Tracing the general structure of Galactic molecular clouds using Planck data: I. The Perseus region as a test case
Orlin Stanchev, Todor V. Veltchev, Jens Kauffmann, Sava Donkov, Rahul Shetty, Bastian Körtgen, Ralf S. Klessen
aa r X i v : . [ a s t r o - ph . GA ] M a y Mon. Not. R. Astron. Soc. , 1–13 (2015) Printed 12 September 2018 (MN L A TEX style file v2.2)
Tracing the general structure of Galactic molecular cloudsusing
Planck data: I. The Perseus region as a test case
Orlin Stanchev ⋆ , Todor V. Veltchev , , Jens Kauffmann , Sava Donkov , RahulShetty , Bastian K¨ortgen and Ralf S. Klessen University of Sofia, Faculty of Physics, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria Universit¨at Heidelberg, Zentrum f¨ur Astronomie, Institut f¨ur Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Ger-many Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany Department of Applied Physics, Technical University, 8 Kliment Ohridski Blvd., 1000 Sofia, Bulgaria Universit¨at Hamburg, Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
Accepted 2015 May 04. Received 2015 May 04; in original form 2013 December 13
ABSTRACT
We present an analysis of probability distribution functions (pdfs) of column density indifferent zones of the star-forming region Perseus and its diffuse environment based onthe map of dust opacity at 353 GHz available from the
Planck archive. The pdf shapecan be fitted by a combination of a lognormal function and an extended power-law tailat high densities, in zones centred at the molecular cloud Perseus. A linear combinationof several lognormals fits very well the pdf in rings surrounding the cloud or in zonesof its diffuse neighbourhood. The slope of the mean density scaling law h ρ i L ∝ L α is steep ( α = − .
93) in the former case and rather shallow ( α = − . ± .
11) inthe rings delineated around the cloud. We interpret these findings as signatures oftwo distinct physical regimes: i) a gravoturbulent one which is characterized by nearlylinear scaling of mass and practical lack of velocity scaling; and ii) a predominantlyturbulent one which is best described by steep velocity scaling and by invariant forcompressible turbulence h ρ i L u L /L , describing a scale-independent flux of the kineticenergy per unit volume through turbulent cascade. The gravoturbulent spatial domaincan be identified with the molecular cloud Perseus while a relatively sharp transitionto predominantly turbulent regime occurs in its vicinity. Key words:
ISM: clouds - ISM: structure - submillimetre: ISM - turbulence - meth-ods: data analysis - methods: statistical
One of the tasks of the High Frequency Instrument (HFI;see Lamarre et al. 2010) operating on the
Planck satel-lite is to map the spatial distribution of Galactic molec-ular and diffuse gas. The data on dust emission at fre-quency of 353 GHz allow for tracing the general struc-ture of molecular clouds (MCs) and star-forming regionsin a wide range of spatial scales – from vast zones thatinclude MCs and their neighbourhood at scales of tensof parsec through their diffuse envelopes down to theirdense and clumpy central zones. Such a study would con-tribute to the numerous attempts of different authors tomodel and explain the MC structure through analysis ofthe column density distribution (Lombardi, Alves & Lada2010; Ballesteros-Paredes et al. 2012), of discrete, hierarchi- ⋆ E-mail: o [email protected] cal clumps (Kauffmann et al. 2010a,b) or combining bothapproaches (Beaumont et al. 2012).The Perseus region is particularly interesting for thestudy of general MC structure due to several reasons: i) itis compact, apparently axisymmetric; ii) it does not over-lap with other fore- or background clouds or cloudlets, dueto its high galactic latitude ( b ∼ − ◦ ); iii) it is associ-ated with several sites of recent or active star-formation;iv) the whole region has been extensively investigated (seeBally et al. 2008, for a review).The shape of the column density probability distribu-tion function (hereafter, N -pdf) contains important infor-mation about the physics of star-forming regions. Dust ex-tinction (Kainulainen et al. 2009; Lombardi, Alves & Lada2011) as well dust continuum studies (e.g. Schneider et al.2013) demonstrate that the N -pdf in Galactic MCs is nearlylognormal in some cases or can be fitted through a combi-nation of lognormal and power-law functions, in other cases. c (cid:13) Stanchev et al.
Figure 1.
The selected probe zones, drawn on the
Planck dust opacity map: (left) the ‘Perseus region’ (PR), containing Perseus MC(the central ellipse) and a high-extinction ‘core’; (right) zones of diffuse gas (labelled; see also Table 1) in the neighbourhood of the PR(unlabelled) .
A purely lognormal N -pdf is typical for turbulent medium(e.g. V´azquez-Semadeni 1994) as the standard deviation de-pends on the Mach number and the forcing of turbulence(Federrath, Klessen & Schmidt 2008). On the other hand,numerical simulations of self-gravitating medium indicatethat the development of a power-law (PL) tail is to be ex-pected at high densities and at timescales, comparable tothe free-fall time (Klessen 2000; Kritsuk, Norman & Wagner2011; Federrath & Klessen 2013; Girichidis et al. 2014).Therefore, the N -pdf can be used as an effective researchtool to disentangle complex issues like scaling laws in turbu-lent MCs, energy equipartitions in star-forming regions andtheir relation to cloud evolutionary status.In this Paper we study and compare the behaviour ofthe N -pdf in embedded zones of Perseus region, containingPerseus MC, and in zones of its diffuse vicinity. The per-formed analysis allows for the reconstruction of the densityscaling law and for the description of the physical regime inthe Perseus region. We used the component map of dust opacity at frequency353 GHz (angular resolution: 1 . ′
72 per pixel) available fromthe
Planck archive to extract a large area containing thePerseus MC. Dust emission data are more appropriate for http://irsa.ipac.caltech.edu/data/Planck/release 1/all-sky-maps/previews/HFI CompMap DustOpacity 2048 R1.10 Table 1.
Statistics of the selected elliptical zones.Zone Nr. R h N i R Massof pixels [ pc ] [ 10 cm − ] [ 10 M ⊙ ] Embedded zones in the ‘Perseus region’ ‘Core’ 4663 5.3 10.1 3.41 (the MC) 14498 9.1 7.4 7.82 32299 13.6 5.2 12.03 57339 18.1 3.9 16.04 89649 22.6 3.1 20.05 123812 26.5 2.8 25.06 (the PR) 163351 30.4 2.5 30.0
Diffuse zones in the neighborhood d1 155132 30.4 1.0 14.7d2 155121 30.4 0.8 10.0d3 147628 30.4 0.9 10.0d1 ∪ d2 ∪ d3 318479 41.4 0.9 19.6 our study than dust extinction maps for two major rea-sons. First and foremost, extinction mapping depends ondetection of background stars and thus suffers from selec-tion effects. Precise photometry of weak stellar sources ispossible in zones of low to moderate extinction which biasesextinction maps towards lower values. Consequently, massesof fragments of fixed size can be essentially underestimated(cf. Sec. 2.2.2 in Kauffmann et al. 2010b). Second, Planck data offer the opportunity to probe regions of very high col-umn densities, not resolved in dust extinction maps due toa loss of background stars.Recent research indicates that a distance gradient exists c (cid:13)000
Diffuse zones in the neighborhood d1 155132 30.4 1.0 14.7d2 155121 30.4 0.8 10.0d3 147628 30.4 0.9 10.0d1 ∪ d2 ∪ d3 318479 41.4 0.9 19.6 our study than dust extinction maps for two major rea-sons. First and foremost, extinction mapping depends ondetection of background stars and thus suffers from selec-tion effects. Precise photometry of weak stellar sources ispossible in zones of low to moderate extinction which biasesextinction maps towards lower values. Consequently, massesof fragments of fixed size can be essentially underestimated(cf. Sec. 2.2.2 in Kauffmann et al. 2010b). Second, Planck data offer the opportunity to probe regions of very high col-umn densities, not resolved in dust extinction maps due toa loss of background stars.Recent research indicates that a distance gradient exists c (cid:13)000 , 1–13 eneral MC structure using Planck data: I. -4 -3 -2 -1
1 10 1 10 A r ea f r a c t i on N( τ ) [ 10 cm -2 ]A V n=-2.08 (Zone 6)Zone 6 (the whole PR)Zone 4Zone 2Zone 1 (the MC) 0.1 1 10 0.1 1 10N( τ ) / 〈 N 〉 R A V ‘Core’, R =5.3 pcZone 1, R =9.1 pcZone 2, R=13.6 pcZone 4, R=22.6 pcZone 6, R=30.4 pc Figure 2.
Column density pdfs in the PR (Fig. 1): (left) N is in absolute units, the lognormal fits of all pdfs (dashed) and the fit of thePL tail in the largest zone (solid) with its slope n are given; (right) N is normalized to its mean value in the considered zone; the spanof N PLlow is plotted with vertical shaded area. Only some of the pdfs are shown for clarity. across the Perseus region (Schlafly et al. 2014). Therefore allspatial sizes used in this Paper were calculated adopting amean distance D = 260 ±
40 pc to the Perseus MC tak-ing into account the parallax estimates to its western part(Hirota et al. 2011) and the variety of other ones to differentparts of it (for review, see Bally et al. 2008).The selected zones to derive the N -pdf are shown inFig. 1 while general physical information on them is pro-vided in Table 1. The zone labelled ‘Perseus region’ (here-after, PR) contains six embedded ellipses of increasing sizeand identical orientation angle, following the morphology ofPerseus MC (Fig. 1, left). The central ellipse ( N & cm − is concentrated in Zone ii region G159.6-18.5 (Bally et al. 2008). To compare the contribution of dif-fuse gas (atomic and molecular) to the N -pdf, we selectedadditionally three zones in the western neighbourhood of PRwith the same areas like the largest elliptical zone ( N to visual extinction A V . The value of N (H i ) /A V which is frequently used in different works is 1 . × cm − (Bohlin et al. 1978). It has been obtained fromphotometry of OB stars, with an upper limit in the chosen sample E ( B − V ) ≃ . m A V . . m
5. In view of this limi-tation, transformations to column density in MC fragmentsof high extinction (typically populating the PL tail of the N -pdf) should be considered with caution. Therefore, in thisPaper, we derive N (H) directly from dust opacity τ at 353GHz, provided from Planck data. As evident from Fig. 6 inPlanck Collaboration 19 (2011), there is a very good linearcorrelation between dust opacity and total hydrogen columndensity N (H) obtained from H i
21 cm maps, except for therange 0 . × N (H) × cm − wherein undetected(‘dark’) molecular gas may affect the results. Neglecting thelatter effect, we adopted a linear conversion formula fromdust opacity to hydrogen column density: N (H) = C τ + C , (1)where coefficients C and C were obtained from fitting the N (H)-binned representation of the correlation at columndensities below 0 . × cm − (see Fig. 6 in Planck Collab-oration 19, 2011). The possible uncertainty of the calculated N (H) is about a factor of 2 which is not significant for theanalysis presented in this Paper. The derived N -pdfs in the PR and in its embedded zones(Fig. 1, left) are plotted in Fig. 2 (left). Apparently, theirshape can be described by a combination of a lognor-mal function around the peak and a power-law (PL) tailwith slope varying in a narrow range between − . − .
1. Such behaviour is in agreement with other obser-vations of nearby MC complexes (Kainulainen et al. 2009;Federrath & Klessen 2013; Schneider et al. 2014a,b) or withresults from numerical simulations of self-gravitating clouds c (cid:13) , 1–13 Stanchev et al. (Kritsuk, Norman & Wagner 2011). The slope and the lowercolumn-density limit N PLlow of the PL tail were calculatedthrough the method
Plfit (Clauset et al. 2009) which ex-tracts these characteristics from analysis of the unbinned observational data. In this way we avoid subjective estima-tions – note, that the obtained values N PLlow (Table 2) arelower than one would put ‘by eye’ (say, at ∼ × cm − ;Fig. 2, left).Normalisation to the mean column density h N i in eachconsidered zone yields N -pdfs with practically identicalshape and N PLlow / h N i (Fig. 2, right), for a wide range of effec-tive radii. Physical interpretation of this result is suggestedin Sect. 4. The N -pdf in each diffuse zone exhibits a complex shapewith several discernible peaks in whose vicinity the distri-bution is apparently lognormal (Fig. 3, left). The same be-haviour is found when one considers the N -pdf in any ofthe rings outlined by the boundaries of embedded ellipticalzones in the PR, excluding the PL tail (Fig. 3, right). The column-density distribution in any of these regions ofdiffuse gas can be represented as sum of lognormal functionsof typelgn i ( N ; a i , N i , σ i ) = a i p πσ i exp − [ lg( N/N i ) ] σ i , where the total number m of components (1 i m ) spanstypically from a few to dozen and the parameters N i , σ i and a i are obtained from the following fitting procedure of thewhole distribution: • Prominent local peaks , i.e. of considerable width andheight, are located and guess values N (0) i are assigned tothem. It is appropriate to start from such peaks with broadwings (large σ i ) since they have greater statistical weight. • The distribution is fitted in close vicinities of thesepeaks through single lognormals with fitting parameters N (1) i , σ (1) i and a (1) i . • The total fitting function P mi =1 lgn i is composed. Thenthe whole distribution is fitted and next approximations ofthe fitting parameters N (2) i , σ (2) i and a (2) i are obtained. • Small local peaks and inflexion points are located andthe corresponding number of lognormal components lgn i isadded to the fitting function, increasing m . The previousthree steps are repeated and a next set N (3) i , σ (3) i and a (3) i is calculated for all components. • In case the achieved approximation of the distributionis not satisfactory, a few further components could be added.The latter have small statistical weights and contribute onlyfor small local improvements of the fit. • Now the fitting-parameters’ space ( N i , σ i , a i ) ( i =1 , ..., m ) is severely restricted and the total fitting func-tion P mi =1 lgn i is fitted via NLLS Levenberg-Marquardt al-gorithm until the χ criterion for fit’s goodness (at 95% con-fidence level) is satisfied. All obtained fits in the diffuse zones and in the ringsof the PR are plotted in the Appendix A, Figs. A1 and A2.The achieved fits’ goodness is given in Table A1. A possible interpretation of this decomposition of the N -PDF in a region to lognormals is that domains of varioustypical column densities N i and velocity dispersions (relatedto σ i ; see e.g. Sect. 3.6 in Federrath et al. 2010) contribute toit. Each domain is constituted of non-overlapping fragmentsof the considered region. Through the third parameter a i ofa given component, we may define the effective size L i of adomain and hence introduce the notion of spatial scale . Eachscale has its own lognormal column-density pdf lgn i ( N ) andfractional area R lgn i ( N ) / P i R lgn i ( N ). Then the effectivesizes of scales included in a zone of diffuse gas with totalarea S and effective radius R = p S/π are derived by: L i = s a i P i a i R . (2)We stress that spatial scale is a rather abstract quantitywhich includes, in general, a set of fragments with varioussizes (down to single pixels) and shapes. The notion is illus-trated in Fig. 4 which represents one possible spatial real-ization of four scales “detected” through decomposition ofthe N -pdf in an exemplary diffuse region. The larger scales L and L (large a i / P i a i ) consist of fragments spanningbroad column density ranges (wide N -pdfs) and have dif-ferent typical densities N i . On the other hand, the smallerscales L and L (smaller a i / P i a i ) are more homogeneous(narrow N -pdfs), with minor variations of N .The N -pdfs in the PR cannot be represented as com-binations of lognormals but note that their column densityranges almost entirely fall into the one, corresponding to thePL tail of the largest zone R as spatial scales. It will be shown be-low that all zones in the PR (except the cloud ‘core’) obeya mass scaling law with identical slope. The analysis presented below is based on several reasonablephysical assumptions. Some of them are discussed and jus-tified in Donkov, Veltchev & Klessen (2011), in the frame-work of a statistical approach for description of MC struc-ture at an early evolutionary stage. Here we briefly sum-marise them as follows:(i) Power scaling laws of velocity dispersion and mean den-sity (often called “first and second Larson’s laws”; Larson1981), as well of mass: u L ∝ L β , β > . (3) h ρ i L ∝ L α , (4) M L ∝ L γ . (5)The natural relation M L = h ρ i L L leads to: c (cid:13) , 1–13 eneral MC structure using Planck data: I. -3 -2 -1 A r ea f r a c t i on N( τ ) [ 10 cm -2 ]A V d1 -3 -2 -1 τ ) [ 10 cm -2 ]A V Ring 6-5
Zone 6Zone 5
Figure 3.
Examples of N -pdf (open symbols) decomposition to lognormals (dotted) in a diffuse zone (left) and in a ring (right) delineatedby the boundaries of two successive elliptical zones in the PR (cf. Table 1). In the latter case, the location of the lower boundary N PLlow of the PL tail is indicated (dashed). The embedded graph demonstrates how the N -pdf in a considered ring results from N -pdfs of theelliptical zones (filled symbols). γ = α + 3 . (6)(ii) An equipartition relation between gravitational and tur-bulent energy of type E kin ∼ f | W | , where f is a constant .The existence of such energy equipartition at different spa-tial scales and evolutionary stages in a MC is observationallyverified and is explained by the coupling of various physicalprocesses in the interstellar medium to each other (see Sect.1 in Hennebelle & Falgarone 2012). It is supported as wellby numerical simulations (V´azquez-Semadeni et al. 2007).Hence one obtains a relation between the scaling indicesof mass and velocity dispersion:12 M L u L ∝ G M L L ⇒ u L ∝ (cid:18) M L L (cid:19) ∝ L γ − , i . e . β = ( γ − / h ρ i L as a quotient of the mean column density and the scale. Ata scale L i in the diffuse rings and zones, characterized by alognormal N -pdf with peak N i , h N i is simply N i . On theother hand, N -pdfs in zones of the PR are asymmetric andin this case we use the estimates of h N i from Table 1 (col.4) to calculate h ρ i R ≈ h N i /R .The mean-density scaling laws in the diffuse rings, inthe elliptical zones and in the test simulation (see Sect. 4.3)are derived from the N -pdfs via a weighted least-squares fit-ting described in Appendix B. These scaling laws and thoseof velocity dispersion hold in the inertial range of a super-sonic turbulent cascade (Donkov, Veltchev & Klessen 2011).Therefore it is instructive to check the behaviour of the fluxof kinetic energy per unit volume through the cascade incase of compressible turbulence, defined as: Note that the value of f is not important for this analysis,e.g. we do not assume gravitational boundedness ( f .
5) orvirialization of the considered volume.
Figure 4.
Schematic representation of spatial scales L i in our ap-proach. The border of a considered diffuse region is drawn (whiteline). The positions and shapes of the fragments are arbitrary andrepresent one possible spatial realization of the scales. h ρ i L u L τ L = h ρ i L u L L/u L = h ρ i L u L L (8)where τ L is the turbulent crossing time at scale L . The pos-sible scaling of this quantity is part of the physical analysispresented below. We suggest that the N -pdfs as well the underlying ρ -pdfsreflect two types of physical regime which govern the gen-eral structure of MCs. The PL tail corresponds to a regimein which gravity determines structure whereas the pdf part c (cid:13) , 1–13 Stanchev et al.
Table 2.
Parameters of the N -pdfs derived in the embeddedelliptical zones of the PR. Column densities are given in units[10 cm − ]. Notation: N max - peak of the distribution, n - slopeof the PL tail, f PL ( S ) - PL area fraction, f PL ( M ) - PL massfraction.Zone N max σ N N PLlow n f PL ( S ) f PL ( M )‘Core’ 5.97 0.14 5.95 − .
67 0.88 0.841 4.51 0.18 5.22 − .
84 0.85 0.772 3.35 0.17 3.90 − .
83 0.83 0.723 2.15 0.12 2.85 − .
85 0.82 0.714 1.81 0.12 2.39 − .
82 0.83 0.715 1.80 0.13 1.98 − .
93 0.84 0.736 1.89 0.16 1.85 − .
09 0.85 0.74 that can be decomposed to lognormal components corre-sponds to a regime with a prevailing role of turbulence. Thisconcept is consistent with the fact that a well-developedPL tail, spanning about one order of magnitude or more,is present in pdfs of regions containing the very MC (Fig.2) while a pdf with lognormal components and without PLtail is characteristic of the rings in the PR and of the diffusezones in its vicinity (Figs. A1 and A2). The derived meandensity scaling laws in the suggested two physical regimesalso differ significantly – they are illustrated in Fig. 5. Letus examine the emerging picture closely. N -pdf As already commented in Sect. 3.2, the embedded zones inthe PR can be considered as subscales of the largest Zone N -pdfs in these zones, normal-ized to the corresponding mean column densities h N i R , turnout to be strikingly identical (Fig. 2, right, and Table 2),with the same width σ N , peak N max / h N i R and lower bound-ary N PLlow / h N i R of the PL tail. (Only the MC ‘Core’ exhibitsan exception from this trend.) This result might be inter-preted by assuming that the Perseus MC is self-gravitatingand its general structure is described by power-law surface-and volume-density profiles.Let a region with area S and volume V , centredat L = 0, is characterised by a surface-density profileΣ( L ) = Σ ( L/ − t and a volume-density profile ρ ( L ) = ρ ( L/ − p . This yields scaling laws of column density h N i ∝ L α N ( α N = − t ) and mean density h ρ i L ∝ L α ( α = − p ), where those quantities are averaged over the en-tire profile. Then, if the column- and volume density pdfs inthe region display PL tails, one gets for their slopes n and q , respectively: p N (ln(Σ / Σ )) ≡ dSd ln(Σ / Σ ) ∝ Σ − /t = ⇒ n = − /t (9) p (ln( ρ/ρ )) ≡ dVd ln( ρ/ρ ) ∝ ρ − /p = ⇒ q = − /p (10)As demonstrated by Kritsuk, Norman & Wagner(2011), the relation between the volume-density and the surface-density profiles in isotropic medium is:Σ( L ) = + ∞ Z −∞ ρ (( L + x ) / ) dx = ... ∝ L − p + ∞ Z (1 + ( x/L ) ) − p/ d ( x/L ) , (11)where the integral is converging for 1 p ∞ . Obviously, t = p − − ( α + 1).Now the derived mean density scaling index α = − . n = − .
15, in a very good agreementwith the value − .
08 obtained directly from the observa-tional N -pdf (Fig. 2, left). On the other hand, one gets fromequation 6 a mass-size scaling index γ = 1 .
07. Recallingassumption (ii) for equipartition of energies, we derive a ve-locity scaling law of vanishing slope (cf. equation 7): u L ∝ L ( γ − / ∝ L β , β ≈ .
04 (12)This surprising result is consistent with the dendro-gram analysis of cloud fragments from numerical simula-tions of self-gravitating cloud (Shetty et al. 2010, see Fig.2 in Stanchev et al. 2013). Its physical meaning is that thesum of the specific kinetic and gravitational energies at eachscale within the density span of the pdf tail is an invariant:12 u L − G M L L = inv( L ) . (13)Note that this relation is not equivalent to the assumedequipartition of energies at each scale, although derived fromit. The kinetic and gravitational energy terms contain as wellthe mass which is scale-dependent.From equation 13 one obtains for the flux of kinetic en-ergy h ρ i L u L /L ∝ L − . This finding means that the kineticflux increases strongly at small scales which can be naturallyexplained with matter acceleration due to gravity. Our resultis very different from the one of Galtier & Banerjee (2011)who derived from theoretical considerations of compress-ible isothermal turbulence h ρ i L u L /L = inv( L ) for purelysolenoidal forcing and h ρ i L u L /L ∝ L / for strong compres-sive forcing. The latter relation was confirmed as well byFederrath (2013) from a number of high-resolution numeri-cal simulations of compressible turbulence without gravity.It is evident that the structure of the PR centred on PerseusMC cannot be modelled taking into account only interstellarturbulence; self-gravity is an essential factor in constructingthe correct physical picture. The derived mean density scaling law in the rings of the PRis shown in Fig. 5, top (red symbols). It is shallower than theclassical “second Larson’s law” with α = − . α ≃ − .
80, one gets γ = 2 .
20 and β ≈ .
60. Then – incomparison with the analysis in the previous Section, – wefind here a scale-invariant flux of the total energy per unitvolume: h h ρ i L u L − G h ρ i L M L L i L/u L = inv( L ) . (14) c (cid:13)000
60. Then – incomparison with the analysis in the previous Section, – wefind here a scale-invariant flux of the total energy per unitvolume: h h ρ i L u L − G h ρ i L M L L i L/u L = inv( L ) . (14) c (cid:13)000 , 1–13 eneral MC structure using Planck data: I.
100 1 10 M ean den s i t y [ c m - ] α =-1.93, γ =1.07 α =-0.77 ± M ean den s i t y [ c m - ] Effective size [pc] α =-1.25 ± α =-1.29 ± ∪ d2 ∪ d3 Figure 5.
Power-law scaling of mean density: (Bottom) in thediffuse zones outside the PR (small circles) and in their union(large circles); (Top) in the PR (filled circles) and in its diffuserings (open circles). The slope in the former case was derivedexcluding the cloud ‘core’ (square).
In particular, it follows that the flux of kinetic energyper unit volume is also invariant: h ρ i L u L L = inv( L ) , (15)in agreement with simulations of interstellar isothermalturbulence without gravity and with mixed forcing byKritsuk et al. (2007). Mean density in the outer diffuse zones is found to scale withan index about the Larson’s value (Fig. 5, bottom). The con-sideration of their union d1 ∪ d2 ∪ d3 yields the same resultwith a larger dispersion. We attribute the different scalinglaw in comparison with the PR rings to the actual variety ofdistances. All spatial scales and, respectively, mean densitiesin the diffuse zones outside the PR were calculated adopting D = 260 pc (to the Perseus MC) which could be far fromthe real values for a number of lognormal components, inview of the large sizes of the zones and the remoteness ofsome parts of them from the cloud. Table 3.
Summary of the used simulation.Box size 256 pcSimulation code
FLASH
Boundary conditions periodicInitial temperature T = 5 000 KInitial density 1 cm − Turbulence decayingMach number 0.4Magnetic field (aligned along the X axis) 3 µ GRefinement levels 11Maximum resolution 0 .
03 pcJeans length 8 grid cells
For a numerical test of the presented results, we ran a mag-netohydrodynamical simulation of MC evolution includinggravity performed with a
FLASH code (Fryxell et al. 2000).The simulation properties are summarised in Table 3. Thecloud formation was modelled through convergence of twolarge-scale cylindrical streams, each 112 pc long, with a ra-dius of 64 pc. They are given an initial supersonic inflowvelocity (isothermal Mach number of 2) in a warm neutral,initially homogeneous medium and collide at the centre ofthe numerical box. Turbulence is not driven continuouslyand, at late evolutionary stages, is due to fluid motions,driven by gravity. Feedback by sink particles is not includedin the simulation.Similar numerical set-up was chosen in the works ofV´azquez-Semadeni et al. (2007) and Banerjee et al. (2009).The former study showed the establishment of equipartitionbetween gravitational and kinetic energy (cf. our physical as-sumption (ii) in Sect. 4.1) in dense gas domains ( >
50 cm − )and at evolutionary stages t >
10 Myr. The latter work re-vealed that such condensations form mainly in knots of in-tersecting filamentary structures and those located in thecentral region of the flow collision plane host sink particles.Now, if one rotates the flow collision plane from a line-of-sight angle θ = 0 ◦ (face-on view) to θ & ◦ , it shouldbe expected that a cloud similar to Perseus MC will be “ob-served”: an elongated, quasi-axisymmetric dense region withsome filamentary ‘trunks’ in its near vicinity (cf. our Fig. 1with Fig. 1, right in Banerjee et al. 2009). Indeed, this is thepicture on the column-density maps obtained from our sim-ulation at evolutionary stages t >
15 Myr, plotted in Fig. 6,left.
We delineated a rectangular region of the same effective sizelike the PR (Zone N -pdfsin it, varying the line-of-sight angle in the range 70 θ ◦ . Some of the obtained distributions are shown in Fig.6, middle. The PL-tail range as detected by Plfit increasesto about one order of magnitude at t >
18 Myr. The slope | n | gets shallower with evolution but stabilizes gradually atvalues 2 . − . . θ . ◦ , in excellent agreementwith our result for the PR (Fig. 2). Below some critical value θ ≃ ◦ the PL tail becomes very steep and too short, i.e. N PLlow increases discontinuously by a factor 2-3. On the other c (cid:13) , 1–13 Stanchev et al.
Figure 6.
Column-density pdfs (middle) and scaling laws of mean density (right) at three evolutionary stages of MC evolution (left).All column-density maps are produced for a fixed line-of-sight angle θ = 84 ◦ and the considered rectangular region is indicated withsolid white line. Shaded areas in the middle panel denote the location of the lower boundary N PLlow of the PL-tail. The pdfs for criticalvalues of θ are plotted with dashed and dotted lines (see text). hand, nearly side-on views ( θ & ◦ ) are characterised bymuch longer and shallower PL tails: | n | is close to unity.Considering only the non-PL part of the pdf ( N PLlow ), we decomposed it to lognormal components throughthe procedure described in Sect. 3.2.1. All obtained fits inthe selected region, at three different evolutionary stages, areplotted in the Appendix A, Fig. A4, while the achieved fits’goodness is given in Table A2. We derived the mean-densityscaling law, adopting the definition of h ρ i L from Sect. 4.1.The results for the corresponding late evolutionary stagesare illustrated in Fig. 6, right.Evidently, the mean density scaling index α ≃ − γ ≃ . β ≃ . 5, respectively. Hence, the totalenergy per unit volume turns out to be scale-invariant:12 h ρ i L u L − G h ρ i L M L L = inv( L ) , (16)while the kinetic energy flux per unit volume scales weekly h ρ i L u L /L ∝ L − / . This behaviour is closer to the ex-pected one for compressible turbulent medium with purelysolenoidal forcing (Galtier & Banerjee 2011) but demon- strates that gravity nevertheless affects gas dynamics. Theresult might be affected as well by the presence of magneticfields in our simulation. The use of a numerical simulation allows for a direct studyof the volume-density pdfs which correspond to those ofcolumn-density. An example of such ρ -pdf of the gas phasewith densities > − is shown in Fig. 7, top. The Plfit procedure derives a very steep PL tail ( q ∼ − . 7) spanningabout an order of magnitude and starting close to the distri-bution maximum. However, the statistics at high densities isvery poor: a few dozens out of 175000 pixels have densitiesover 800 cm − , which are typical for dense regions in MCs.High-resolution simulations of self-gravitating turbulent me-dia (e.g. Collins et al. 2011; Kritsuk, Norman & Wagner2011) produce ρ -pdfs whose PL tail starts at least 1 . ρ -pdf has not been resolved in our simulation. One is notable to derive the actual slope but the data hint at a valuenot far from q = − . q ≃ − . c (cid:13) , 1–13 eneral MC structure using Planck data: I. -5 -4 -3 -2 -1 10 100 1000 V o l u m e f r a c t i on Density [cm -3 ] q=-1.5 10 100 1000 1 10 M ean den s i t y [ c m - ] Effective size [pc] α =-1.33 ± Figure 7. Pdf and scaling of volume density in the rectangularregion delineated in Fig. 6, left. Top: Density distribution at t =20 . θ = 84 ◦ ) with the lower limit of the PL tail from the Plfit procedure (arrow) and the presumed actual one (dashed).The presumed slope of the PL tail is shown for reference. Bottom: Scaling law of mean density, averaged over three evolutionarystages. yields a PL-tail slope of the N -pdf n ≃ − α = − N -pdfs (Fig. 6, middle).Ignoring the presumed PL-tail (at densities & 600 cm − ), we applied the procedure of lognormal decom-position (Sect. 3.2.1) to the rest of the ρ -pdf. (The resultsare given in Appendix A.) Thus the notion of spatial scales,introduced in Sect. 3.2.2, was retained as well in the 3Dcase; with the necessary correction of the coefficient in equa-tion 2 to ( a i / P i a i ) / . The peak value ρ i of each lognor-mal component was adopted as mean density of the cor-responding spatial scale. The number of lognormal compo-nents at each of the considered three evolutionary stages16 . t . ∼ − . ∼ − . − . ± . 26 is steeper but – in view of the large data scatter– in general agreement with the result from the simulational N -pdf analysis. Our analysis hints at the existence of different physicalregimes in two spatial domains associated with the PR:gravoturbulent in the Perseus MC and its vicinity, describedby equation 13, and predominantly turbulent in the diffuseneighbourhood, described by equations 14 and 15. Belowwe propose an idea to estimate the effective size L gt of thegravoturbulent domain and of the transition zone betweenthe domains.Let L and L are the scaling variables in the gravotur-bulent and in the predominantly turbulent domain, respec-tively. For simplicity, the transition zone is considered asan extended shell encompassing the gravoturbulent domain,with effective size L tr , i.e. with thickness d tr = q L + L − L gt . The velocity dispersion v L in this diffuse shell obeys theinvariant flux of kinetic energy (equation 15) and the meandensity scales with index α ≃ − . 80 (see Sect. 4.2.2). Onthe other hand, adopting γ ≃ β ≃ M L = M ( L/ u L = const( L ) there.It is safe to assume that the time of consideration dt is very small in comparison to the turbulent crossingtime τ L in the predominantly turbulent domain and, ac-cordingly, the mass increase dM L of the gravoturbulentdomain is negligible (Hennebelle & Falgarone 2012, Sect.2.4.1). A further reasonable approximation is that the massinflow is stationary: dM L /dt = const( t ) > E tot = M L u L / − GM L / L is: dE tot dt = 12 u L dM L dt − G M L L dM L dt + 35 G M L L dLdt . (17)where the first two terms in the right-hand side accountstraightforwardly for mass accretion and the last one – forthe size change associated with the mass increase.In view of the negligible dM L , the main contributor to dE tot /dt is the derivative of the kinetic energy of the pre-dominantly turbulent domain: | dE tot /dt | ≈ v L | dM L /dt | .Now, replacing dL/dt = (1 /M ) dM L /dt in equation 17 andusing | dM L /dt | ≈ | dM L /dt | , one obtains:12 v L = (cid:12)(cid:12)(cid:12) u L − G M L L (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( f − 1) 35 G M (cid:12)(cid:12)(cid:12) , recalling our basic assumption (ii) for equipartition E kin = f | W | at each scale. From the derived mean density scalinglaw in the PR (Fig. 5, blue symbols) one calculates directly M = M L =1 pc = h ρ i L =1 pc . pc ∼ M ⊙ . Thus we arriveat an expression to estimate L tr , assessing the coefficient f in the gravoturbulent domain and the coefficient in thevelocity scaling law v L = v ( L / β in the predominantlyturbulent domain: (cid:18) L tr (cid:19) β = 65 | f − | v G M , β ≃ . 60 (18)Typical values of f and v in star-forming regionsare provided from the extensive recent study of the virialparameter α vir in MCs and embedded MC fragments byKauffmann, Pillai & Goldsmith (2013). Those authors de-fine α vir = g ( p ) 2 E kin / | W | , where g ( p ) is a geometrical fac-tor depending on the density gradient with index p . They c (cid:13) , 1–13 Stanchev et al. E ff e c t i v e s i z e [ p c ] fVirial parameter ‘Core’Zone 1Zone 2 Figure 8. Modelled sizes of the gravoturbulent domain L gt (solid line) and of the transition zone d tr (grey area) as func-tions of f , fixing v = 0 . β = 0 . 60 in the turbu-lent domain. The locations of the ‘Core’, Zone 1 and Zone 2in the limiting cases of f (minimal value - open symbols; maxi-mal value - filled symbols) are calculated according to Fig. 1 inKauffmann, Pillai & Goldsmith (2013); the bars show the exten-sion of the transition zones. See text. found 0 . . α vir . . . v . . to few times10 M ⊙ which correspond (through M ) to spatial scales1 − 30 pc considered in this Paper. For a variety of possibledensity profiles an appropriate mean value of the geometri-cal factor is g ( p ) ≃ . . . f . . 7, depending on the mass of the MC fragment.Assuming that mass in the transition zone still scaleslinearly, like in the gravoturbulent domain, one gets an equa-tion for L gt : M ( d tr ) = M d tr ∝ L α tr h ( L gt + d tr ) − L i , α ≃ − . 80 (19)The obtained modelled values of L gt and d tr as func-tions of f (or of the virial parameter), for a fixed v =0 . M L = M ( L/ N -pdf char-acteristics (Fig. 2, right) and the density scaling law (Fig. 5)in the PR considered as a whole, while these quantities in theset of rings are typical for diffuse, predominantly turbulentregions. It is interesting also to note that the column-densityrange N & cm − in the gravoturbulent domain (see thepdf of Zone The derivation of density scaling laws could be sensitiveto various factors. Here we examine briefly three of them:map resolution, uncertainties due to the distance gradientto the PR and uniqueness of N -pdf decomposition. Our re-sults were obtained by use of the original map resolutionwhich corresponds to ∼ . ψ around the small axis. Such geometry would affect the ef-fective radii R and sizes L i (see eq. 2) by factor cos ψ and,respectively, would shift the zero point of the derived den-sity scaling laws (Fig. 5) but would not affect the scalingindices.Next, one might question the uniqueness of N -pdf de-composition to a small number of lognormals. Is it possibleto fit the observational column-density distribution by sumof another set of lognormals which shows up different phys-ical features? Such arbitrariness of the suggested approachis severely limited through its starting steps: 1) setting thenumber of components equal to the number of prominentlocal peaks of the N -pdf and, 2) choosing such guess val-ues of ( N i , σ i , a i ) which nearly reproduce the location ofeach peak and the N -pdf shape in its vicinity. Finally asimultaneous Levenberg-Marquardt optimisation of all 3 N parameters is performed to obtain the fitting curve. On theother hand, the termination of the fitting procedure dependson estimation of the fit goodness – in principle, the lattercan be improved by adding one or two further componentsto account for single tiny fluctuations of the distribution.However, such additional component(s) would be of smalleffective size (small a i / Σ i a i ) in comparison with the bulkof components and could slightly affect the scaling of meandensity (cf. Fig. 5). The defined statistical weights (equa-tion B1) also exclude essential changes of the mean densityscaling index with adding up small components. It may seem surprising that the derived mean-density scal-ing law in the diffuse rings α ≃ − . 80 differs substantiallyfrom the one in the PR whereas – expressed in terms ofmass-size relationship ( γ ≃ . 2, equation 6), – it resem-bles that of a large sample of Galactic MCs with sizes froma few to several tens pc studied by Roman-Duval et al. c (cid:13) , 1–13 eneral MC structure using Planck data: I. (2010). In fact, such behaviour is to be expected if onedifferentiates internal general structure of individual cloudsfrom consideration of a sample of clouds. Total masses ofMCs correlate well with their sizes following a single powerlaw with 2 ∼ γ . . γ within a cloudincreases reversely to the scale L . The latter phenomenonwas discovered by Lombardi, Alves & Lada (2010, see Fig.2 there) from a dust extinction study of 11 Galactic MCs andexplained within a statistical model of general MC structure(Donkov, Veltchev & Klessen 2011), assuming equipartitionof energies at each scale, characterized by its own pdf. Themean-density scaling law in the PR derived in this Paperconfirms those results extending their applicability to thevicinity of the cloud itself (Fig. 5; L > 10 pc) while thevalue of γ in the dense cloud regions (within the ‘Core’)might increase. A further investigation by use of Planck data, including other MCs with different general structureand various physical conditions (e.g. star-forming activity),would shed light on this important issue.On the other hand, the diffuse rings evidently obeya mass scaling law like that for single MCs. The power-law index γ ≃ . not to con-nected regions with well-defined boundaries for which sur-face energy terms are comparable to the volume ones. Never-theless, it is noteworthy that the mean-density scaling law ofabstractly defined scales – both from observational data onthe PR and from the numerical simulations (Sect. 4.3) – areclose, within uncertainties, to α ≃ − . α ≃ − h ρ i L u L /L (equation8). The latter quantity is scale-independent in the diffuserings of the PR, scales as L − / in the diffuse componentsof the simulational pdfs and scales as strongly as L − in theelliptical zones of the PR. That points to the increasing roleof gravity in the gas dynamics. Possibly, the assumed threephysical regimes hint at different self-similar gas structure. The density scaling laws in the PR are derived only by useof the procedure for N -pdf decomposition, without any ref-erence to the velocity scaling law. The relation between thetwo scaling laws relies essentially on our second physicalassumptions for equipartition between gravitational and ki-netic energy (Sect. 4.1). Introducing the latter, we aimedto explore the physics which governs the column-densitystatistics in star-forming regions and not to derive velocity-size relationships out of column density data. Nevertheless,a comparison of the (indirectly) estimated scaling index β with observational and numerical studies might be instruc-tive.The linewidth-size relations in the Perseus MC, ob-tained by Shetty et al. (2012) from CO COMPLETE sur-vey data, are plotted in Fig. 9. The considered objects arehierarchical clumps, extracted by use of the Dendrogram algorithm (Rosolowsky et al. 2008). Therefore they are veryappropriate for comparison to our embedded elliptical zones.Unfortunately, their size range overlaps with our study onlywithin Zone L & h β i is in a rough agreement with the estimation from resultsin this Paper for the ‘Core’ and Zone . . f . . d tr of thetransition zone between gravoturbulent and predominantlyturbulent regimes depends mainly on the balance between E kin and | W | in the former regime and of coefficient v ofthe velocity scaling law in the latter one. The typical valuesof d tr are about 10 % of size of the gravoturbulent domain,irrespective of the degree of gravitational boundedness. c (cid:13) , 1–13 Stanchev et al. 1 10 L i ne w i d t h [ k m / s ] Size [ pc ]525 ’Core’ Zone 1 β ( α )=0.21 (this work) 〈 β 〉 =0.32 (Shetty+ 2012) Figure 9. Velocity-size diagram of large embedded (dendrogram)structures in the Perseus region studied by Shetty et al. (2012),with their unique scaling laws (dotted). The index β in the rangeof overlapping with the present Paper is estimated through thedensity scaling index α and shown for comparison. Finally, we briefly consider the possible effect of magneticfields which were not taken into account in the adopted phys-ical frame (Sect. 4.1). Detailed comparison between cloudevolution from simulations in non-magnetized and in moder-ately magnetized medium shows that the magnetic supportdelays the gravitational collapse whereas the correspondingstages of the N -pdf evolution are identical, differing only bya small time factor (Hennebelle et al. 2008). About the timewhen gravitational collapse of the whole cloud is to start(i.e. when equipartition between gravity and turbulence isachieved), the shape of the column-density pdf in both casesis a combination of lognormal function plus power-law tailat N & few times 10 cm − (see Fig. 1 in Hennebelle et al.2008). Therefore one could not expect that the presence ofmagnetic fields would affect generally the presented physicalanalysis, given than the exact evolutionary stage of the PRis not known. In this Paper, we presented an analysis of the column-density pdfs in the star-forming region Perseus and its dif-fuse environment using the dust opacity map at 353 GHzavailable from the Planck archive. Due to its high galac-tic latitude and approximately axisymmetric form, free ofcontaminating fore- or background cloudlets, the region isan appropriate test case for studying the general structureof molecular clouds and the physical processes that governtheir evolution. On the other hand, the Planck data ondust emission offer the opportunity to probe regions of veryhigh extinction, not resolved on dust extinction maps.Our main conclusions are as follows:1. The pdf shape can be fitted: i) by a combination of alognormal function and an extended power-law tail at highdensities, in zones centred at the molecular cloud Perseus;and, ii) by a linear combination of several lognormals, inrings surrounding the cloud or in zones of its diffuse neigh-bourhood. In the first case, the power-law tail of the pdfin the largest zone includes the pdf density ranges of allothers (embedded) zones whose sizes may be considered as subscales of this zone. In the second case, each lognormalcomponent of a given pdf is interpreted as a contributionof a separate spatial scale in the turbulent cascade. In thatway, the notion of spatial scales L is introduced which al-lows for derivation of power-law scaling laws of mean density h ρ i L ∝ L α and mass M L ∝ L γ in the studied regions.2. The derived scaling laws in zones centred on thePerseus MC and those in zones of its diffuse vicinity or neigh-bourhood differ substantially: α = − . 93 ( γ = 1 . 07) in theformer case and α = − . ± . 11 ( γ = 2 . 23) in the lattercase. Assuming an equipartition relation between gravita-tional and kinetic (turbulent) energy at each spatial scaleand a power-law scaling of velocity dispersion u L ∝ L β , thisresult bears evidence of two distinct physical regimes in thePerseus region:- gravoturbulent, characterized by nearly linear scalingof mass and practical lack of velocity scaling.- predominantly turbulent, characterized by a steep ve-locity scaling ( β ≃ . 60) and by invariant for compressibleturbulence h ρ i L u L /L , describing a scale-independent fluxof the kinetic energy through turbulent cascade.General identification of these regimes with spatial domainsin the Perseus region is physically sustained in point 5 bel-low.3. The mean-density scaling law α ≃ − 1, derived bothfrom column- and (within the a large data scatter) volume-density pdfs from the performed magnetohydrodynamicalsimulations with gravity, is in excellent agreement with theclassical study of Larson (1981). This is striking since theresult was obtained for abstractly defined spatial scales –not for connected regions in MCs, delineated by use of someclump-finding technique.4. The obtained three different physical invariants (resp.,scalings of h ρ i L u L /L ) in the PR, its diffuse rings and in thesimulated cloud region seem to point to three different typesof self-similar structure.5. Modelling the gravoturbulent spatial domain as cen-trally symmetric, with effective size L gt , and the predomi-nantly turbulent domain as its extended shell, we estimate L gt and the thickness of the transition zone d tr . For rea-sonable values of the virial parameter, one obtains L gt ∼ − 11 pc and d tr ∼ . − . 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Decomposition of the N -pdf (open symbols) in thediffuse zones to lognormals (dotted). The fitting function is plot-ted with solid line. APPENDIX A: DECOMPOSITION OF THEPDF: LOGNORMAL COMPONENTS AND FITGOODNESS Below we show the pdf decomposition to lognormal compo-nents in the diffuse zones (Fig. A1) and in the rings out-lined by the boundaries of embedded elliptical zones in thePerseus region (Fig. A2). For comparison, the pdf decompo-sitions in the simulated region at three different evolutionarystages (cf. Fig. 6) are plotted in Fig. A4. The goodness of theobtained fits of observational and numerical pdfs are givenin Tables A1 and A2, respectively. We studied the possibleeffect of insufficient resolution on the pdf, applying the tech-nique of moving median smoothing on the original Planck map: the value of N in each pixel was replaced by the medianvalue in a square containing the neighbor pixels. A typical re-sult is shown in Fig. A3. As expected, a lower map resolutionwould strengthen local peaks in N (i.e. the densities withhigher probability) – the pdf becomes less smoothed andcould require one or two additional components to achievea satisfactory fit goodness. However, this additional compo-nents do not influence practically the derived density scalinglaw (slope changes of a few hundredths).The decompositions of the volume-density pdf in thesimulated region at three different evolutionary stages areplotted in Fig. A5 and the goodness of the correspondingfits is given in Table A3.Evidently, the goodness of the obtained fits is satisfac-tory – in most cases (except for Ring 3-2 in the PR andat t = 16 . 24 Myr in the simulations), the χ value is con-sistent with the two-sided test at 95% confidence level. Asone can expect, the PL tail in simulational N -pdfs grows interms of column-density range and statistical weight withthe MC evolution (Fig. A4). In contrast, the density-scalinglaw derived from the non-PL part does not change, withslope α ∼ − Table A1. Goodness of the obtained fits to N-pdfs in the ringsof the PR and in the diffuse zones. Notation: N bin – number ofbins, m – total number of lognormal components, DF = degrees offreedom ( N bin − m ), χ , low (up) - lower (upper) critical value ofthe χ -distribution at 95% confidence level. The χ values, whichare consistent with the two-sided test χ , low χ χ , up ,are put in bold.Region N bin m DF χ χ , low χ , up Rings in the PR Diffuse zones in the neighborhood d1 55 5 40 ∪ d2 ∪ d3 60 7 39 Table A2. Goodness of the obtained fits to N-pdfs at three evolu-tionary stages of simulated MC evolution. The line-of-sight angleis denoted by θ . Other notation is the same like in Table A1. θ N bin m DF χ χ , low χ , up t = 16 . 24 Myr82 58 12 22 t = 18 . 59 Myr80 31 8 7 1.55 2.17 14.0781 32 11 11 t = 20 . 19 Myr81 33 10 3 APPENDIX B: DERIVATION OF DENSITYSCALING LAW THROUGH A WEIGHTED FIT What should be an appropriate and physically reliable mea-sure for the statistical weight w i of each scale L i ? Refer-ring the reader to the suggested procedure (Sect. 3.2.1), wepoint out that larger components have major contributionsto the pdf fit while small components lead to minor localimprovements. These contributions are proportional to theparameter a i and must be taken into account as one derivesthe scaling law of h ρ i . Hence, the chosen estimate of thestatistical weight of a scale is: c (cid:13)000 What should be an appropriate and physically reliable mea-sure for the statistical weight w i of each scale L i ? Refer-ring the reader to the suggested procedure (Sect. 3.2.1), wepoint out that larger components have major contributionsto the pdf fit while small components lead to minor localimprovements. These contributions are proportional to theparameter a i and must be taken into account as one derivesthe scaling law of h ρ i . Hence, the chosen estimate of thestatistical weight of a scale is: c (cid:13)000 , 1–13 eneral MC structure using Planck data: I. Table A3. Goodness of the obtained fits to volume density pdfsat three evolutionary stages of simulated MC evolution and line-of-sight angle θ = 84 ◦ . The notation is the same like in TableA1. t [ Myr ] N bin m DF χ χ , low χ , up -4 -3 -2 -1 A r ea f r a c t i on A V Ring 2-1 10 -4 -3 -2 -1 V Ring 3-210 -4 -3 -2 -1 A r ea f r a c t i on Ring 4-3 1 10 -4 -3 -2 -1 N( τ ) [ 10 cm -2 ]Ring 5-410 -4 -3 -2 -1 A r ea f r a c t i on N( τ ) [ 10 cm -2 ]Ring 6-5 Figure A2. Decomposition of the N -pdf (open symbols) to log-normals in rings delineated by the boundaries of two successiveelliptical zones in the PR (cf. Table 1). The locations of the lowerboundaries N PLlow of the PL tail are indicated (dashed). Other no-tation is the same like in Fig. A1. -4 -3 -2 -1 1 1 A r ea f r a c t i on N( τ ) [ 10 cm -2 ]A V Ring 5-4 (original map) 1 10 -4 -3 -2 -1 τ ) [ 10 cm -2 ]A V Ring 5-4 (smoothed map) Figure A3. Illustration of the effect of map smoothing (mimick-ing lower resolution) on the pdf decomposition. See text. -2 -1 A r ea f r a c t i on θ =82 ° t=16.24 Myr10 -2 -1 A r ea f r a c t i on N [ 10 cm -2 ] θ =84 ° -2 -1 θ =85 ° -2 -1 N [ 10 cm -2 ] θ =87 ° -3 -2 A r ea f r a c t i on θ =80 ° t=18.59 Myr10 -3 -2 A r ea f r a c t i on N [ 10 cm -2 ] θ =81 ° -3 -2 θ =82 ° -3 -2 N [ 10 cm -2 ] θ =84 ° -3 -2 -1 A r ea f r a c t i on θ =81 ° t=20.19 Myr10 -3 -2 -1 A r ea f r a c t i on N [ 10 cm -2 ] θ =83 ° -3 -2 -1 θ =84 ° -3 -2 -1 N [ 10 cm -2 ] θ =85 ° Figure A4. Decomposition of the N -pdf (open symbols) to log-normals in the simulated regions at three evolutionary stages ofMC evolution (cf. Table A2). w i ( L i ) ∝ a i / X i a i ∝ σ i / X i σ i . (B1)Examples of fits derived in the diffuse rings of thePerseus region and in the simulated cloud are shown in Fig.B1. Evidently, the adopted weighting yields similar slopesto those calculated from standard NLLS procedure. c (cid:13) , 1–13 Stanchev et al. -3 -2 -1 10 100 1000 A r ea f r a c t i on t=16.24 Myr10 -3 -2 -1 A r ea f r a c t i on t=18.59 Myr10 -3 -2 -1 10 100 1000 A r ea f r a c t i on Density [ cm -3 ] t=20.19 Myr Figure A5. Decomposition of the volume-density pdf (open sym-bols) to lognormals in the simulated regions at three evolutionarystages of MC evolution (cf. Table A3). 200 400 600 800 1000 5 10 15 20 M ean den s i t y [ c m - ] Perseus region (diffuse rings) 101001000 1 10 α =-0.77 ± α =-0.82 ± M ean den s i t y [ c m - ] Effective size [pc] 1 10 101001000Effective size [pc] α =-1.00 ± α =-1.00 ± Figure B1. Correlations between effective size and mean densityin linear (left) and log-log plot (right). Fits derived by standardNLLS procedure (dotted line) and by adopted weighting (solidline) are shown. c (cid:13)000