The Dynamical State of Filamentary Infrared Dark Clouds
aa r X i v : . [ a s t r o - ph . GA ] J a n Draft version November 8, 2018
Preprint typeset using L A TEX style emulateapj v. 11/12/01
THE DYNAMICAL STATE OF FILAMENTARY INFRARED DARK CLOUDS
Audra K. Hernandez
Department of Astronomy, University of Florida, Gainesville, FL 32611, USA;[email protected]fl.edu
Jonathan C. Tan
Departments of Astronomy & Physics, University of Florida, Gainesville, FL 32611, USA;[email protected]fl.edu
Draft version November 8, 2018
ABSTRACTThe dense, cold gas of Infrared Dark Clouds (IRDCs) is thought to be representative of the initialconditions of massive star and star cluster formation. We analyze CO J = 1 − CO, Σ , with those derived from mid-infrared small median filter extinctionmapping, Σ
SMF , by Butler & Tan. After accounting for molecular envelopes around the filaments, wefind approximately linear relations between Σ and Σ
SMF , i.e. an approximately constant ratioΣ / Σ SMF in the clouds. There is a variation of about a factor of two between the two clouds. Wefind evidence for a modest decrease of Σ / Σ SMF with increasing Σ, which may be due to a systematicdecrease in temperature, increase in importance of high CO opacity cores, increase in dust opacity, ordecrease in CO abundance due to depletion in regions of higher column density. We perform ellipsoidaland filamentary virial analyses of the clouds, finding that the surface pressure terms are dynamicallyimportant and that globally the filaments may not yet have reached virial equilibrium. Some local regionsalong the filaments appear to be close to virial equilibrium, although still with dynamically importantsurface pressures, and these appear to be sites where star formation is most active.
Subject headings:
ISM: clouds, dust, extinction — stars: formation introduction Massive, high column density Infrared Dark Clouds (IRDCs), typically identified as being opaque against the Galacticbackground at ∼ µ m, are thought to contain the sites of future massive star and star cluster formation (e.g. Rathborneet al. 2006), since their densities ( n H ∼ > cm − ) and mass surface densities (Σ ∼ > . − ) are similar to regionsknown to be undergoing such formation activity (Tan 2007). Studies of molecular line emission from IRDCs can helpdetermine their kinematics. In particular, we would like to know if they are gravitationally bound, if they are near virialequilibrium and if there is evidence for coherent gas motions that might indicate that IRDC formation involves convergingatomic flows (Heitsch et al. 2008) or converging molecular flows from cloud collisions (Tan 2000).In this study we use CO J = 1 − l = 34 . ◦ , b = 0 . ◦ , d = 3 . l = 35 . ◦ , b = − . ◦ , d = 2 . CO-derived mass surface densities, Σ , with small median filter (SMF) mid-infrared (MIR) (8 µ m) extinction mappingderived mass surface densities, Σ SMF , using the method of BT09 applied to the
Spitzer
Infrared Array Camera (IRAC)band 4 images of the Galactic plane taken as part of the Galactic Legacy Mid-Plane Survey Extraordinaire (GLIMPSE)(Benjamin et al. 2003). We consider systematic errors in each of these methods, which is necessary before analyzing largersamples of clouds. We are also able to look for evidence of changing CO abundance with column density, e.g. due topossible depletion of CO at high densities. We then perform a virial analysis of the clouds to determine their dynamicalstate.There have been a number of other studies comparing CO derived mass surface densities with those from othermethods. For example, Goodman, Pineda & Schnee (2009) compared near infrared (NIR) dust extinction, far infrared(FIR) dust emission and CO line emission in the Perseus giant molecular cloud (GMC), probing values of Σ up to ∼ .
02g cm − (i.e. up to A V ≃ CO emission was a relatively unreliable tracer of mass surface density, perhaps due to threshold, depletion andopacity effects. Our study probes higher values of Σ, from ∼ .
01 to ∼ .
05 g cm − , and compares CO emission withMIR extinction in order to investigate these processes.Battersby et al. (2010) used CO emission, MIR extinction and FIR dust emission methods to measure Σ and mass ofclumps in 8 IRDCs, one of which is IRDC F of our study. They did not present a specific comparison of Σ with othermethods, although derived clump masses were in reasonable agreement. Their sample also included MIR-bright regions,associated with ultra-compact HII regions, for which the MIR extinction method cannot be applied. As we describebelow, our approach differs in a number of ways, including by focusing on filamentary and mostly quiescent regions ofIRDCs for which the MIR extinction method is most reliable and which are likely to be closer to the initial conditions ofthe massive star and star cluster formation process. We note that while IRDC F in particular does contain some regions1
Fig. 1.— (a) Left panel: Dependence of τ ν with T B,ν (eq. 2) assuming T ex = 10 , ,
20 K (red dotted, blue solid, green dashed lines,respectively). (b) Right panel: Dependence of d N / d v with T B,ν assuming T ex = 10 , ,
20 K (red dotted, blue solid, green dashed lines,respectively). of quite active star formation, including an ultra-compact H II region, here we have concentrated on its more quiescentportions. mass surface density estimation from COWe evaluate the column density of CO molecules, d N , in a velocity interval d v from their J = 1 → N ( v )d v = 8 πQ rot Aλ g l g u τ ν (cid:20) − exp (cid:18) − hνkT ex (cid:19)(cid:21) − , (1)where Q rot is the partition function, A = 6 . × − s − is the Einstein coefficient, λ = 0 . g l = 1 and g u = 3 are the statistical weights of the lower and upper levels, τ ν is the optical depth of the line at frequency ν , i.e.at velocity v , T ex is the excitation temperature (assumed to be the same for all rotational levels). For linear molecules,the partition function is Q rot = P ∞ J =0 (2 J + 1)exp( − E J /kT ex ) with E J = J ( J + 1) hB where J is the rotational quantumnumber and B = 5 . × s − is the rotational constant. Thus for CO(1-0) we have E J /k = 5 .
289 K. For J = 1, Q rot = 4 . , . , .
906 for T ex = 10 , ,
20 K.The optical depth is determined via T B,ν = hνk [ f ( T ex ) − f ( T bg )] (cid:2) − e − τ ν (cid:3) , (2)where T B,ν is the brightness temperature at frequency ν , f ( T ) ≡ [exp( hν/ [ kT ]) − − , and T bg = 2 .
725 K is thebackground temperature. T B,ν is derived from the antenna temperature, T A , via T A ≡ ηf clump T B,ν , where η is the mainbeam efficiency ( η = 0 .
48 for the GRS) and f clump is the beam dilution factor of the CO emitting gas, which we assumeto be 1 for the IRDCs we are studying. However, it should be noted that the BT09 MIR extinction maps of the IRDCsdo show that there is structure on scales smaller than the angular resolution of the GRS survey. Given the observed T B,ν and for an assumed T ex we solve equations (1) and (2) for τ ν and thus d N / d v (see Fig. 1). For a given T B ,ν , thisfigure allows us to judge the sensitivity of the derived column density per unit velocity to temperature uncertainties.While we use the above formulae, which account for optical depth, to calculate our column density estimates, forconvenience we also state their behavior in the limit of optically thin conditions. We have T B,ν = ( hν/k )[ f ( T ex ) − f ( T bg )] τ ν so that d N ( v )d v = 1 . × Q rot f ( T ex ) − f ( T bg ) [1 − exp( − hν/kT ex )] − T A /Kηf clump cm − km − s (3) → . × T A /Kηf clump cm − km − s ( T ex = 15 K) . (4)The last coefficient changes to (0 . , . × for T ex = 10 ,
20 K.Devine (2009) estimates a temperature of 19 K for cloud F based on VLA observations of NH (1 ,
1) and (2 , ′′ , with sampling every 22 ′′ . The velocity resolution is 0.22 km s − (Jackson etal. 2006). From a morphological examination of the CO emission in l, b, v space and comparison to the MIR extinctionmaps of BT09 we identify the velocity range of the gas associated with each IRDC filament (see Figs. 2, 3 and 4). Forcloud F we consider associated gas to be at LSR velocities 48 −
65 km s − and for cloud H at 40 −
50 km s − . The total CO column is then evaluated over the velocity range of the cloud N = R dN .To convert from N to total mass surface density Σ we first assume n n = 6 . R gal kpc + 18 . R gal is the galactocentric radius. For clouds F and H we estimate R gal = 5 . , .
89 kpc,respectively (assuming R gal , = 8 . n /n = 52 ,
55, respectively. For simplicity we adopt n /n = 54 for both clouds. We then assume n n H2 = 2 . × − , (6)similar to the results of Lacy et al. (1994). The observed variation in this abundance in GMCs is about a factor of two(Pineda, Caselli, & Goodman 2008), and this uncertainty is a major contributor to the overall systematic uncertainty inour estimate of Σ . Thus our assumed abundance of CO to H is 3 . × − andΣ = 1 . × − N cm − g cm − , (7)assuming a mass per H nucleus of µ H = 2 . × − g.The assumption of LTE breaks down if the density of the gas is lower than the effective critical density, βn crit , i.e. thecritical density of CO (J=1-0), n crit = 1 . × cm − , allowing for radiative trapping with escape probability β = e − ¯ τ ,where we approximate ¯ τ as the column density weighted mean value of τ ν . Note, for a spherical cloud we would set ¯ τ equalto the average of τ ν /
2, but we use τ ν for these clouds with a filamentary geometry since we expect τ ν seen along our line ofsight to be relatively small compared to other viewing angles. Heyer et al. (2009) argued sub-thermal excitation of COrotational levels may be common in GMCs, causing lower values of line emission than expected under LTE conditions andthus causing estimates of Σ based on LTE assumptions to be underestimates. The IRDC filaments we are considering aregenerally of higher density than the typical GMC volumes considered by Heyer et al. (2009). For the average and highestintensity spectra in each IRDC (see Fig. 2), we evaluate β (see Table 1). We also estimate n H2 , assuming a filamentline-of-sight thickness that is equal to its observed width. A comparison of n H2 with βn crit shows that the densities areclose to or greater than the effective critical densities, thus justifying the assumption of LTE conditions. Furthermore, weexpect that the typical density at a given location is greater than our estimated values due to clumping on angular scalesthat are smaller than the 46 ′′ resolution of the CO observations. Such clumping is apparent in the MIR extinction maps(Figs. 3 & 4). comparison Σ and Σ SMF : possible trends in temperature, co depletion and dust opacity with
ΣThe morphologies of IRDCs F and H are shown in Figures 3 and 4, respectively. The CO emission is more extendedthan the structure traced by the BT09 MIR extinction maps, which are derived by comparing the observed 8 µ m intensityat the cloud position with the expected background intensity interpolated from nearby regions. Uncertainties in thisestimation of the background lead to a lack of sensitivity of the MIR extinction maps to mass surface density contrasts∆Σ . .
013 g cm − .Since there is a significant amount of molecular gas in the surrounding “envelope” regions around the filaments, tocompare the ratio of Σ SMF and Σ in the filaments we must first subtract the contribution to Σ from the envelopes.To do this we consider orthogonal strips across each IRDC filament and measure CO emission on either side (see Figs. 3& 4). The spectra of these “off-source”, envelope regions are corrected for optical depth effects, as described above,averaged and then subtracted from the central “on-source”, filament region (see Fig. 5). We find the envelope-subtracted
Table 1Escape Probabilities and Critical Densities
IRDC Spectrum N Σ β n H2 βn crit n H2 / ( βn crit )(see Fig. 2) (10 cm − ) (g cm − ) (cm − ) (cm − )F mean high mean high Fig. 2.—
Column density profiles with velocity for IRDCs F (left) and H (right). The dashed lines shows the profiles assuming opticallythin emission, while the solid lines show the profiles after correction for optical depth. A temperature of 20 K was assumed for IRDC F and15 K for IRDC H (see text). For each cloud the upper, offset profile is that of the average for the cloud and the lower profile is that of thehighest column density position. spectra are generally narrower than the total. We also find that the size of negative residuals created in the subtractionprocess are relatively small. However, the fact that in some cases we see quite significant variation in the envelope spectrafrom one side of the filament to the other, suggests that this is one of the major sources of uncertainty in measuring themolecular emission properties of these embedded filaments. After this procedure, we are now in a position to comparethe envelope-subtracted values of Σ with those derived from the SMF extinction mapping method of BT09.We also note that the MIR extinction derived values of Σ suffer from their own systematic uncertainties, includingcorrections due to foreground dust emission (BT09), scattering in the IRAC array (Battersby et al. 2010), adopted dustopacities and dust to gas ratios. However, saturation due to large 8 µ m optical depth and/or an insufficiently subtractedforeground is not an important source of error for our present study, since the values of Σ SMF are in any case reducedwhen the maps are smoothed to the lower resolution of the CO data.As noted by BT09, the MIR extinction mapping technique fails for locations where there are bright MIR sources. If theintensity of the source is greater than the background model, then formally an unphysical, negative value of Σ is returnedby this method. In the analysis of BT09, negative values of Σ are allowed up to a certain threshold value to account fornoise-like, approximately Gaussian, fluctuations in the background intensity. For the more extreme fluctuations causedby bright sources, BT09 set Σ = 0. Thus the effect of a bright MIR region within an IRDC is to cause the extinctionmapping method to underestimate the true mass surface density. For most point-like MIR sources, this effect is quiteminor after the extinction maps are averaged to the 22.14 ′′ pixel scale of the CO observations. However, in IRDC F weidentify two regions (indicated in Fig. 3), which are significantly affected by bright MIR sources and exclude them fromthe subsequent analysis.The values of Σ (after envelope subtraction) and Σ SMF for each pixel in the filament regions of IRDCs F and H arecompared in Figure 6. We note that both these measures of Σ are independent of the distance to the cloud. Consideringjust the data with Σ and Σ
SMF > .
01 g cm − , the best fit power law relation Σ / g cm − = A (Σ SMF / g cm − ) α has α = 0 . ± . , . ± .
28 and A = 0 . ± . , . ± .
069 for IRDCs F and H, respectively.Over the range 0 . < Σ SMF / g cm − < .
07 we find that Σ ≃ Σ SMF to within a factor of ∼ CO abundance, dust opacities orenvelope subtraction method for each IRDC compared to our adopted values and methods. The dispersion may reflectlocal systematic variations and errors introduced by measurement noise.In both IRDCs the ratio Σ / Σ SMF decreases with increasing mass surface density as measured by Σ
SMF . We gaugethe significance of this trend by noting that the above values of α for IRDCs F and H differ from unity by 1.3 and 2.4standard deviations, respectively, assuming errors are distributed normally.There are several physical processes that may be causing such a trend. A systematic temperature decrease from thelower column density, outer regions of the filaments to the higher column density centers would lead us to systematically Fig. 3.—
Morphology of IRDC F.
Top left: Spitzer
GLIMPSE IRAC 8 µ m image, with linear intensity scale in MJy sr − . The horizontalline shows a scale of 3 ′ . The image has 1.2 ′′ pixels and the PSF has a FWHM of 2 ′′ ˙ Top middle:
Mass surface density, Σ
SMF , with linearintensity scale in g cm − , derived from the previous image using the small median filter MIR extinction mapping method of Butler & Tan(2009). The inner, red rectangle (centered at l = 34 . ◦ , b = 0 . ◦ , with P.A.= +60 ◦ and size 0 . ◦ by 0 . ◦ ) along the filament showsthe “on source” region we consider to contain the main filamentary structure of the IRDC. The outer, blue rectangle extends to “off source”regions we consider to be representative of the surrounding GMC envelope. These rectangles are divided into 7 orthogonal strips to aid inthe separation of components of CO emission from the filament and GMC envelope. Top right:
The same extinction map convolved with aGaussian of 46 ′′ FWHM to match the resolution of the CO maps and pixelated to 22 ′′ on the same grid as the GRS survey image. The twoblack hatched squares show regions with unreliable measures of Σ SMF because of the presence of bright MIR sources. These are excluded fromthe comparison with Σ . Bottom left:
Integrated intensity map of CO(1-0) emission over the full velocity range of − −
135 km s − of the GRS survey, with linear intensity scale in K km s − . Bottom middle:
Integrated intensity map of CO over the velocity range of48 −
65 km s − , i.e. the gas we believe is associated with the IRDC, with linear intensity scale in K km s − . Bottom right:
Mass surfacedensity of the filament derived from CO emission, Σ , with linear intensity scale in g cm − . underestimate Σ in the latter, while having little direct effect on dust opacities (although there may be an indirecteffect via formation of ice mantles, see below). For the temperature ranges we expect to be present, i.e. from ∼ , which may trace different conditions tothose of the CO), this effect becomes important for values of T B,ν ∼ > SMF positions in IRDCs F and H in Fig. 6. A systematic temperature decrease of 5 K in the highΣ regions could remove much of the observed trend.Another possibility is that our corrections for the optical depth of the CO emission are systematically underestimatedat the higher column density positions. This would be expected if a significant amount of mass is contained in unresolveddense cores. The largest optical depth corrections in the highest column density positions presently lead to an increasein the estimated column by factors ∼ <
2. Future higher angular resolution CO and C O observations of these filamentsare required to investigate this issue further.Depletion of CO molecules onto dust grains is known to occur in cold, high volume density gas (Caselli et al. 1999).This process could systematically reduce the gas phase CO abundance in the high Σ regions of the IRDCs. For depletionto be fully responsible for the observed trends in IRDCs F and H would require about a factor of 2 depletion as Σ
SMF
Fig. 4.—
Morphology of IRDC H.
Top left: Spitzer
GLIMPSE IRAC 8 µ m image, with linear intensity scale in MJy sr − . Thehorizontal line shows a scale of 3 ′ . The image has 1.2 ′′ pixels and the PSF has a FWHM of 2 ′′ ˙ Top middle:
Mass surface density,Σ
SMF , with linear intensity scale in g cm − , derived from the previous image using the small median filter MIR extinction mappingmethod of Butler & Tan (2009). The inner, red rectangle (centered at l = 35 . ◦ , b = − . ◦ , with P.A.= +62 . ◦ and size0 . ◦ by 0 . ◦ ) along the filament shows the “on source” region we consider to contain the main filamentary structure of theIRDC. The outer, blue rectangle extends to “off source” regions we consider to be representative of the surrounding GMC envelope.These rectangles are divided into 4 orthogonal strips to aid in the separation of components of CO emission from the filament andGMC envelope. Top right:
The same extinction map convolved with a Gaussian of 46 ′′ FWHM to match the resolution of the COmaps and pixelated to 22 ′′ on the same grid as the GRS survey image. Bottom left:
Integrated intensity map of CO over the fullvelocity range of − −
135 km s − of the GRS survey, with linear intensity scale in K km s − . Bottom middle:
Integrated intensitymap of CO over the velocity range of 40 −
50 km s − , i.e. the gas we believe is associated with the IRDC, with linear intensityscale in K km s − . Bottom right:
Mass surface density of the filament derived from CO emission, Σ , with linear intensityscale in g cm − . increases from 0.01 to 0.05 g cm − . Because of depletion, CO is not expected to be an ideal tracer of the densest, coldestparts of IRDCs. High resolution studies of other tracers, such as NH , will likely be needed to measure the kinematics ofthese regions.The depletion of CO via formation of CO ice mantles on dust grains would also have some effect on the MIR opacitiesof these grains, thus affecting our measurement of Σ SMF . The grains would become larger and absorption features dueto pre-existing water ice mantles may become obscured. The Ossenkopf & Henning (1994) grain models show a ∼ µ m opacity, κ µ m , going from bare grains to those with thick ice mantles. The BT09 estimates of Σ SMF assumed a constant value of κ µ m consistent with the thin ice mantle model of Ossenkopf & Henning (1994). Thus theobserved decrease in the ratio of Σ / Σ SMF with increasing mass surface density could be caused by thickening of grainice mantles, causing us to systematically overestimate Σ
SMF .Another possible explanation for a trend of decreasing Σ / Σ SMF with increasing mass surface density is activechemical fractionation (Langer et al. 1980, 1984; Glassgold, Huggins, & Langer 1985; Visser, Van Dishoeck, & Black2009), which enhances the abundance of CO in regions where C + is present via the via the ion-molecule exchange Fig. 5.—
Subtraction of IRDC envelopes. (a) Top left:
Filament and envelope of IRDC F, with the 7 sets of spectra (top to bottom)corresponding to the 7 orthogonal strips (top left to bottom right) shown in Figure 3. In each, the dotted, red line shows the total COcolumn density distribution, including optical depth corrections, from the filament region, which includes an assumed contribution from“envelope” material along the line of sight. The long dashed and dot-dashed, blue lines show the total CO column density from thenorthern and southern envelope regions, respectively. (b) Top right:
Filament and envelope of IRDC H, with the 4 sets of spectra (top tobottom) corresponding to the 4 orthogonal strips (top left to bottom right) shown in Figure 4. The line styles have the same meaning asin (a). (c) Bottom left:
For the same strips as in (a), we subtract the average of the northern and southern envelope spectra (short dashedblue lines) from the filament (dotted red line), to leave an estimate of the material in the filament (solid, black line). The vertical solid lineindicates the mean velocity. (d) Bottom right:
Same analysis and labels as (c) applied to IRDC H.
Fig. 6.— (a) Top Left:
Direct comparison of Σ (after envelope subtraction) and Σ
SMF in IRDC F. The crosses show locationswhere both Σ and Σ
SMF > .
01g cm − . The dots show locations of lower surface densities. The uncertainties in the individualmeasurements are assumed to be 15% plus a systematic error of 0.01 g cm − . The dotted line shows the one to one linear relationand the long dashed line shows the best-fit offset linear relation. The solid line shows the best-fit power law relation (see text). (b) Top Right: Same as (a) but for IRDC H. (c) Bottom Left:
Logarithm of the ratio of Σ to Σ
SMF as a function of Σ
SMF forIRDC F, with the same symbol notation as in (a). The cross in the upper-right corner indicates typical estimated uncertainties.The solid (dashed) arrow shows the effect on the highest Σ position of reducing the assumed temperature from 20 K to 15 K(10 K). (d) Bottom Right:
Same as (c) but for IRDC H. The arrow shows the effect on the highest Σ position of reducing theassumed temperature from 15 K to 10 K. reaction (Watson, Anicich, & Huntress 1976), CO + C + → C + + CO+ 35 K. FUV irradiation maintains a relativelyhigh abundance of C + in the outer regions of the cloud, with A V . CO to COcan be enhanced by factors of ∼
10. For A V &
2, the enhancement factor is only about 20% (e.g. for a model with n H = 10 cm − and T=20 K; Glassgold et al. 1985). For IRDCs F and H, the filaments are typically embedded inside“envelope” gas with Σ ≃ .
01 g cm − , i.e. N H ≃ × cm − . The trend of decreasing Σ / Σ SMF is observed tooccur as Σ
SMF increases to ∼ .
05 g cm − . CO fractionation enhancements should not be varying in this regime by largeenough factors to explain our observed results. It should also be noted that even in the lower-column density regions, ifthe FUV flux is large enough, then isotope selective photodestruction of CO compared to the self-shielded CO (Bally& Langer 1982) may reverse the fractionation-produced enhancement of the abundance of CO to CO. the dynamical state of the irdc filaments IRDC Masses from CO Emission and MIR Dust Absorption
We calculate IRDC masses using the observed mass surface densities and angular sizes, and by assuming near kinematicdistances, since the clouds are seen in absorption against the Galaxy’s diffuse MIR emission. We adopt the kinematicdistances of Simon et al. (2006), who assumed the Clemens (1985) rotation curve. This leads to a distance of 3.7 kpc forIRDC F and 2.9 kpc for IRDC H. We assume uncertainties of 0.5 kpc, which could result, for example, from line of sightnoncircular motions of ∼ − . Temperature uncertainties of 5 K would lead to Σ uncertainties of ∼ SMF due to foreground correction and background interpolation uncertainties(BT09). Then, for IRDC F, the CO-derived mass assuming T = 20 K in the on-source filament region after envelopesubtraction is M = 3300 ± M ⊙ and the dust extinction mass is M SMF = 1900 ± M ⊙ . For IRDC H, we find M ( T = 15 K) = 370 ± M ⊙ and M SMF = 580 ± M ⊙ . For our calculations involving the total mass of theclouds we take averages of the above estimates while still assuming 20% uncertainties in the averaged values of Σ. Thuswe adopt M = 2600 ± M ⊙ for IRDC F and M = 480 ± M ⊙ for IRDC H. These results are summarized in Table 2.4.2. Ellipsoidal Cloud Virial Analysis
Following Bertoldi & McKee (1992, hereafter BM92), we consider an ellipsoidal cloud with radius R normal to the axisof symmetry and size 2 Z along the axis. The aspect ratio is defined as y ≡ Z/R , while R max and R min are the semimajorand semiminor axes of the ellipse obtained by projecting the cloud onto the plane of the sky.IRDCs F and H both have relatively thin, filamentary morphologies: we set R max /R min = 3 . , .
78, respectively, i.e.the same elongation as the rectangular regions we consider to define the filaments (Figs. 3 & 4). Given these morphologies,we expect the symmetry axes of the clouds to be close to the plane of the sky. Thus for both we adopt a fiducial valueof the inclination angle between the cloud symmetry axis and the line of sight of θ = 60 ◦ . We assume R = R min and Z = R max / sin θ , so y = 3 . , .
06 for IRDCs F and H, respectively. An uncertainty of 15 ◦ in inclination would cause ∼
15% uncertainties in y . It is also useful to introduce a geometric mean observed radius, R obs ≡ ( R max R min ) / , whichis also related to R via, R obs = R cos / θ [1 + ( y tan θ ) ] / . BM92 also define R m as the mean value of R obs averaged overall viewing angles, but for our individual clouds we will express quantities in terms of R and R obs . It should be noted thatthe treatment of these IRDC filaments as simple ellipsoids is necessarily approximate, and we consider the filamentaryanalysis of section 4.3 to be more accurate. Table 2IRDC Ellipsoidal Virial Analysis
Cloud property IRDC F IRDC Hd (kpc) 3 . ± . . ± .
5R (pc) 1 . ± .
25 0 . ± . R obs (pc) 3 . ± .
48 1 . ± . y ≡ Z/R . ± . . ± . (g cm − ) 0 . ± .
003 0 . ± . SMF (g cm − ) 0 . ± . . ± . (env) (g cm − ) 0 . ± .
004 0 . ± . M ( M ⊙ ) 3300 ± ± M SMF ( M ⊙ ) 1900 ±
640 580 ± M ( M ⊙ ) 2600 ±
870 480 ± a a W (10 erg) − . ± . − . ± . σ (km/s) 1 . ± .
15 1 . ± . t s = 2 R/σ (Myr) 2 . ± . . ± . T (10 erg) 16 . ± . . ± . T ( A ) (10 erg) 14 ± . ± . T ( B ) (10 erg) 73 ±
30 4 . ± . α ≡ σ R/ ( GM ) 1 . ± .
73 3 . ± . t s ≡ R/σ , then it should obey the equilibrium virial equation (McKee & Zweibel 1992):0 = 2(
T − T ) + M + W. (8)Here, T is the clump kinetic energy, T is the kinetic energy resulting from the surface pressure on the clump, M is themagnetic energy associated with the cloud, and W is the gravitational binding energy, which for an ellipsoidal cloud is(BM92) W = − a a GM R , (9)where for a power-law density distribution ρ ∝ r − k ρ , a = (1 − k ρ / / (1 − k ρ /
5) and a = arcsinh( y − / ( y − / (10)for prolate clouds. Note, our definition of W and a differs slightly from BM92 since we do not need to consider R m .We adopt k ρ = 1 (based on a study of the density profiles in IRDCs - Butler & Tan, in prep.) so that a = 10 /
9. Forour measured values of y , we have a = 0 . , .
71 for IRDCs F and H, respectively. For the mass of the cloud we takethe average values of the estimates from MIR dust extinction and CO line emission. Using these values we estimate W = − (11 . , . × erg for IRDCs F and H, respectively (see Table 2). The uncertainties in W are relatively largegiven the measurement errors of Σ and R .The clump kinetic energy is T = (3 / M σ , where σ is the average total 1D velocity dispersion, which we derive fromthe CO line emission (counting only those parts of the envelope-subtracted spectra with positive signal greater than orequal to one standard deviation of the noise level) including corrections for the molecular weight of CO and our adoptedcloud temperatures. We estimate that we measure σ to a 10% accuracy. We find T = (16 . , . × erg for IRDCs Fand H, with uncertainties at about the 40% level.The surface term for the kinetic energy is T = (3 / P V . We estimate this term in two ways. First (method A), wecan measure the mass surface density of the surrounding molecular cloud from the CO emission of the envelope regions.We find Σ (env) = 0 . , . − for IRDCs F and H, respectively. We scale these values by M/M ,i.e. 0.79 and 1.30. If the envelope is self-gravitating, it has mean internal pressure P (env) = 1 . G Σ (env), adaptingthe analysis of McKee & Tan (2003) with parameters f g = 1 (i.e. fully gas dominated), φ geom ≡ R / ( R Z ) = 1 . φ B = 2 . α vir = 1.Thus setting P = P (env) = (2 . , . × − cgs for IRDCs F and H and with the cloud volume V = 4 πR Z/
3, wefind T ( A ) = (14 , . × erg.Second (method B), we estimate the density in the envelope region, assuming it has a cylindrical, annular volumewith outer radius 2 R and inner radius R . For IRDCs F and H, we find densities of ρ = 4Σ(env) / (3 πR ) = (1 . , . × − g cm − , equivalent to n H (env) = (610 , − . We again scale these values by M/M , i.e. 0.79 and 1.30for IRDCs F and H, respectively. We then equate P = ρ (env) σ (env), where σ (env) is the velocity dispersion of theenvelope gas (we find 3 . , .
89 km s − for IRDCs F and H) and evaluate T ( B ) = (3 / V P → (73 , . × erg, with V = (4 / πR Z .These results indicate that, for both IRDCs, the surface pressure term of the virial equation is comparable to or muchlarger than the internal kinetic term, although the uncertainties are large. Assuming T ≥ T , then for virial equilbriumto be maintained would require M ≡ (1 / π ) R V a ( B − B ) dV ≥ − W , where B is the magnetic field strength far from thecloud and V a is a volume that extends beyond the cloud where the field lines have been distorted by the cloud. Assuming V a is the volume of the envelope regions and assuming negligible B , we find B ≥ , µ G for IRDCs F and H. If amore realistic value of B = 10 µ G is adopted (Crutcher et al. 2010), then we find B ≥ , µ G. Thus relatively modestmagnetic field enhancements could stabilize the clouds.Bertoldi & McKee (1992) define a dimensionless virial parameter, α ≡ σ R m / ( GM ) to describe the dynamical stateof clouds. We adopt a slightly revised definition α ≡ σ R/ ( GM ) = a a T / | W | , with a defined as above. For IRDCs Fand H we find α = 1 . , .
17. While these values are quite close to unity, especially for IRDC F, which is commonly takento infer that self-gravity is important, this is somewhat misleading since the value of a is quite small for these elongatedclouds and the surface pressure terms seem to be quite important.It is possible that these IRDCs have not yet reached virial equilibrium if their surroundings are evolving on timescales . t s ∼ a few Myr. The mean velocity of the CO emitting gas in the north and south envelope regions is 55.72,56.10 km s − , respectively, for IRDC F, and 44.96, 44.28 km s − , respectively, for IRDC H. The north/south velocitydispersions are 3.81/3.38 km s − for F and 1.69/2.07 km s − for H. The north/south Σ ’s are 0.040/0.031 g cm − forF and 0.050/0.051 g cm − for H. The ratios of north/south pressures (estimated by method B, ∝ Σ σ ) are thus 1.64 and0.65 for IRDCs F and H. The pressures appear to be fairly similar on the different sides of the filaments, although theuncertainties are such that variation at the level of about a factor of two could be present.14.3. Filamentary Cloud Virial Analysis
Fiege & Pudritz (2000, hereafter FP00) present a virial analysis of filamentary clouds. They derived the followingequation satisfied by pressure-confined, nonrotating, self-gravitating, filamentary (i.e. lengths ≫ widths) clouds threadedby helical magnetic fields that are in virial equilibrium: P P = 1 − mm vir (cid:18) − M l | W l | (cid:19) . (11)Here P is the external pressure at the surface of the filament, P = ρσ is the average total pressure in the filament, m is the mass per unit length, m vir ≡ σ /G is the virial mass per unit length, M l is the magnetic energy per unit length,and W l = − m G is the gravitational energy per unit length.We divide IRDCs F, H into 7, 4 orthogonal strips (see Figs. 3 & 4) with angular widths 1.70 ′ , 0.955 ′ along the filaments,respectively. Assuming a fiducial value of the inclination angle between the cloud symmetry axis and the line of sight of θ = 60 ◦ , these correspond to physical lengths along the filaments of 2.11, 0.930 pc, respectively.In Table 3, for each strip in IRDCs F and H, we list the values of Σ , Σ SMF , M (calculated from the mean of thesevalues of Σ), m , ρ , ¯ v , σ , m vir , P , Σ (env), ρ (env) (calculated after scaling Σ (env) by 0 . + Σ SMF ) / Σ ),¯ v (env), σ (env) and P (env) ≡ ρ (env) σ (env). We equate P = P (env).Following FP00, in Fig. 7 we plot P /P versus m/m vir . The range of models considered by FP00 allows for positivevalues of M l / | W l | (i.e. poloidally-dominated B-fields that provide net support to the filament against gravitationalcollapse) and negative values (i.e. toroidally-dominated B-fields that provide net confinement of the filament). In allcases, P /P ≤
1. In contrast, we find all of the filament regions have P /P >
1, i.e. the pressures in the envelope regionsappear to be greater than in the filament. This echoes the results from the ellipsoidal virial analysis, which found largesurface pressure terms. Assuming our measurements of pressures are reliable, e.g. are not being adversely affected bysystematic effects due to, for example, our assumed filament and envelope geometry, then these results imply that thefilaments have not yet reached virial equilibrium.Very large values of P /P are inferred for strips F1, F2 and F3. These are consistent with the filament and envelopespectra shown in Fig. 5, which reveal a relatively weak filament and relatively strong and varying envelope velocity profiles.Strips F6, F7 and H2 have the smallest values of P /P .
2. Examining the IRAC 8 µ m images (Figs. 3 & 4), thereis some indication that these are the sites of relatively active star formation (especially F7 and H2). Star formationrequires gravitationally unstable conditions in the filament, i.e. regions where self-gravity starts to dominate over externalpressure. Our results indicate that this also requires the local region of the filament to reach approximate virial equilibrium,although surface pressure terms still remain dynamically important, i.e. m/m vir is significantly less than unity. Given ourmeasurement uncertainties and the fact that the observed regions do not have P /P <
1, we are not able to determinewhether the field geometries are more dominated by poloidal or toroidal components. We note that FP00’s conclusionthat observed filaments are dominated by toroidal fields depends on their assumption that the filaments are in virialequilibrium and on their choice of P , which was not directly measured for most of the sources they considered.We caution that if independent molecular clouds are present along these lines of sight and with similar velocities to thefilaments, then this may cause us to overestimate the velocity dispersion and pressure in the envelope regions around thefilaments. From the spectra shown in Fig. 5 we do not expect this is occurring in IRDC H, since the envelope spectrashare a very similar velocity range as the filament. The situation in IRDC F is less clear cut, since there appears to be abroader, offset component that dominates more in the envelope region. conclusions We have compared measurements of mass surface density, Σ, in two IRDC filaments based on CO observations, Σ ,with those derived from MIR extinction mapping, Σ
SMF , finding agreement at the factor of ∼ Table 3IRDC Filamentary Virial Analysis
Cloud property F1 F2 F3 F4 F5 F6 F7 F tot
H1 H2 H3 H4 H tot Σ (10 − g cm − ) 0.218 0.338 0.811 1.31 2.29 2.87 2.58 1.47 0.864 1.67 1.56 1.21 1.33Σ SMF (10 − g cm − ) 0.476 0.311 -0.0432 0.862 1.37 1.62 1.03 0.823 1.38 1.94 1.97 3.28 2.09 M ( M ⊙ ) 114 107 126 357 601 737 593 2640 78.5 126 124 157 478 m ( M ⊙ pc − ) 54.0 50.7 59.7 169 285 349 281 178 84.4 135 133 169 128 ρ (10 − g cm − ) 3.29 3.09 3.64 10.3 17.4 21.3 17.1 10.9 22.0 35.1 34.6 44.0 33.3¯ v (km s − ) 59.19 59.34 58.81 58.26 58.30 57.77 57.77 58.39 45.09 45.20 45.13 44.78 45.07 σ (km s − ) 1.36 1.27 1.07 1.44 1.59 1.86 2.27 1.46 1.52 1.34 1.03 0.995 1.20 m vir ( M ⊙ pc − ) 856 751 536 962 1170 1610 2400 986 1070 840 494 460 669 P (10 − cgs) 6.05 4.99 4.19 21.3 43.8 73.8 88.3 23.1 50.7 63.4 36.8 43.6 47.9Σ (env) (10 − g cm − ) 1.61 1.83 1.94 2.14 1.81 1.95 2.46 1.96 2.31 1.86 2.00 2.35 2.12 ρ (env) (10 − g cm − ) 18.7 12.9 6.72 13.0 10.6 11.2 12.6 11.2 45.3 30.4 34.2 65.9 41.2¯ v (env) (km s − ) 56.03 55.84 55.78 56.59 56.04 55.65 55.72 55.95 44.97 44.78 44.53 44.40 44.67 σ (env) (km s − ) 3.45 3.48 3.95 3.91 3.55 3.43 3.56 3.65 1.85 1.91 1.89 1.86 1.89 P (env) (10 − cgs) 223 156 105 199 133 132 159 149 155 110 122 229 148 Fig. 7.— P /P versus m/m vir for strips in IRDCs F (open squares joined by dotted line for F1 to F7) and H (open triangles joined bydashed line for H1 to H4). The smooth curves show the conditions satisfied by equation (11) for M l / | W l | < M l / | W l | = 0(dotted line), and M l / | W l | > P /P may indicate that most of the regions in IRDCs F and Hhave not yet established virial equilibrium. Those strips with the lowest values of P /P , i.e. F6, F7, and H2, appear to be undergoing moreactive star formation (see text and Figs. 3 & 4). / Σ SMF with increasing Σ may be due to a systematic decrease in temperature, increase in the contribution ofunder-resolved high optical depth regions, increase in dust opacity, or decrease in CO abundance due to depletion inregions of higher column density. Future studies that spatially resolve the temperature structure and MIR dust absorptionproperties can help to distinguish these possibilities.We have then used the kinematic information derived from CO to study the dynamical state of the IRDCs. Inparticular we have evaluated the terms of the steady-state virial equation, including surface terms, under the assumptionof ellipsoidal and filamentary geometries. In both cases we find evidence that the surface pressure terms are importantand possibly dominant, which may indicate that the filaments, at least globally, have not yet reached virial equilibrium.These results would be consistent with models of compression of dense gas in colliding molecular flows, e.g. GMCcollisions. Tan (2000) proposed that this mechanism may trigger the majority of star formation in shearing disk galaxies.The expected collision velocities are ∼
10 km s − . It is less clear whether colliding atomic flows, (e.g. Heitsch et al. 2008),which form the molecular gas after shock compression of atomic gas, would also produce such kinematic signatures: recallthat we are inferring large surface pressures based on CO emission from the envelopes around the IRDC filaments.Recent observations of extended, parsec-scale SiO emission, likely produced in shocks with velocities &
12 km s − inIRDC H by Jim´enez-Serra et al. (2010) may also support models of filament formation from converging flows. However,we caution that the observed extended SiO emission is very weak and may also be produced by multiple protostellaroutflow sources forming in the IRDC (see Jim´enez-Serra et al. 2010 for further discussion).Our resolved filamentary virial analysis also indicates that the regions closest to virial equilibrium (strips F6, F7 andH2) are those which have initiated the most active star formation. This would be expected if models of slow, equilibriumstar formation (Tan et al. 2006; Krumholz & Tan 2007) apply locally in these regions. In this case, these dense regionsthat have become gravitationally unstable, perhaps due to the action of external pressure and/or converging flows, thenpersist for more than one local dynamical time and so are able to reach approximate pressure and virial equilibrium withtheir surroundings. In this scenario, they are stabilized by the ram pressure generated by protostellar outflow feedbackfrom the forming stars (Nakamura & Li 2007).We thank Michael Butler for providing the MIR extinction maps used in this analysis. We also thank Peter Barnes,Crystal Brogan, Michael Butler, Paola Caselli, James Jackson, Izaskun Jim´enez-Serra, Thushara Pillai and Robert Simonfor helpful discussions. The comments of an anonymous referee led to improvements in the paper. AKH acknowlegessupport from a SEAGEP Dissertation Fellowship. JCT acknowledges support from NSF CAREER grant AST-0645412and NASA Astrophysics Theory and Fundamental Physics grant ATP09-0094. REFERENCESBattersby, C., Bally, J., Jackson, J. M. et al. 2010, ApJ, 721, 222Bally, J., & Langer, W. D. 1982, ApJ, 255, 153Bertoldi, F., & McKee, C. F. 1992, ApJ, 395, 140 (BM92)Butler M. J., & Tan J. C. 2009, ApJ, 696, 484 (BT09)Carey, S. J., Clark, F. O., Egan, M. P., Price, S. D., Shipman, R. F.,& Kuchar, T. A. 1998, ApJ, 508, 721Carey, S. J., Feldman, P. A., Redman, R. O., Egan, M. P., MacLeod,J. M., & Price, S. D. 2000, ApJ, 508, 721Caselli, P., Walmsley, C. M., Tafalla, M., Dore, L., & Myers, P. C.1999, ApJ, 523, L165Clemens, D. P. 1985, ApJ, 295, 422Crutcher, R. M., Wandelt, B, & Falgarone, E. 2010, in
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