overa timescale of 1 Myr (Heitsch & Hartmann 2008), con-sistent with this estimate.The long-range nature of gravity makes it very difficultto avoid the generation of supersonic velocity fields andthe production of filaments and other dense structures inclouds with many Jeans masses (Burkert & Hartmann2004; Bate et al. 2002, 2003; V´azquez-Semadeni et al.2007; Heitsch et al. 2008b; Smith et al. 2009).The mass, spatial extent, and typical column densitiesof the Pipe are not very different from those of the Tau-rus molecular cloud. It is conceivable that the Pipe isrepresentative of how Taurus would have appeared 1 -2 Myr ago. The average filament line density in Tau-rus is estimated to be roughly twice the critical value(Hartmann 2002), which is consistent with the active starformation in the region. Blindly applying the mass infallrate calculated above to Taurus would imply a buildupfrom ∼ ⊙ pc − to the observed ∼ ⊙ pc − on atimescale of 1 Myr.Finally, one may consider the future evolution of thePipe nebula with regards to the star formation efficiency.While we have shown that besides the gas dynamics,the gravity-driven model can naturally explain the originof the strong fragmentation as a key ingredient for theobserved low star formation efficiency, we have not dis-cussed how the remaining molecular gas can be preventedfrom eventually collapsing onto the dense filament. Inlow-mass star-forming regions this could be achievedby e.g. magnetic support of the diffuse molecular gas(Heyer et al. 2008; Price & Bate 2009), or by Galactictidal disruption (Ballesteros-Paredes et al. 2009a,b).We thank the referee for comments and questions thathelped to improve our presentation. This work was sup-ported in part by NSF grant AST-0807305 and by theUniversity of Michigan. Computations were performedat the National Center for Supercomputing Applications(AST 060034). J.P.-B. acknowledges support from grantUNAM-PAPIIT IN110409. This work has made use ofNASA’s Astrophysics Data System. REFERENCESAlves, J., Lombardi, M., & Lada, C. J. 2007, A&A, 462, L17Andr´e, P., Belloche, A., Motte, F., & Peretto, N. 2007, A&A, 472,519Audit, E. & Hennebelle, P. 2005, A&A, 433, 1Ballesteros-Paredes, J. 2006, MNRAS, 372, 443Ballesteros-Paredes, J., G´omez, G. C., Loinard, L., Torres, R. M.,& Pichardo, B. 2009a, MNRAS, 395, L81Ballesteros-Paredes, J., G´omez, G. C., Pichardo, B., &V´azquez-Semadeni, E. 2009b, MNRAS, 393, 1563Ballesteros-Paredes, J., Hartmann, L., & V´azquez-Semadeni, E.1999a, ApJ, 527, 285Ballesteros-Paredes, J., Klessen, R. S., Mac Low, M.-M., &Vazquez-Semadeni, E. 2007, in Protostars and Planets V, ed.B. Reipurth, D. Jewitt, & K. Keil, 63–80Ballesteros-Paredes, J., V´azquez-Semadeni, E., & Scalo, J. 1999b,ApJ, 515, 286Bate, M. R., Bonnell, I. A., & Bromm, V. 2002, MNRAS, 332,L65 —. 2003, MNRAS, 339, 577Bernes, C. 1979, A&A, 73, 67Burkert, A. & Hartmann, L. 2004, ApJ, 616, 288Clark, P. C., Klessen, R. S., & Bonnell, I. A. 2007, MNRAS, 379,57Covey, K. R., Lada, C., Muench, A., & Rom´an-Z´u˜niga, C. 2009,in Bulletin of the American Astronomical Society, Vol. 41,Bulletin of the American Astronomical Society, 399Dobbs, C. L. & Bonnell, I. A. 2007, MNRAS, 376, 1747Elmegreen, B. G. 1979, ApJ, 231, 372—. 2007, ApJ, 668, 1064Field, G. B., Blackman, E. G., & Keto, E. R. 2008, MNRAS, 385,181Forbrich, J., Lada, C. J., Muench, A. A., Alves, J., & Lombardi,M. 2009, ArXiv e-prints, 0908.4086Goldreich, P. & Kwan, J. 1974, ApJ, 189, 441G´omez, G. C., V´azquez-Semadeni, E., Shadmehri, M., &Ballesteros-Paredes, J. 2007, ApJ, 669, 10420 Heitsch et al.