The Mass-Size Relation from Clouds to Cores. I. A new Probe of Structure in Molecular Clouds
J. Kauffmann, T. Pillai, R. Shetty, P. C. Myers, A. A. Goodman
aa r X i v : . [ a s t r o - ph . GA ] F e b Accepted to the Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 11/10/09
THE MASS-SIZE RELATION FROM CLOUDS TO CORES. I. A NEW PROBE OF STRUCTURE INMOLECULAR CLOUDS
J. Kauffmann , T. Pillai , R. Shetty , P. C. Myers & A. A. Goodman Draft version October 30, 2018
ABSTRACTWe use a new contour-based map analysis technique to measure the mass and size of molecularcloud fragments continuously over a wide range of spatial scales (0 . ≤ r/ pc ≤ Subject headings:
ISM: clouds; methods: data analysis; stars: formation INTRODUCTIONSome of the most fundamental properties of molec-ular clouds are the mass and size of these clouds andtheir substructure. Today, these properties are well con-strained: we know the masses and sizes of dense cores inmolecular clouds ( . . . &
10 pc) containing the cores (e.g., Williams et al. 1994,Cambr´esy 1999, Kirk et al. 2006; see Williams et al. 2000for definitions of cores, clumps, and clouds). We do,however, not know much about the relation between themasses and sizes of cores, clumps, and clouds: tradition-ally, every domain is characterized and analyzed sepa-rately. As a result, it is still not known how the coredensities (and thus star-formation properties) relate tothe state of the surrounding cloud.In principle, the relation between the mass in cloudstructure at large and small spatial scales is described bymass-size relations. Larson (1981) presented one of thefirst studies of such relations. He concluded (in his Eq.5) that the mass contained within the radius r obeys apower-law, m ( r ) = 460 M ⊙ ( r/ pc) . . (1)Most subsequent work refers to this relation as “Lar-son’s 3 rd law”, and replaces the original result with m ( r ) ∝ r (e.g., McKee & Ostriker 2007). This “law jens.kauff[email protected] Initiative in Innovative Computing (IIC), 60 Oxford Street,Cambridge, MA 02138, USA Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138, USA present addresses: Jens Kauffmann, NPP Fellow, Jet Propul-sion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109,USA; Thushara Pillai, California Institute of Technology, MC249-17, 1200 East California Boulevard, Pasadena, CA 91125,USA; Rahul Shetty, Zentrum f¨ur Astronomie der Universit¨at Hei-delberg, Institut f¨ur Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany of constant column density” (with respect to scale, r ) is now considered one of the fundamental prop-erties of molecular cloud structure (e.g., reviews byBallesteros-Paredes et al. 2007, McKee & Ostriker 2007,Bergin & Tafalla 2007). This relation has, however,never been re-examined comprehensively on the basisof up-to-date data. It is, e.g., not clear whether re-cent dust extinction and emission work is consistent with m ( r ) ∝ r .Further, the limitations of available structure identi-fication schemes (such as CLUMPFIND; Williams et al.1994) forced past studies to break cloud structure mapsup into discrete fragments. These fragments typicallyhave a size slightly larger than the map resolution. Asa consequence, the cloud structure is only probed ina narrow spatial domain; the largest spatial featuresin a given map are, for example, usually not charac-terized. Today, approaches permitting automatic ex-amination of a continuous range of spatial scales areavailable. Rosolowsky et al. (2008b), in particular, pro-vide software for such studies (their dendrogram anal-ysis); our work would be impossible without the workby Rosolowsky et al.. Such software permits derivationof spatially more comprehensive mass-size relations thanpossible in the past.In this series of papers, we combine contemporary col-umn density observations of high sensitivity with a newdata analysis technique to examine the mass-size rela-tion in molecular clouds for a continuous range of spatialscales of order 0 .
01 to 10 pc. We rely on extinction mapsof molecular clouds (here: Ridge et al. 2006), as well asmaps of dust emission (Enoch et al. 2006). Following theterminology of Peretto & Fuller (2009), we define “cloudfragments” in the maps as regions enclosed by a contin-uous column density contour, and derive their mass andsize at various contour levels. This is implemented usingalgorithms introduced by Rosolowsky et al. (2008b).The first two papers in this series establish our analysisapproach (part I, the present paper) and explore several Kauffmann et al.clouds in the solar neighborhood ( .
500 pc; part II).Section 2 of the present paper describes our cloud frag-ment extraction and characterization scheme. In Sec. 3we provide a first idea how basic physical properties andobservational limitations affect the mass-size measure-ments. This includes a comparison to results obtainedusing the CLUMPFIND algorithm. A detailed discussionof analysis uncertainties is presented in Sec. 4. These areexplored using data for the Perseus Molecular Cloud. Wealso explain how dust emission and extinction data canbe combined for a given cloud.Section 5.1 briefly describes how the new map analysisscheme might help to advance star formation research.As we describe there, it will help to jointly analyse datataken at different spatial resolution. This is a key fea-ture in the age of multi-wavelength and multi-resolutionstudies. We conclude with a summary in Sec. 5.2. METHOD2.1.
Processing of Contour Maps
Consider some column density map containing a num-ber of local maxima, as sketched in Fig. 1 (a). Inthis map, cloud fragments can be identified as struc-tures bound by some continuous contour. To charac-terize these, we pick one of the column density maxima,and measure the mass, m , and area, A , contained withineach column density contour containing this peak. Here,we use the effective radius, r = ( A/π ) / , (2)to quantify A . By following the trends from contour tocontour, as shown in Fig. 1 (b), one can construct a mass-size relation for every cloud fragment. Strictly speaking,we do thus construct mass-area diagrams. In practice,however, r is arguably a more intuitive variable than A .Some contours may contain several local maxima.They represent composite fragments. To give an examplebased on Fig. 1, the region bound by the dark-green con-tour consists of the two regions marked in lighter shadesof green, and the magenta contour contains the fragmentsmarked in yellow and dark-green. Merging of two frag-ments occurs at the first column density contour contain-ing both of the fragments. In our analysis, we enforcethat only two fragments can merge at a time.During a merger, mass and size jump discontinuouslyfrom the pre-merger situation, m i and r i , to their post-merger value, m = m A + m B and r = ( r + r ) / .This yields gaps in the mass-size relation (Fig. 1 [b]).The merger information is preserved during processing,so that it is possible to look up which fragments are con-tained within others.Our map characterization scheme is thus closely re-lated to the one employed by Peretto & Fuller (2009).Like us, these authors measure sizes and masses forregions bound by lines of constant column density.The schemes differ in the number of contours consid-ered: we use a very large number (10 to 10 ), wherePeretto & Fuller only consider two contours per cloud.In practice, we use the dendrogram (i.e., tree analysis)code presented by Rosolowsky et al. (2008b) for auto-matic processing of the maps. A minimum significantcontour has to be set for every region; emission below this limit is not characterized here. It is further neces-sary to specify a minimum contrast between peaks, inorder to identify significant central maxima for the ob-jects. These threshold column densities and contrasts arelisted separately for every region in the following. Theidentified maxima are required to be spaced by more thanone spatial resolution element. Again, this parameter isnoted separately for every map. In this work, all sourceswith an effective diameter (i.e., 2 r ) smaller than twicethe map resolution are rejected and are treated as partsof enveloping objects.In the terminology of Rosolowsky et al., our mass mea-surement approach (i.e., integrate column density withinsome contour) corresponds to their “bijection paradigm”.Such mass measurements, e.g. made towards a densecore, always give a sum over several spatially overlap-ping components (e.g., some fraction of the dense core,plus some fraction of its envelope). This is not a prob-lem, if properly taken into account in the later analysis.Other choices are possible (e.g., the “clipping paradigm”of Rosolowsky et al.), but they are less intuitive andeven harder to model. Also, most previously existingdata has been published effectively adopting the bijec-tion paradigm. 2.2. Mass Estimates
As a first example, below we present a mass-sizeanalysis of Perseus, based on column densities derivedfrom 2MASS-derived extinction data. As explained byRidge et al. (2006), the map is derived in terms of mag-nitudes of visual extinction, A V . We convert this to H column densities using the relation N H = 9 . × cm − ( A V / mag) (3)(Bohlin et al. 1978). Mass surface densities, Σ, can thenbe derived as Σ = µ H m H N H , (4)where µ H = 2 . molecule (Kauffmann et al. 2008) and m H is the weight of the hydrogen molecule. Inpractice, Σ = 0 .
047 g cm − ( N H / cm − ) =226 M ⊙ pc − ( N H / cm − ), or A V =227 . / g cm − ). The mass is then derivedby integrating the mass surface density, m = R Σ d A .For this we adopt a Perseus distance of 260 pc (Cernis1993).Goodman et al. (2009b) present a comparison of col-umn density tracers for the Perseus region. After study-ing column density maps based on dust extinction (from2MASS data), dust emission (from IRAS imaging), andline emission data (from large field CO [1–0] mapping),they conclude that extinction-based maps provide thebest available information on a cloud’s spatial mass dis-tribution. Extinction-based column densities deviate by ∼
25% from those derived from other tracers (after re-moving global offsets between estimates, e.g. due to thechoice of dust opacities). The true column density is sup-posedly in between these estimates. If every tracer has asimilar scatter with respect to the true column density,this scatter would then be ∼ / / ≈
18% for alltracers. Extinction-based estimates of the column den-sity do probably deviate by a lower amount from the truehe Mass-Size Relation from Clouds to Cores. I. A New Probe of Cloud Structure 3 a) input map b) mass-size data
Fig. 1.—
Fundamental concept of mass-size measurements. Panel (a) shows an example column density map for the Perseus molecularcloud, where labels refer to individual star-forming regions. Such maps usually contain several local maxima ( numbered crosses in map).We pick one of these maxima, and draw contours at constant column density around this peak ( map inset ). For each contour, we measuremass and size. These measurements can then be placed in a mass-size plot, as marked by crosses in panel (b). When a map containsseveral maxima, it can be divided into cloud fragments where contours just contain a single maximum ( light green and yellow boundariesin panel [a]), two of these ( dark green boundary), or even more ( magenta contour). In this sense, mass-size measurements for contourscontained in one of these areas are related. These relations are highlighted by colored lines in panel (b); the color refers to the map regionfrom where the measurements are taken, and numbers indicate the central column density peak of each fragment. When two fragmentsblend into a combined one that contains both, mass and size measurements jump discontinuously from those of the individual fragmentsto those for the combined one. These jumps are indicated by dotted lines . value. Here, we thus adopt a systematic uncertainty of .
15% for extinction-based mass estimates in Perseus.Young stars embedded in the clouds can further biasextinction observations, given their red intrinsic colors.This bias is particularly significant towards clusters, suchas NGC1333 and IC348 in Perseus. We do not excludethese regions from our study, but measurements towardsthe clusters should be interpreted with particular cau-tion. 2.3.
Example Map
Figure 2 shows the aforementioned extinction map forPerseus. The two most prominent star-forming regions inPerseus, NGC1333 and IC348, manifest as extended ex-tinction structures in this map. Fragments of very smallsize (e.g., . . ′ , corresponding to ≈ . × M ⊙ . The properties from con-tours containing the two major stellar clusters, NGC1333and IC348, are highlighted by bold black lines. As onemay naively expect, the cloud fragments enclosing theseclusters are, at given radius, the most massive fragmentswithin the cloud complex. This analysis also reveals frag-ments that are, again at given radius, much less massive Fig. 2.—
Example column density map for Perseus, presented interms of visual extinction, A V . The map is taken from Ridge et al.(2006). Contours are drawn in steps of 1 mag, starting at 2 mag.
Labels indicate the rough position of the star forming regionsNGC1333 and IC348. than the regions containing the clusters. As discussedin the next paragraphs, the column density sensitivityof the map sets a radius-dependent lower limit to themasses that can be detected in a given map. Fragmentsof a mass much lower than those shown here may thuswell exist in Perseus. Eventually, at sufficiently large ra-dius, most of the contour-bound objects found merge intoa single fragment containing essentially all of the Perseus Kauffmann et al.
Fig. 3.—
Mass-size relation for the Perseus molecular cloud.Compared to Fig. 1, we only keep the lines connecting relatedmeasurements. The top panel describes the data in detail.
Boldsolid lines indicate data for fragments either containing NGC1333or IC348.
Light solid lines show observations for regions not as-sociated with these clusters. The dashed line gives the sensitivitylimit of the analysis. In the bottom panel we present the data inthe context of reference mass-size relations.
Dotted lines indicatemean H column densities, h N H i , respectively the mass-size re-lation for a singular hydrostatic equilibrium sphere supported byisothermal pressure from gas at 10 K temperature (see Sec. 3.1 forboth). molecular cloud.In the processing of the map, we have used a minimumthreshold extinction of 2 mag. This limit is much largerthan the noise level of 0 . N H , min (in Eq. 5; note that h N H i ≥ N H , min ), gives sensitivity limits like the oneshown in Fig. 3. The minimum contrast between peaksis required to be the noise level times a factor 3, i.e.1 . ′ ; we dotherefore require the maxima to be separated by at least3 pixels (7 . ′ ′ are rejected as being unphysical. PROPERTIES OF MASS-SIZE DATASome mathematical and physical laws governing theproperties of mass-size data have to be heeded, if a mean-ingful interpretation of the observations is desired. Herewe list the most fundamental of these.3.1.
Reference Relations
A number of reference mass-size relations can help tonavigate within the observational data more intuitively.They are in part derived assuming a spherical geome-try for cloud fragments. The assumption of sphericalsymmetry may not be appropriate, though. This caveatshould be kept in mind when using the following refer-ence relations.Mass and size measurements can be used to calculatethe mean mass surface (or column) density of a frag-ment, h Σ i = m ( r ) /A ( r ). Conversely, one can draw linesof constant mass surface density, m ( r ) = h Σ i π r . Con-version to column density, and substitution of the afore-mentioned constants, yields m ( r ) = 71 M ⊙ ( h N H i / [10 cm − ]) ( r/ pc) . (5)These lines are drawn in most mass-size plots presentedhere (e.g., Fig. 3). Equation (5) implies one of the mostimportant properties of mass-size data: since columndensity decreases with increasing radius , the observedmass-size relations must be flatter than m ( r ) ∝ r .Equation (5) can actually be used to transform our re-sults into the analytical diagram presented by Tan (2007;column-density vs. mass). Our study goes beyond thework by Tan (2007) in that it systematically populatesthe parameter space with observational data.For spherical clouds, the mean density is h ̺ i = m ( r ) / (4 / π r ). The corresponding mass-size relationreads m ( r ) = 4 / π h ̺ i r , or m ( r ) = 282 M ⊙ ( h n H i / [100 cm − ]) ( r/ pc) , (6)where we substitute the density of H molecules, n H = ̺/ ( µ H m H ).In part II of this series, we shall study spherical power-law density profiles, ̺ ( s ) ∝ s − k (where s is the radius),as models for the observed mass-size relations. We showthat ̺ ( s ) ∝ s − k ⇔ m ( r ) ∝ r − k . (7) By definition, this is always the case for our source character-ization scheme. We start off from a column density peak and thenconsider, by design, regions that increase with size when loweringthe threshold column density. he Mass-Size Relation from Clouds to Cores. I. A New Probe of Cloud Structure 5The slope of the mass-size relation is therefore relatedto the slope of the density law. A density profile ̺ ( s ) ∝ s − is often adopted to describe dense cores (seeDapp & Basu 2009 for a discussion). This gives a mass-size relation m ( r ) ∝ r . In hydrostatic spheres supportedby isothermal pressure, mass, size, and gas temperatureare related by m ( r ) = 2 . M ⊙ (cid:18) T g
10 K (cid:19) (cid:18) r . (cid:19) (8)(see Kauffmann et al. 2008, Eq. 13). For gas tempera-tures T g = 10 K, one obtains the model relation drawnin most mass-size diagrams of this paper (e.g., Fig. 3).We stress that we obtain two-dimensional mass-sizerelations from column density maps. These are relatedto, but not identical with, mass-size laws obtained fromthree-dimensional density maps. This is illustrated bythe experiments conducted by Shetty et al. (2010, sub-mitted) who use the fragment identification techniquealso used by us. Their analysis is based on three-dimensional numerical simulations of turbulent clouds.As part of their experiments, they fit power-laws (sim-ilar to Eq. 1) to their mass-size data. For their par-ticular set of simulations, the exponent derived in thisfashion is similar to the number of dimensions used formass measurements (i.e., 3 when based on density, and2 when based on column density). This underlines thatthe number of dimensions considered has to be kept inmind. Note, though, that these details do not compro-mise mass-size measurements as a tool for cloud structureanalysis. Observed mass-size laws unambiguously sum-marize actual cloud structure. Only their interpretation is sensitive to the assumed geometry.3.2.
Relation to CLUMPFIND-like Results
The approach chosen here constitutes one of severalpossible choices to measure the mass and size of ob-jects in maps. Another popular approach is to use theCLUMPFIND algorithm (Williams et al. 1994). Thismethod uses contours to break emission up into sev-eral objects, just as done here. CLUMPFIND-extractedboundaries do, however, not necessarily follow contours.This is a major difference to our method, where ob-jects are always bound by some column density contour.Based on this fact alone, CLUMPFIND and our approachwill thus extract very different objects. Goodman et al.(2009a) illustrate this problem comprehensively.Further, CLUMPFIND does not allow for hierarchicalstructure, i.e., fragments containing other fragments. Forexample, CLUMPFIND will never determine the integralproperties of the entire cloud, since the cloud is usuallybroken up into many independent fragments. The rela-tion of CLUMPFIND-results to cloud hierarchy is stud-ied by Pineda et al. (2009).Figure 4 compares mass and size measurements fromCLUMPFIND to those from our approach . Both char-acterizations are based on our Perseus extinction map.To initiate CLUMPFIND, we choose contours as close aspossible to the parameters used in our contour-based seg-mentation (Sec. 2.3); the contour spacing is 1 . The CLUMPFIND results were kindly provided by J.E. Pineda.
Fig. 4.—
CLUMPFIND ( crosses ) and dendrogram mass-size re-sults ( lines ) for the extinction map presented in Fig. 2. The twoapproaches yield broadly similar, but not directly related results. segmentation yields objects that are only remotely re-lated to the regions extracted by our method. To startwith, CLUMPFIND extracts 52 peaks enclosed by in-dividual clumps, where our method only identifies 17significant peaks. This discrepancy is a consequence ofCLUMPFIND’s relaxed peak identification criterion: ev-ery local peak encircled by a continuous contour is con-sidered significant, independent of the depth of the col-umn density dip towards the next peak. There is thusno well-defined correspondence between regions identi-fied by the different algorithms.To characterize the relation between objects extractedby different methods in some sense, we inspect the struc-ture around the 17 significant column density peaksfound by our method. For every such peak, we iden-tify the CLUMPFIND object containing this peak. Themass of the latter clump can then be compared to ourresults, taken at the clump’s radius. Because of mergingof objects, our method does not provide a mass mea-surement for every possible radius (consider, e.g., theevolution of mass-size measurements starting from peak3 in Fig. 1[b]). For those cases where mass measurementsexist for the CLUMPFIND-derived radius, we find thatCLUMPFIND gives masses of order 55% to 95% of themasses derived by our approach.In summary, mass and size measurements fromCLUMPFIND are thus broadly compatible with our re-sults. By this we mean that the CLUMPFIND-derivedmass-size measurements reside in the space spanned byour own measurements. There is, however, no good cor-respondence on a detailed level. PERSEUS IN DETAIL Kauffmann et al.Given the tools derived above, it is now possible tocharacterize the mass-size relation for Perseus. While do-ing so, we also evaluate uncertainties affecting our anal-ysis. In a final step, we extend the analysis to cloudsother than Perseus.4.1.
Mass Uncertainty
To explore the impact of noise on mass measurements,we run Monte-Carlo experiments in which we add arti-ficial Gaussian noise of root-mean-square (RMS) ampli-tude σ ( m i ) to the map before structure characterization(where m i is the mass per pixel). Such trials are pre-sented in Fig. 5(a). Comparison of the derived mass-sizedata to the one from the original map does then revealthe impact of noise. In our experiments, we find that σ ( m ) m = 6 σ ( m i ) m rr beam (9)is an upper limit to the noise-induced mass changes, σ ( m ). In this, r beam is the beam radius and the nu-merical constant does slightly depend on the number ofpixels per beam.Equation (9) follows from Gaussian error propagationof m = Σ i m i . The numerical constant is increased bya factor 3, though, to provide a strict upper limit touncertainties. To test for non-Gaussian sources of error,we validate Eq. (9) by comparing the original data withthose derived from maps with additional noise. For anygiven structural branch with r < ∼
1% has to be included to capture non-Gaussian sources of error.In our study, the relative uncertainty of column densityestimates (i.e., σ ( m i ) /m i ) is ≤ / . . m > π m i, min ( r/r beam ) (in ourmap, the beam contains, by area, π pixels). Substitu-tion of these parameters in Eq. (9) give uncertaintiesof 19% for the smallest extracted features (for which r = 2 r beam ) located just at the sensitivity limit. Since σ ( m ) /m ∝ h N H i − r − (via substitution of Eq. [5]),larger regions well above the detection threshold will suf-fer lesser uncertainties, ≪ r . Aftersmoothing, the mass contained in this region will besmeared our over an area of radius r ′ ∼ r ini + r k . In ( a ) i m p a c t o f n o i s e ( b ) i m p a c t o f s m oo t h i n g Fig. 5.—
Impact of noise ( top ) and limited spatial resolution( bottom ) on mass-size measurements.
Note that the size and massscale differ between the panels.
Noise and resolution are exploredusing artificial maps created by adding noise to ( top ), respectivelysmoothing of ( bottom ), the observed data for Perseus (Sec. 4.1).Differences between the input data and the properties of the artifi-cial maps are due to these biases. The original input data is drawnin black . yellow lines in the top panel show mass-size relations frommaps with additional noise similar to the observed one (0 . Red and yellow lines in the bottom panel present mass-size rela-tions for maps with a resolution worsened to r beam , f /r beam , i = 3 / /
2, respectively (and features with r < r beam , f removed).At given radius, both biases just induce moderate changes in mass(Sec. 4.1). this, r k = ( r , f − r , i ) / / (ln(2)) / is the effec-tive radius of the smoothing kernel required to go fromthe initial to the final beam radius of the map duringsmoothing, r beam , i → r beam , f . Very approximately, aftersmoothing the mass retained in the initial area will be oforder of the fraction of the initial area to the one aftersmoothing, r / ( r + r k ) .he Mass-Size Relation from Clouds to Cores. I. A New Probe of Cloud Structure 7After rearrangement (and using r k ≪ r ), we find thatthe smoothing-induced mass bias should obey a relationof the form∆ mm ≈ . . · (cid:20) − r k /r ) (cid:21) . (10)To allow for more realistic source geometries, the nu-merical constant must be derived from experiments withactual data. We use the smoothing experiments shownin Fig. 5(b) for this purpose. With these parameters,Eq. (10) describes the relative mass bias for all structurebranches in our example map.In our study, we only consider regions with radii largerthan twice the effective beam radius, r > r beam . Sub-stitution of this (with r beam , i = 0, since the true dis-tribution on the sky has infinite resolution) in Eq. (10)shows that the mass of regions with a radius similar tothe beam diameter may be underestimated by not morethat 20%, and by less for larger fragments.4.2. Combining Dust Emission and Extinction Data
Our Perseus extinction map has a spatial resolutionof 5 ′ (0 . . . Basic Concept
For Perseus, Enoch et al. (2006) present a Bolocammap of dust emission in Perseus at 1 . ′′ , correspondingto 0 .
04 pc. At these wave lengths, the continuum emis-sion of dust at temperature T d is optically thin, and sothe observed dust continuum emission intensity, I ν , is adirect measure of the column density along a given lineof sight, I ν = µ H m H κ ν N H B ν ( T d ) (11)(see Kauffmann et al. 2008 for full details), where B ν isthe Planck function, and the dust opacity, κ ν , is eval-uated per total gas mass. Column density maps fromdust emission can then be contoured and analyzed as de-scribed before (Sec. 2) to derive mass-size data for thefragments in the dust emission map.Figure 6 combines the Bolocam-derived mass-size mea-surements with the extinction-based ones. Spatially,they are separated by a gap, since Bolocam ( . ′ ) and2MASS extinction maps ( & ′ ) probe different spatialscales. The extinction-identified fragments do, however,contain the Bolocam-derived ones. For example, shad-ing in Fig. 6 highlights Bolocam-detected fragments con-tained in the extinction peak harboring NGC1333.4.2.2. Relative Calibration of Dust Opacities
Bianchi et al. (2003) studied B68 to examine the re-lation of mass estimates from dust extinction and emis-sion. They find relative differences by factors of 1 to ∼ .
35, when adopting opacities for Ossenkopf & Henning(1994) dust grains with thin ice mantles that coagulatefor 10 yr at 10 cm − density (see Kauffmann et al.2008, Table A.1, for numerical values). This is con-sistent with previous studies of this subject, includingsources as diverse as diffuse clouds and galaxies (see ref-erences in Bianchi et al. 2003). If one fixes the opacity used for extinction observations, it thus appears that thethe Ossenkopf & Henning (1994) model opacities near1 mm wave length are too large by an average factor1 . ≈ . / .In our emission-based column density estimates,we do therefore basically adopt the aforementionedOssenkopf & Henning (1994) opacities for dust emissionobservations near 1 mm wave length, but scale thesedown by a further factor 1.5 to bring mass estimatesfrom extinction into harmony with those from dust emis-sion. As seen in Fig. 6, this procedure yields reasonableresults, since the dust emission and extinction observa-tions for NGC1333 match within less than a factor 2(i.e., when comparing the masses of the most massiveBolocam-detected fragment to the one of the least mas-sive 2MASS-identified fragment). Incorrect assumptionsabout dust temperatures may cause most of this offset(see below).To some extent, correction factors between massesfrom dust emission and extinction are also influencedby spatial filtering affecting bolometer observations (Sec.4.2.4). Future studies need to address this problem inmore detail. Still, the aforementioned scaling factoraligns dust emission and extinction studies, which is theonly aspect needed in our present study.4.2.3. Dust Temperatures
For Perseus, Rosolowsky et al. (2008a) estimate gastemperatures between 9 K and 18 K. Assuming thatdust and gas temperatures are similar, we do thereforeadopt a temperature of (12 . ± .
5) K in our mass esti-mates. This temperature uncertainty results in a relativemass uncertainty of ∼
20% for Perseus. A few sourcesmay have dust temperatures, and mass biases, outside ofthis range. 4.2.4.
Impact of Spatial Filtering
Like all other upcoming, present, and pastground-based bolometer-derived dust emission maps(Kauffmann et al. 2008), the Bolocam maps of Perseusare not sensitive to structure larger than someinstrument-dependent spatial scale. In its impact,this problem is similar to the spatial filtering in inter-ferometric imaging. For Bolocam, Enoch et al. (2006)show the filtering scale to be of order 1 ′ to 2 ′ radius.Quantitatively, this removal of large-scale structure hasan influence opposite to the impact of smoothing (Sec.4.1): here, the relative mass-loss increases with spatialscale. This bias has to be considered carefully whenusing emission-based mass-size measurements for anal-ysis. Obviously, the true mass will be larger than theobserved value. Filtering in bolometer maps is unfor-tunately too complex to be characterized in a compactfashion; see Kauffmann et al. (2008) for a few rules ofthumb. For the particular case of the Bolocam maps ofPerseus, Enoch et al. (2006) report losses ≤
10% for radii ≤ ′ , but do not sufficiently explore larger objects.4.3. Global Trends in Maximum Mass for given Size
The mass-size tendencies seen, e.g., in Fig. 6 suggestto describe these trends with power laws of the form m ( r ) = m ( r/r ) b . (12) Kauffmann et al. Fig. 6.—
Mass-size data for Perseus ( solid lines ). The data are from dust extinction maps probing large spatial scales, and Bolocam dustemission observations sensitive to small ones. The left panel highlights the nature of the data; dashed lines give the instrument-dependentsensitivity limits.
Circles and the dotted line indicate calculation of the global mass-size slope (Sec. 4.3). The middle panel highlights slopesderived by matching power-laws to parts of the data. Tangential slopes (Sec. 4.4) are derived in the same fashion, but for infinitesimalradius ranges. The right panel presents Perseus in the context of the reference mass-size relations from Sec. 3.1. Here, black solid lines highlight Bolocam-detected fragments in NGC1333, as well as the extinction-probed fragments containing this cluster. Other data aredrawn using grey solid lines . In this, b is the slope of the relation, and m is the in-tercept. As shown in Fig. 6, such laws trace, e.g., themass-size relation of the most massive fragments in the0 . ≤ r/ pc ≤ . ≤ r/ pc ≤ ± m max ,observed at radii of r sm = 0 .
05 pc and r lg = 3 . b glob = ln[ m max ( r lg ) /m max ( r sm )]ln[ r lg /r sm ] . (13)As illustrated in Fig. 6, this slope is defined such thatEq. (12) connects the mass and size measurements for b = b glob . In Perseus, b glob = 1 . ± .
14, where weevaluate the uncertainty very conservatively by scalingthe emission-based mass in Eq. (13) up and down by afactor 2.In Perseus, at the chosen scale of r = 0 .
05 pc (i.e.,40 ′′ ), the Bolocam dust emission map is not significantlyaffected by spatial filtering. Similarly, at r = 3 pc (40 ′ ),the extinction map is not biased by presence of stellarclusters. The measurements of b glob are thus immune tothese influences.4.4. Local Trends in Mass
We now turn to slopes of infinitesimal tangents. Aswe explain below, the Bolocam data are not suited forthis analysis, since they suffer from too strong spatialfiltering. 4.4.1.
Method and Uncertainties
In some cases, one may wish to use Eq. (12) to describetangents to the data on spatial scales smaller than theone for which we calculate global slopes. This is demon-strated in Fig. 6 using tangents to the most massive frag-ments seen at given radius. In particular, it can be de-sirable to fit infinitesimal tangents to mass-size trends.For these, the slope reads b ( r ) = d ln( m [ r ′ ])d ln( r ′ ) (cid:12)(cid:12)(cid:12)(cid:12) r ′ = r . (14)Figure 7(a) shows slopes derived from mass and size dif-ferences between consecutive contours. As seen in the fig-ure, these data are rather noisy. We do therefore smooththe measurements. To do this, at radius r , we replaceslope and radius by their respective unweighted arith-metic means, as derived within a smoothing kernel ofwidth [ r, . · r ] (for a given cloud fragment, not permit-ting mergers). This yields the data shown in Fig. 7(b).Sometimes, however, the smoothing kernel is not filledwell, and the data are still noisy. Thus, we do finallyremove all data where the kernel is not filled to at least2 /
3. Figure 7(c) shows this final result.The impact of noise can be estimated by propagatingthe mass uncertainties due to noise (Eq. 9) within theslope calculations. This yields σ ( b ) b ≤ σ ( m i ) m rr beam . (15)Because of the aforementioned smoothing, the numer-ical constant must be calibrate with our noise exper-iments (as done for Eq. [9]). To obtain an estimateof the expected uncertainties, we can repeat the massand size substitutions done in the discussion of Eq.(9). This gives maximum uncertainties ∼ σ ( b ) /b ∝ h N H i − r − (see discussion of Eq. [9]), the un-certainties are small for larger regions well above the de-he Mass-Size Relation from Clouds to Cores. I. A New Probe of Cloud Structure 9 a ) p l a i nd i ff e r e n ce s b ) s m oo t h e d c ) s m oo t h e d a nd fi l t e r e d Fig. 7.—
Calculation scheme for slopes ( top to bottom ) and im-pact of noise ( black vs. yellow lines ). We start with slopes directlycalculated from mass and size differences for successive contours(panel a). These measurements are subsequently smoothed (panelb), and data are removed where the smoothing kernel is not filled(panel c). The black lines hold for the observed Perseus extinctiondata. As in Fig. 5, yellow and black lines indicate results frommaps with and without artificial noise, respectively. The dottedline indicates the upper limit on slopes inherent to our method, b < tection threshold. In practice, uncertainties <
10% are areasonable estimate for well-detected regions warrantingdetailed study.The slope difference due to smoothing is given by thefirst derivative of the mass bias due to smoothing (Eq.10) with respect to the radius. Including the usual cali-bration of numerical constants, we constrain the absolutesmoothing-induced error to∆ b ≤ . . r k /r (1 + r k /r ) . (16)In a few cases, however, the error can be larger by afactor 2. In this paper, we reject regions with a diametersmaller than twice the beam diameter. Substitution ofthis limit (i.e., r k /r < /
2) into Eq. (16) implies thatslopes are overestimated by a number smaller than 0.1due to resolution.As mentioned in Sec. 4.2.4, bolometer maps suffer fromspatial filtering. This has an impact opposite to the in-fluence of smoothing, and artificially shallow slopes aremeasured from such maps. Since the filtering is verystrong in Bolocam maps, we do not use these for thederivation of tangential slopes.
Fig. 8.—
Impact of limited spatial resolution on slope measure-ments. See Figs. 5(b) and 7 for explanations of mark-up.
Results for Perseus
Figures 7(c) and 8 show tangential slopes for Perseus.The tangential slopes in the 1 ≤ r/ pc ≤ b = 1 . .
7. At a given radius, theslopes for different fragments do often differ by more thantheir uncertainty. Also, in a given fragment the slope canchange significantly with respect to radius. This meansthat it is not possible to describe an entire cloud by a sin-gle tangential slope. The observed tangential slopes bearno obvious relation to the global slope ∼ .
56 derived forthe same radius range (Fig. 6).A slope of ≥ b < SUMMARY & OUTLOOK5.1.
Utility of and Outlook for Mass-Size Studies
As we have shown above, our map characterizationscheme yields reliable measurements of mass and size.From these, mass-size slopes and intercepts can be de-rived. Below, we describe how these data contribute tocritical fields of star formation research.First of all, this approach permits a continuous char-acterization of cloud structure across a large range ofspatial scales. This is just a desirable feature of anydata analysis method, independent of the exact natureof the later analysis. The need for such a procedureled Rosolowsky et al. (2008b) to the development of the“dendrogram technique”.In star formation research, the basic mass-size mea-surements permit to compare fragment masses at a givenspatial scale. Consider the classical case in order to seethe advantage: usually, “cores” and “clumps” extractedfrom maps differ in their size. In this case, it is not clearwhat differences in masses mean, even if just a singlecloud is considered.Spatially continuous cloud characterizations becomeparticularly useful when comparing observations for dif-ferent molecular clouds. Usually, every cloud is stud-ied at a different physical resolution (i.e., pc per pixel).0 Kauffmann et al.In the classical case, mass measurements will thus usu-ally refer to vastly different spatial scales. With ourapproach, however, all scales are probed, and differentclouds can be compared at the same physical scale. Thisis extensively employed in part II, where we study a sam-ple of clouds.The general utility of measurements of mass and size isknown since long. For example, one can compare the ac-tual to virial masses or, more generally, masses predictedby theoretical cloud models. Equation (8), for example,relates model fragment masses and gas temperatures. Weshall not discuss such considerations here in detail.A property uniquely constrained by our method aremass-size slopes; these can only be measured via a scale-independent method. In particular, this gives access tothe density structure of molecular clouds. For simplemodels of cloud structure, the mass-size slope is, e.g.,directly related to the slope of the density profile (Eq. 7).Such work on large-scale structure in molecular clouds isurgently needed, since cloud density profiles are presentlynot known on scales & . Summary
This work studies the internal structure of molecularclouds by breaking individual cloud complexes up intoseveral nested fragments. For these, we derive massesand sizes, as e.g. outlined in Fig. 1. Effectively, we per-form a “dendrogram analysis” of a two-dimensional map,as introduced by Rosolowsky et al. (2008b).The present paper establishes the method via a de-tailed analysis of the Perseus Molecular Cloud. Othersolar neighborhood molecular clouds ( .
500 pc; the PipeNebula, Taurus, Ophiuchus, and Orion) are discussed inthe next paper of this series (part II).Power-laws of the form m ( r ) = m ( r/ pc) b , with slope b and intercept m , prove useful to quantify the rela-tions between mass, m , and size, r (i.e., the effectiveradius). Sections 4.3 and 4.4 discuss two different ap-proaches to define and measure the slope. We use global slopes to measure the relation between structure at smalland large scales. This is done by connecting the mass-sizemeasurements of the most massive fragments at small(0 .
05 pc) and large radius (3 . Tangential slopes , on the other hand, are calculatedinfinitesimally at a given spatial scale (Eq. 14). The un-certainties in these properties are examined in Secs. 4.1and 4.2 (for mass and intercept), respectively Secs. 4.3and 4.4 (for slopes).We conclude that our mass, slope, and intercept mea-surements provide a reliable method to characterizecloud structure. Our approach enables a continuousand reliable characterization of cloud structure in the0 . . r/ pc .
10 spatial range. This is not possi-ble using previous methods, since these are usually bi-ased towards a particular spatial scale (see, e.g., theCLUMPFIND analysis in Fig. 4). Such comprehensivepictures of star-forming regions can be used to develop amore complete theoretical understanding of global cloudstructure (Sec. 5.1).A first observational and theoretical exploitation ofthis method is presented in part II of this series. Wecharacterize, for example, the typical parameter space forsolar neighborhood molecular clouds not forming mas-sive stars. Based on this, we chart a potential mass-sizethreshold for the formation of massive stars. Mass-sizeslopes are used to constrain large-scale density gradientswithin molecular clouds.We are grateful to Nicolas Peretto, who served as aconsiderate and knowledgeable referee who helped tosignificantly improve the quality of the paper. Thisproject would not have been possible without helpfrom Erik Rosolowsky. His dendrogram analysis code(Rosolowsky et al. 2008b) was instrumental for our anal-ysis. We thank Jaime Pineda for his help withthe CLUMPFIND experiments presented in Fig. 4.Enoch et al. (2006) contributed maps to the presentstudy. We are grateful for their help. This work wasin part made possible through Harvard Interfaculty Ini-tiative funding to the Harvard Initiative in InnovativeComputing (IIC).10 spatial range. This is not possi-ble using previous methods, since these are usually bi-ased towards a particular spatial scale (see, e.g., theCLUMPFIND analysis in Fig. 4). Such comprehensivepictures of star-forming regions can be used to develop amore complete theoretical understanding of global cloudstructure (Sec. 5.1).A first observational and theoretical exploitation ofthis method is presented in part II of this series. Wecharacterize, for example, the typical parameter space forsolar neighborhood molecular clouds not forming mas-sive stars. Based on this, we chart a potential mass-sizethreshold for the formation of massive stars. Mass-sizeslopes are used to constrain large-scale density gradientswithin molecular clouds.We are grateful to Nicolas Peretto, who served as aconsiderate and knowledgeable referee who helped tosignificantly improve the quality of the paper. Thisproject would not have been possible without helpfrom Erik Rosolowsky. His dendrogram analysis code(Rosolowsky et al. 2008b) was instrumental for our anal-ysis. We thank Jaime Pineda for his help withthe CLUMPFIND experiments presented in Fig. 4.Enoch et al. (2006) contributed maps to the presentstudy. We are grateful for their help. This work wasin part made possible through Harvard Interfaculty Ini-tiative funding to the Harvard Initiative in InnovativeComputing (IIC).