The Effects of Inhomogeneities within Colliding Flows on the Formation and Evolution of Molecular Clouds
TThe Effects of Inhomogeneities within Colliding Flows on theFormation and Evolution of Molecular Clouds
Jonathan J. Carroll-Nellenback , Adam Frank , Fabian Heitsch [email protected] Received ; accepted Department of Physics and Astronomy, University of Rochester, Rochester, NY 14620 Department of Physics and Astronomy, University of North Carolina Chapel Hill, ChapelHill, NC 27599 a r X i v : . [ a s t r o - ph . GA ] A p r ABSTRACT
Observational evidence from local star-forming regions mandates that star for-mation occurs shortly after, or even during, molecular cloud formation. Models ofthe formation of molecular clouds in large-scale converging flows have identifiedthe physical mechanisms driving the necessary rapid fragmentation. They alsopoint to global gravitational collapse driving supersonic turbulence in molecularclouds. Previous cloud formation models have focused on turbulence genera-tion, gravitational collapse, magnetic fields, and feedback. Here, we explore theeffect of structure in the flow on the resulting clouds and the ensuing gravita-tional collapse. We compare two extreme cases, one with a collision between twosmooth streams, and one with streams containing small clumps. We find thatstructured converging flows lead to a delay of local gravitational collapse (“starformation”). Thus, more gas has time to accumulate, eventually leading to astrong global collapse, and thus to a high star formation rate. Uniform converg-ing flows fragment hydrodynamically early on, leading to the rapid onset of localgravitational collapse and an overall low sink formation rate.
Subject headings: instabilities — gravity — turbulence — methods:numerical —stars:formation — ISM:clouds
1. Introduction
The concept of flow-driven cloud formation (V´azquez-Semadeni et al. 1995; Ballesteros-Paredes et al. 1999; Hartmann et al. 2001) can explain two observational constraints onhow molecular clouds form stars, derived from local star-forming regions: first, all localmolecular clouds are observed to be forming stars, and second, the stellar age spreads are 3 –on the order of 1 −
2. Method, Initial Conditions, and Parameters
To model a finite molecular cloud forming in a collision of two flows, we use theadaptive-mesh-refinement code AstroBEAR 2.0 to solve the equations of hydrodynamicsincluding self-gravity and equilibrium cooling. For a detailed discussion of AstroBEAR,see Carroll-Nellenback et al. (2013). Poisson’s equation is solved with HYPRE (Falgout &Yang 2002). We used a non-split CTU integrator following Gardiner & Stone (2008), and 5 –the sink particle implementation discussed by Federrath et al. (2010).We performed two simulations of 40 pc diameter flows with a mean density n = 1 . − , colliding head-on at v = 8 .
25 km s − for a period of 30 Myr. The flowscollide in the y-z plane within a box that is 62 . × ×
100 pc in size. We use a basegrid of 40 × ×
64 cells with 5 additional levels of refinement for an effective resolution of1280 × × ≈ .
05 pc. The flows combine to give a massflux of ˙ M = 665 M (cid:12) Myr − and a ram pressure of P ram = 10472 K cm − . We used anideal equation of state at γ = 5 /
3, with a mean particle mass χ = 1 .
27 and a parametrizedcooling function S that includes heating terms consistent with Inoue & Inutsuka (2008)though modified to give lower temperatures 10 K at higher densities ( n > cm − ) toaccount for UV shielding (Ryan & Heitsch in prep). S = n ( − Γ + n Λ)erg cm − s − Γ = 2 × − = 1 . × exp (cid:0) − T +1000 (cid:1) + 1 . × − √ T exp (cid:16) − . . ,T − (cid:17) (1)The combined heating and cooling results in a thermal equilibrium pressure for eachdensity. This curve can be seen in the dashed line of figure 1. Note the dashed line onlyextends to densities of 10 cm − to avoid confusion at higher densities - but the equilibriumcurve can be seen at higher densities in the distribution itself because the thermal timescalesare much smaller than any other time scale at those densities, and thus the gas lines upwith the equilibrium curve. In the “Smooth” simulation, the inflowing gas has a uniformdensity of 1 . − and a thermal equilibrium pressure of 4931 K cm − . For the “Clumpy”simulation, the mean inflow density is also 1 . − , yet the flow contains many smallclumps of radius 0 .
55 pc and a density of 15 . − , placed randomly in a smooth lowerbackground density of 0 .
25 cm − . Both the clumps and the low density background are inpressure equilibrium at 6857 K cm − and both are stable to thermal instabilities, though 6 –they are not in thermal equilibrium with each other. The high density contrast χ = 60 . f = 0 .
05. The clump radiuswas chosen to be much less than the Jeans length at the clump density and pressure( L J = 43 . . k − .
3. Results
All of the following analysis focuses on a hockey-puck shaped region that is 40 pc indiameter and 10 pc thick, centered on the interface between the two flows. This region isoutlined in figures 2 and 3. Figure 1 shows the joint probability distribution in logarithmicdensity-pressure space for both runs at times 10 . , .
1, and 27 . In the Smooth model (left column), material enters the collision region on theequilibrium curve at 1 . − which coincides with the peak in the equilibrium coolingcurve. As the material collides with the oppositely directed material it is initially compressedadiabatically up to the flow ram pressure at 1 . × cm − . It then cools and compressesonto the thermal equilibrium curve. With time, more material piles up at higher densities.Eventually, self-gravity takes over at the highest densities and compresses this materialfurther, above the ram pressure provided by the flow. At this point, gas collapses and formsa core, or is being accreted by an existing core. The core formation or accretion explainsthe lack of material at densities above ≈ . cm − . 7 –Fig. 1.— Density-weighted joint probability distribution function for density vs. pressure forthe Smooth run (left) and Clumpy run (right), at 10 . , .
1, and 27 . T ≡
10 K. 8 –The Clumpy flow on the other hand (right column) has material entering the collisionregion at both 0 .
25 cm − and 15 . − . Additionally, some mixing occurs between theclumps and the background flow which causes the thick band of material below the thermalequilibrium curve. At these densities, the thermal time scales are longer than the dynamicaltime [see Fig. 3 of Heitsch et al. (2008b)] - so this material does not equilibrate beforecolliding with the oppositely directed flow. The low density background appears to alsocompress adiabatically though not to as high pressures as the Smooth run. At the interfacebetween the two flows there are three possible types of interactions due to the two densitiespresent in the flow: (a) For background-background collisions, the ram pressure will be1 / v = 16 . − , resulting in a rampressure equal to that in the Smooth model, at 1 . × K cm − . (c) Finally, head-onclump-clump collisions, though rare, can produce pressures 15 . Figures 2 and 3 show column densities taken along the flow axis (left) and normal tothe flow axis (right) for the Smooth and Clumpy runs respectively. Also shown are theboundaries of the “hockey puck” region used for the following analysis. The Smooth runexhibits the usual filamentary structure due primarily to the non-linear thin shell instability[NTSI, (Vishniac 1994; Hueckstaedt 2003; Heitsch et al. 2005; V´azquez-Semadeni et al.2006)], and at later times gravity. Also visible is material which has been “splashed”radially outward from the collision region due to the high ram pressures. The NTSI focusesmaterial into various nodes and by 10 . . . / t cc = χr c v w (Klein et al. 1994) where v w is the ‘wind velocity’ as seen by the clump and χ is the density contrast. Since the clump isitself traveling into an oppositely directed flow, v w = 2 v and the distance the clump willtravel will be of order D = t cc v = χr c . . (2)If the clump survives for a few clump crushing times, it will travel distances of ≈
10 pcbefore being destroyed. This explains the more extended interaction region in the upperright panel of figure 3. Later in time, the clumps pass through a denser wall of materialthat has built up and they are also pulled back by gravity - so the extent of the collisionregion shrinks over time. While the Smooth run has formed nine isolated cores by 20 . . Myr, the entire region isundergoing rapid global collapse and a dense group of 20 cores has formed again near thecenter of the potential well. 10 –Fig. 2.— Column density in units of cm − projected parallel (left) and perpendicular (right)to flow axis, at 10 . , .
1, and 27 . − , projected parallel (left) and perpendicular (right)to flow axis, at 10 . , .
1, and 27 . To generate spectra we first take a cube of size d = 40 .
625 pc, centered on thesimulation domain. The data within the cube is then windowed with w ( r ) = cos( πr/d ) : r < = d/
20 : r > d/ . . . Figure 5 shows the evolution the energy densities at large and small scales. We choosethe clump shredding distance D = 4 . This excess kineticenergy on small scales suppresses local collapse (remember that the clumps themselves aregravitationally stable) but cannot prevent global collapse - while in the Smooth run, the
14 – higher degree of kinetic energy on large scales resists global collapse but not local collapse.
Another way to see this is that the shocks in the Smooth run will fragment quickly due tothe thermal instability. Yet, the velocity dispersion between the fragments will be small, atleast smaller than for the Clumpy run (see Fig. 6, right), thus forming structures that aremore or less coherent in velocity space. Thus, local collapse is seeded. For the Clumpy run,thermal instability does not play much of a role, and gas accretion onto the clumps due tocooling or gravity is negligible within the (dynamical) timescales considered. Thus, localcollapse is suppressed, while global collapse sets in once enough mass in clumps has beenassembled.
The left panel of Figure 6 shows the growth in total mass within the collision regionas well as the theoretical upper bound (dotted line) using the mass flux ˙ M . Both runscollect mass at the inflow rate ˙ M for the first two Myr, after which the growth rate drops.For the Smooth run, material is being splashed radially outwards, and at later times,some of the NTSI fingers develop past the analysis region (see Fig. 2). Eventually, after15 Myr, material is falling back in from the edges of the analysis region, increasing the masscollection rate again. In the Clumpy run, some of the clumps exit the analysis region onthe far side after 2 Myr. The overall mass collection rate slowly increases after that, withmaterial falling back in, and eventually collapsing globally.The Smooth run begins forming cores at 10 Myr, and by 25 Myr, the rate of totalmass growth and core mass growth have become equal. This implies that material is beingaccreted by the cores at the same rate it is entering the collision region. The Clumpy run(Figure 3) does not begin to form cores until 20 Myr, but then quickly accretes gas at arate higher than the mass flux into the region. This suggests a degree of global collapse not 15 – E ( k ) k/k min k -2 E ( k ) k/k min Fig. 4.— Kinetic energy (left) and gravitational energy spectra (right) for both runs andthree times as indicated in the diagram. The vertical line indicates the clump shreddingdistance (eq. 2). E ne r g y D en s i t y ( x K / cc ) E ne r g y D en s i t y ( x K / cc ) Fig. 5.— Plots showing time evolution of mean kinetic and gravitational energy densitiessplit between large ( > D , see eq. 2) and small scales for both the Clumpy run (left) and theSmooth run (right). 16 –present in the Smooth run.Since the gas is being compressed and cooled, the Jeans mass at a given sound speed c s and a mass column density Σ M J = 1 . c s G Σ (3)will drop with time, as shown in Figure 6 (left panel). It levels out once the minimumtemperature of ≈
10 K is reached (this is only obvious in the Clumpy run, dashed lines,for t >
25 Myr. The Jeans mass for the Clumpy run is smaller by at least an order ofmagnitude, because of the clumps at higher densities and lower temperatures. Yet, sincethese clumps do not form a coherent region with
M > M J , local gravitational collapse issuppressed until ≈
20 Myr, and sinks form only once global collapse sets in, indicated bythe increasing slope of the total mass, black dashed line. The onset of global collapse in theClumpy run is also visible in Figure 5 (left), and in the velocity dispersions shown in theright panel of Figure 6.The Smooth run has a substantially larger Jeans mass that does not level out at aminimum within the model run time. Yet, because of the rapid local fragmentation intostructures larger than a local Jeans mass, local collapse (and sink formation) sets in at ≈
10 Myr. There is no signature of global collapse in the velocity dispersions, or in theenergies.Figure 7 shows the distribution of core masses for both runs at 27 . <
100 M (cid:12) consistent with the idea of local collapse.The Clumpy run shows many more high density cores (100 − (cid:12) ) visible in the centerof the potential well (Fig. 3) due to global collapse. 17 – Smooth Total MassSmooth Jeans MassSmooth Core MassClumpy Total MassClumpy Jeans MassClumpy Core MassTheoretical Mass
Time (Myr) M / M סּ Time (Myr) D en s i t y w e i gh t ed v e l o c i t y d i s pe r s i on ( k m / s ) Fig. 6.—
Left:
Mass history against time for the Clumpy and Smooth run. Black linesindicate total mass in the analysis region, dark gray lines trace the Jeans mass, also inthe analysis region, and the light gray lines follow the mass in sinks, tracing local collapse.The dotted line stands for the mass accumulation expected from simple sweep-up. Localcollapse is suppressed in the Clumpy run until ≈
20 Myr, while the Smooth run forms sinksafter ≈
10 Myr.
Right:
Density-weighted velocity dispersion against time, for the Clumpyand Smooth run, again within the analysis region. The Clumpy dispersion is systematicallyhigher until ≈
15 Myr, and increases again once global collapse sets in at 22 Myr. 18 –
In both runs, material injected from the left and right side was marked with a tracer( ρ L and ρ R ) proportional to the density so that the amount of mixing could be investigated.We then define a mixing ratio M R = 2 min ( ρ L , ρ R )max ( ρ, ρ L + ρ R ) . (4)Thus, M R = 0 indicates the presence of only one tracer (or none - as is the case in theambient medium outside of the flow), and
M R = 1 indicates equal amounts of both tracerswith no ambient material mixed in. Since we are confining our analysis to the colliding flowregion, there should be no ambient material present so ρ = ρ L + ρ R - and the definition isequivalent to M R = 2 min ( ρ L , ρ R ) ρ L + ρ R . (5)In the Smooth run, there is a higher mass-fraction of well-mixed material (Fig. 8, leftpanel; note that the total masses in the analysis region at that time are comparable). TheClumpy run tends to have a more spread out distribution of mixing ratios than the Smoothrun. As clumps drive through the opposing stream - they will shed some of their materialand provide varying amounts of mixing. In the Smooth case, the flows interact along a thininterface and it is difficult to get unequal amounts of material from either side in the sameregion. One might expect the Clumpy run to have less well-mixed cores, yet the right panelof figure 8 shows just the opposite. The Smooth run has more cores with lower mixingratios.One possible explanation for this is that the NTSI creates nodes that act to funnelmaterial streaming into the “trough” from only side, while diverting material from theother side. If so, then cores that formed to the left of the collisional mid-plane should be 19 –Fig. 7.— Core mass distribution at 27 . Left:
Mass-weighted mixing ratio (eq. 5) for the gas in the analysis region, at10 . Right:
Mixing ratio of cores at 27 . M B = ρ R − ρ L ρ R + ρ L (6)and plot it against the core’s distance from the y-z mid-plane (Fig. 9). Indeed, cores withnegative offsets (left of mid-plane) tend to have a higher value for the mixing bias so theyhave more right tracer and are comprised of material primarily from the right side, and viceversa. Note that this bias is absent in the cores formed in the Clumpy run, whose cores areclustered around 0.
4. Discussion and Conclusion