Issue 
A&A
Volume 642, October 2020



Article Number  A177  
Number of page(s)  18  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/202038849  
Published online  19 October 2020 
The structure and characteristic scales of molecular clouds
^{1}
Laboratoire d’Astrophysique de Bordeaux, Université de Bordeaux, CNRS, B18N, allée Geoffroy SaintHilaire,
33615
Pessac,
France
email: sami.dib@gmail.com
^{2}
I. Physikalisches Institut, Universität zu Köln,
Zülpicher Straße 77,
50937
Köln,
Germany
^{3}
Istituto di Astrofisica e Planetologia Spazialli, INAF,
Via Fosso del Cavaliere 100,
Roma
00133,
Italy
^{4}
Department of Physics, Faculty of Sciences, Golestan University,
Gorgan
4913815739,
Iran
^{5}
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas,
4150762
Porto,
Portugal
^{6}
Institut de Planétologie et d’Astrophysique de Grenoble, Université Grenoble Alpes, CNRS,
Grenoble,
France
^{7}
Department of Astronomy, University of Massachusetts,
Amherst,
MA
01003,
USA
^{8}
Centre for Star and Planet Formation, the Niels Bohr Institute and the Natural History Museum of Denmark, University of Copenhagen,
Øster Voldgade 57
1350,
Denmark
^{9}
Instituto de Radioastronomía Milimétrica,
IRAM Avenida Divina Pastora 7, Local 20,
18012
Granada,
Spain
Received:
5
July
2020
Accepted:
9
September
2020
The structure of molecular clouds holds important clues regarding the physical processes that lead to their formation and subsequent dynamical evolution. While it is well established that turbulence imprints a selfsimilar structure onto the clouds, other processes, such as gravity and stellar feedback, can break their scalefree nature. The break of selfsimilarity can manifest itself in the existence of characteristic scales that stand out from the underlying structure generated by turbulent motions. In this work, we investigate the structure of the CygnusX North and Polaris Flare molecular clouds, which represent two extremes in terms of their star formation activity. We characterize the structure of the clouds using the deltavariance (Δvariance) spectrum. In the Polaris Flare, the structure of the cloud is selfsimilar over more than one order of magnitude in spatial scales. In contrast, the Δvariance spectrum of CygnusX North exhibits an excess and a plateau on physical scales of ≈0.5−1.2 pc. In order to explain the observations for CygnusX North, we use synthetic maps where we overlay populations of discrete structures on top of a fractal Brownian motion (fBm) image. The properties of these structures, such as their major axis sizes, aspect ratios, and column density contrasts with the fBm image, are randomly drawn from parameterized distribution functions. We are able to show that, under plausible assumptions, it is possible to reproduce a Δvariance spectrum that resembles that of the CygnusX North region. We also use a “reverse engineering” approach in which we extract the compact structures in the CygnusX North cloud and reinject them onto an fBm map. Using this approach, the calculated Δvariance spectrum deviates from the observations and is an indication that the range of characteristic scales (≈0.5−1.2 pc) observed in CygnusX North is not only due to the existence of compact sources, but is a signature of the whole population of structures that exist in the cloud, including more extended and elongated structures.
Key words: stars: formation / ISM: clouds / ISM: general / ISM: structure / galaxies: star formation / galaxies: ISM
© S. Dib et al. 2020
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The interstellar medium (ISM), both in the Milky Way and in external galaxies, exhibits a scalefree nature that extends over many physical scales. This is observed both in the diffuse H I gas (e.g., Elmegreen et al. 2001; Dickey et al. 2001; Dib & Burkert 2005; Begum et al. 2006; Dutta et al. 2009, 2013; Zhang et al. 2012; MivilleDeschênes et al. 2016) and in the molecular phase (e.g., Stutzki et al. 1998; Heyer & Brunt 2004; Heyer et al. 2009; Schneider et al. 2011; RomanDuval et al. 2011; Rebolledo et al. 2015; Panopoulou et al. 2017; Traficante et al. 2018; Hirota et al. 2018; Kong et al. 2018; Dib & Henning 2019; Henschaw et al. 2020). This selfsimilarity is also observed in the spatial distribution of young clusters in galactic disks (e.g., Elmegreen et al. 2006; Gouliermis et al. 2017; Grasha et al. 2019).
Turbulence is ubiquitously observed in all phases of the interstellar gas. It is thought to be the main regulator of the ISM structure and dynamics in cold, neutral gas and, hence, is responsible for setting a selfsimilar behavior in this regime (e.g., Elmegreen & Scalo 2004; Dib et al. 2008; Burkhart et al. 2013). This selfsimilarity can be broken on various scales. This can happen when specific physical processes dominate the injection of energy and momentum in the ISM. In galactic disks, various forms of feedback from massive stars (i.e., ionizing radiation, radiation pressure, stellar winds, and supernova explosions) impart significant amounts of energy and momentum onto the ISM on intermediate scales, that is, ≈50−500 pc (e.g., Dib et al. 2006, 2011, 2013, 2017; Ostriker et al. 2010; Dib 2011; Hennebelle & Iffrig 2014; Hony et al. 2015; Padoan et al. 2016; Ntomousi et al. 2017; Seifried et al. 2020). Some of these scales could be detected as characteristic scales in the ISM. Dib et al. (2009) found that the orientations of molecular clouds in the outer Galaxy are correlated on spatial scales that are on the order of the expected sizes of supernova remnants, which are prevalent in those regions of the Galactic disk. On small scales, particularly within molecular clouds, the selfsimilarity can be broken on physical scales where the selfgravity of the gas becomes important and dictates the motions of the gas (e.g., Dib et al. 2008). When dynamically important, and due to their anisotropic nature, magnetic fields can also play a role in breaking the selfsimilar nature of the gas (e.g., Soler 2019). Scales at which there might be a departure from selfsimilarity are the ones associated with the sizes of filaments and fragments within filaments (e.g., André et al. 2010; Arzoumanian et al. 2011; Hacar et al. 2013; Könyves et al. 2015) as well as scales on which filaments interact, with hubs and ridges forming at the intersection of two or more filaments (e.g., Schneider et al. 2012; Samal et al. 2015; Dewangan et al. 2017; TreviñoMorales et al. 2019; Clarke et al. 2019). Stellar feedback can also perturb the selfsimilarity of the gas on small scales. Russeil et al. (2013) find, in a study of the massive starforming region NGC6334, that characteristic scales around 1–10 pc can be caused by the injection of energy due to expanding H II regions.
Using the deltavariance (Δvariance) spectrum (Stutzki et al. 1998), we analyzed the spatial structure of two Galactic molecular clouds that lie at the extreme ends of what can befound in the Galaxy in terms of their star formation activity, namely the CygnusX North region and the Polaris Flare. The CygnusX North molecular cloud complex is an active region of star formation where many subregions of highmass star formation can be found (e.g., Schneider et al. 2006, 2010; Motte et al. 2007; Reipurth & Schneider 2008; Bontemps et al. 2010; Csengeri et al. 2011; Hennemann et al. 2012; Kryukova et al. 2014; Maia et al. 2016). In contrast, the Polaris Flare is essentially a translucent, nonstar forming cloud (e.g., WardThompson et al. 2010; MivilleDeschênes et al. 2010). In Sect. 2, we briefly summarize the Herschel satellite data that are analyzed in this work, and in Sect. 3 we present and discuss the column density distribution functions of both regions. The Δvariance method is discussed in Sect. 4, and its application to the Herschel satellite maps of CygnusX North and Polaris is presented in Sect. 5. In Sect. 6, we interpret our findings with the help of simulated synthetic observations and discuss the shape of the Δvariance in models ofincreasing complexity. We start with models of pure fractal Brownian motion (fBm) images (Sect. 6.1) and continue to models where individual structures are superimposed on an fBm (Sect. 6.2). We finish with models in which an entire population of structures is superimposed on an fBm and which have sets of properties (such as the size of major axis, elongation, and column density contrast) that are described by parameterized probability distribution functions (Sect. 6.3). In Sect. 7, we discuss different caveats and limitations pertaining to the observations and the models, and in Sect. 8 we summarize our results and conclude.
2 The data: Herschel maps of starforming regions
The observations that are analyzed in this work were performed using the Herschel space observatory (Pilbratt et al. 2010). In particular, we made use of data products from the Herschel Gould Belt Survey (HGBS^{1}, André et al. 2010) for the Polaris Flare region and the Herschel imaging survey of OB Young Stellar objects (HOBYS, Motte et al. (2010)) program for the CygnusX North region. The column density maps were determined from a pixeltopixel greybody fit to the red wavelength of PACS (Poglitsch et al. 2010) observations at 160 μm (11.7′′ angular resolution), and the three SPIRE (Griffin et al. 2010) wavelengths are 250, 350, and 500 μm at the resolutions of 18.2′′, 24.9′′, and 36.3′′, respectively.For the SPIRE data reduction, we used the HIPE pipeline (versions 10 to 13), including the destriper task for SPIRE as well as HIPE and scanamorphos (Roussel 2013) for PACS. The SPIRE maps were calibrated for extended emission. All maps have an absolute flux calibration using offset values determined in Bernard et al. (2010). For the spectral energy distribution (SED) fit, the specific dust opacity per unit mass (dust plus gas) is approximated by a power law cm^{2} g^{−1} with β_{dust} = 2 and the dust temperature and column density left as free parameters. The description of how high angular resolution maps were derived is detailed in Palmeirim et al. (2013). The concept is to employ a multiscale decomposition of the flux maps and assume a constant lineofsight temperature. The final map at 18.2′′ angular resolution is constructed from the difference maps of the convolved surface density SPIRE maps (at 500, 350, and 250 μm) and the temperature information from the color temperature derived from the 160 μm∕250 μm ratio.
The molecular hydrogen (H_{2}) column densities were transformed into visual extinction (A_{V}) using the conversion formula N(H_{2})∕A_{V} = 0.94 × 10^{21} cm^{−2} mag^{−1} (Bohlin et al. 1978). The column density maps for the CygnusX North and Polaris Flare clouds are displayed in Fig. 1 (left and righthand panels, respectively). For CygnusX, column density maps were already presented in Hennemann et al. (2012) and Schneider et al. (2016a,b), and for Polaris in Robitaille et al. (2019). These maps have a lower angular resolution of 36.3″ and cover different areas than those presented in the current study. The Polaris Flare cloud is located at a distance of 140 pc (Falgarone et al. 1998), and hence each pixel on the map corresponds to a spatial size of ≈ 0.002 pc. The CygnusX cloud is located at a distance of 1.7 kpc (Schneider et al. 2006), and, in this case, each pixel corresponds to a spatial size of ≈ 0.025 pc. The maps of CygnusX North and Polaris contain 5740 × 5740 pixels and 3538 × 3164 pixels, respectively. The total physical size covered by the maps of CygnusX North and Polaris is ≈ 143.5 pc × 143.5 pc and 7.07 pc × 6.32 pc, respectively. The full width at half maximum (FHWM) of the beam is sampled with six pixels and thus the spatial resolution for the CygnusX North and Polaris maps are ≈0.15 pc and ≈0.012 pc, respectively.
3 Column density distribution functions
Here, we only wish to highlight the differences between the CygnusX North and Polaris regions in terms of their column density distributions before analyzing the spatial structure of the clouds. Figure 2 displays the column density probability distribution function (NPDF) for CygnusX North (lefthand panel) and Polaris (righthand panel). Both NPDFs resemble those shown in Schneider et al. (2016b) for CygnusX and Schneider et al. (2013) for Polaris. The NPDFs for both regions exhibit a lognormal behavior at low A_{V} (2.5 ≤ A_{V} < 12 for CygnusX North and ≤1 for Polaris) with a significant difference in the position of the peak between the two regions. A fit to the lognormal part of the NPDF yields in CygnusX North and in Polaris. The width of the lognormal is and in CygnusX North and Polaris, respectively. At larger column densities (A_{V} ≥ 12 in CygnusX and A_{V} ≥ 1 in Polaris), the NPDF turns into a powerlaw distribution. In CygnusX, there are two distinct power laws: a steep power law with an exponent of − 4.24 ± 0.15 in the A_{V} range of [12, 30] and a shallower power law with an exponent of − 1.85 ± 0.02 in the A_{V} range [30, 300]. In contrast, the NPDF for Polaris exhibits a single powerlaw tail (PLT) starting from A_{V} ≳ 1 with an exponent of − 3.97 ± 0.13. The exponent of the PLT we find for Polaris matches the one found in other studies for the same cloud (e.g., Alves et al. 2007; Schneider et al. 2013). The NPDF parameters we find for the two regions are only slightly different from what was obtained in earlier studies, and this difference is due to the fact that the considered areas of the clouds are different.
A PLT in the NPDF is connected to the existence of a powerlaw distribution in volume density and is commonly attributed to the effects of the selfgravity of the gas in generating dense structures in the cloud (e.g., Klessen 2000; Dib 2005; Dib & Burkert 2005; Kainulainen et al. 2009; Kritsuk et al. 2011; Ward et al. 2014; Girichidis et al. 2014; Schneider et al. 2015; Donkov & Stefanov 2018; Corbelli et al. 2018; Veltchev et al. 2019). Another interpretation for the origin of the first, steep PLT has been proposed by Auddy et al. (2018, 2019). These authors showed, using numerical simulations of molecular clouds with nonideal magnetohydrodynamics (MHD), that in the case of a magnetically subcritical cloud, a steep PLT (slope ≈ –4) can emerge as a result of gravitational contraction driven by ambipolar diffusion. The second, shallower PLT is only associated with regions of the highest column densities in CygnusX. This was reported for the first time in highmass starforming regions (Tremblin et al. 2014; Schneider et al. 2016b) and interpreted as arising from gravitational collapse of cores with either internal sources (protostars, ultracompact H II regions) that lead to internal ionization compressions, or external compression from the associated H II region. However, the picture is probably more complicated since a second shallower PLT was also detected in lowmass starforming regions (Schneider et al., in prep.). It is not within the scope of this paper to discuss the NPDFs in extensive detail. In summary, and despite the fact that gravity is suspected to be the primary culprit of the formation of both PLTs, we conclude that it currently is not straightfoward to explain the different parts of the PLT as the consequence of a hierarchical gravitational collapse, whereby the first steep PLT can be attributed to the formation of compact structures (filaments or clumps) and the second, shallower PLT to the collapse of dense cores.
Fig. 1 Column density maps of the CygnusX North cloud (left) and the Polaris Flare cloud (right). Column densities are displayed in units of the visual extinction using the conversion cm^{−2}. 

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Fig. 2 Column density distribution function in the CygnusX North cloud (left panel) and the Polaris Flare cloud (right panel). The full red lines in both panels show a fit by a lognormal function in the low column density regime (≲ 5 × 10^{21} cm^{−2} in CygnusX North and ≲1 × 10^{21} cm^{2} in Polaris). The dashed line is a fit to the powerlaw regime that is observed in both regions in the intermediate column density regime, while the tripledot dashed red line in the case of the CygnusX North region is a fit to the shallower power law in the high column density regime. 

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4 Analysis: the Δvariance method
We quantified the structure of molecular clouds using the Δvariance method. The method was originally introduced in Stutzki et al. (1998) and Zielinsky & Stützki (1999) and is a generalization of the Allan variance (Allan 1966). In this work, we used an improved version of the method presented in Ossenkopf et al. (2008a)^{2}.
Here, we briefly present a summary of the main steps and characteristics of the method. For a 2D field A(x, y), the Δvariance on a scale L is defined as being the variance of the convolution of A with a filter function ⊙_{L} such that (1)
For the filter function, Ossenkopf et al. (2008a) recommend the use of a “Mexican hat” that is defined as (2)
where the two terms on the right side of Eq. (2) represent the core and the annulus of the Mexican hat function, respectively, and v is the ratio of their diameters (we used v = 1.5). For a faster and more efficient computation of Eq. (1), Ossenkopf et al. (2008a) performed the calculation as a multiplication in Fourier space, and thus, the Δvariance is given by (3)
where P is the power spectrum of A and is the Fourier transform of the filter function. If P can be described bya power law, and if β is the exponent of the power spectrum, then a relation exists between the exponent of the power law that describes the Δvariance (α) and β (Stutzki et al. 1998), and this is given by (4)
The value of α can be inferred from the range of spatial scales over which the Δvariance displays a selfsimilar behavior and can be tied to the value of β. The error bars of the Δvariance on a given scale are computed from the counting error determined by the finite number of statistically independent measurements in the filtered map and the variance of the variances (i.e., the fourth moment of the filtered map). Characteristic scales are scales at which there is a break of the selfsimilarity and which show up in the Δvariance spectra as break points, peaks, or inflection points. Any underlying selfsimilar behavior of the cloud can be entirely perturbed on many or all physical scales if there is a variety of structures that coexist in the cloud. The Δvariance has been employed to analyze the structure of observed molecular clouds (e.g., Bensch et al. 2001; Campeggio et al. 2004; Sun et al. 2006; Ossenkopf et al. 2008b; Rowles & Froebrich 2011; Schneider et al. 2011; Russeil et al. 2013; Elia et al. 2014) as well as simulated molecular clouds (e.g., Ossenkopf et al. 2001; Mac Low & Ossenkopf 2000; Ossenkopf & Mac Low 2002; Federrath et al. 2009; Bertram et al. 2015). In most cases, the Δvariance has been used to investigate the selfsimilar nature of the clouds and examine whether the slope of the Δvariance in the selfsimilar regime varies from cloud to cloud and, in the case of simulations, whether it depends on the properties of the turbulent motions that are generated in the clouds. However, it has already been demonstrated that the method is capable of detecting break points. Ossenkopf & Mac Low 2002 found, when applying the method to numerical models of molecular clouds where turbulence is driven on various physical scales, that the Δvariance departs from the selfsimilar regime on physical scales where turbulence is injected into the clouds. Using extinction maps, Schneider et al. (2011) found that lowmass starforming clouds have a doublepeak structure in the Δvariance with characteristic size scales around ≈1 and ≈4 pc. They propose that the physical process governing structure formation could be the scale at which either a largescale supernova shock or an expanding H II region sweeping through the diffuse medium are broken at dense clouds, which turns the wellordered velocity into turbulence.
5 Spatialstructure of CygnusX North and Polaris
We applied the Δvariance method to the column density maps of CygnusX North and Polaris. As stated above, these two regions were selected because they are significantly different, both in terms of their column density distribution (i.e., Fig. 2) and their star formation activity. While Polaris harbors a population of starless cores, it is still a region with no ongoing star formation and a modest contrast in column density. On the other hand, the CygnusX North cloud is a region with a much higher star formation rate and a much larger contrast in column densities (see Fig. 2, also Hennemann et al. 2012; Schneider et al. 2016b). The Δvariance spectra for both clouds are displayed in Fig. 3. The Δvariance spectrum of Polaris displays a selfsimilar behavior above the resolution limit, and this selfsimilarity extends for more than one order of magnitude in spatial scales (from ≈0.03 to ≈ 0.6 pc). A powerlaw fit to the Δvariance of Polaris in the range [0.035–0.6] pc yields a value of the powerlaw exponent of α = 0.4 ± 0.003, and this implies a value of β = 2.4. On scales larger than 0.6 pc, the selfsimilarity is perturbed, possibly due to the existence of a large filamentary structure (i.e., the MCLD 123.5+24.9 structure), though substructured, in the region. The scalefree nature of the Δvariance spectrum of Polaris is consistent with earlier findings using the Δvariance technique for the same cloud (Bensch et al. 2001; OssenkopfOkada & Stepanov 2019). However, Bensch et al. (2001) found larger values of β (≈ 3 from observations in the ^{12}CO (J = 2−1) line and β ≈3.2 from observations in the ^{13}CO (J = 1−0) line) when the Δvariance spectrum is fitted over a spatial range that is roughly similar to the one used in this study. The spatial resolution of the observations they used are 2.2′ and 0.78′, respectively, and are lower than the resolution of the observations presented in this work (≈ 0.3′). Ossenkopf 2002 showed that the use of lowJ CO isotopologues leads to somewhat steeper Δvariance spectra than the one corresponding to the underlying column density structure. The exact relative effect of the lower spatial resolution, which effectively smoothes the map and possibly increases the values of β, compared to the role of the optical depths of these molecular tracers in steepening the Δvariance spectrum, is not yet entirely clear.
In contrast to Polaris, the Δvariance of CygnusX North displays a more complex shape with a steep slope above the spatial resolution limit (dashed black lines in Fig. 3) and a broad peak at around ≈ 0.5−0.12 pc. A reasonable assumption to make is that the existence of many smallscale dense structures (e.g., cores, clumps, and filaments) in CygnusX North alters the underlying (i.e., primordial) selfsimilar structure of the gas that had existed before these structures formed. However, it remains an open question whether a massive starforming region such as CygnusX North had, at an earlier stage, a spatial distribution of (column) density similar to that of Polaris. This is a plausible assumption given that, prior to the formation of massive stars in the region, turbulence in the CygnusX North cloud, like in Polaris and elsewhere in the ISM, must have been dominated by shearing motions. Largescale converging flows may be responsible for aggregating gas in specific regions that would be the parental structures of ridges and hubs. Compressive motions due to feedback from massive stars in specific regions of CygnusX North can also modify the spatial distribution of the (column) density field. However, massive star formation is localized in CygnusX North and not distributed across the entire cloud (Beerer et al. 2010). We speculate here that the underlying, “primordial,” structure in the CygnusX North cloud resembled that of Polaris and use this as a working hypothesis. In what follows, we focus our attention on the CygnusX North cloud and adopt the Polaris value of β =2.4 as the exponent of the underlying selfsimilar fBm structure in CygnusX. We explore, using synthetic data, if and how the addition of dense structures with specific properties on top of a cloud with a selfsimilar structure modifies the Δvariance spectrum.
Fig. 3 Deltavariance functions calculated for the CygnusX North cloud (left) and the Polaris Flare cloud (right). The vertical dashed black lines in both panels mark the position of the spatial resolution for each of these two regions. The vertical dashed red lines in the case of Polaris mark the spatial range over which the powerlaw fit is performed. We do not attemptany fit in CygnusX North because the underlying selfsimilar regime is heavily perturbed by the presence of structures (see Sect. 6.3). 

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6 Interpretation
While one of our aims is to understand the structure of the CygnusX North region as revealed by its Δvariance spectrum, a broader goal is to investigate how the existence of compact and dense structures (cores, clumps, and filaments) with diverse characteristics can alter the selfsimilar nature of a molecular cloud and modify the Δvariance spectrum. We first summarize some of the basic properties of fBm images that are known to possess a selfsimilar structure. In a second step, we include discrete structures with specific characteristics on top of an existing fBm and investigate how the inclusion of these structures impacts the shape of the Δvariance spectrum. Lastly, we investigate how the shape of the Δvariance is modified in the presence of an entire population of structures that are characterized by distribution functions of their sizes, elongations, and column density contrasts.
6.1 Fractal Brownian motion maps
Fractal Brownian motion images (Peitgen & Saupe 1988) are often used as a surrogate of ISM maps thanks to their visual similarity with cloud features (e.g., Stutzki et al. 1998; Bensch et al. 2001; MivilleDeschênes et al. 2003; Elia et al. 2014, 2018). A full description of their analytic properties is presented in Stutzki et al. (1998). Here, we simply review their basic properties. Firstly, their radially averaged power spectrum exhibits a powerlaw behavior with an exponent β =E + 2H, where E is the Euclidian dimension (E = 2 for 2D images) and H is the Hurst exponent whose value ranges from 0 to 1. For 2D maps, β can take values between 2 and 4. Secondly, the distribution of the phases of their Fourier transform is completely random. Thus, it is possible to generate fBm maps by defining the value of β and a random phase distribution. If expressed in terms of the fractal dimension, the fractal dimension of an fBm image has been shown to be given by D = E + 1 − H, and this leads to a direct relation between D and β that is given by: (5)
Stutzki et al. (1998) showed that the power spectrum of the (E − 1) projection of an Edimensional fBm is also a power law with the same spectral index (i.e., the same β). Using this property, it is possible to establish the link between the 2D (E = 2) and 3D (E = 3) fractal dimensions. This will be given by: (6)
In this paper, we used fBm images as a reference for selfsimilar structures since they can be obtained with preconditioned statistical properties (e.g., Stutzki et al. 1998; Shadmehri & Elmegreen 2011; Elia et al. 2018). Figure 4 displays three fBm images generated with a value of β = 2.5 and for three resolutions: 250 × 250 pixels (top left), 500 × 500 pixels (top right), and 1000 × 1000 pixels (bottom left)^{3}. The bottomright panel in Fig. 4 displays the Δvariance functions calculated for these three fBm images. The selfsimilar regime is observed in all cases and extends to larger spatial scales for cases with a higher spatial resolution. The fBm images are periodic, and, if we were using a periodic analysis, the Δvariance spectra would be perfect power laws. However, in order to compare them to the observational data, we performed the calculations of the Δvariance with a cut at the map boundaries. This cut has two effects. First, there is a natural limit to the size of any structure so that the Δvariance spectrum flattens at the largest scales. Second, the statistical significance of the structures close to the map boundaries is reduced. This changes the denominator in the normalization of the Δvariance when computing mean properties of the map according to the areatoboundary ratio of maps of different sizes so that the absolute scale of the Δvariance is only comparable in the limit of very large maps. Figure 5 displays the same type of fBm images, but in this case all images have a fixed resolution of1000 × 1000 pixels and the value of β is varied between 2 and 4 (in steps of 0.5). While in Fig. 4 the phase distribution varies from image to image, in Fig 5 the same distribution of phases is kept, so that the basic “shape” of the image remains the same. Increasing the value of β produces a gradual smoothing of the image due to the transfer of power from high to low spatial frequencies. The bottomright panel in Fig. 5 displays the corresponding Δvariance functions for each of these cases. As expected, the Δvariance functions are scalefree powerlaw functions whose exponent is given by α =β − 2.
Fig. 4 fBm images with β = 2.5 and with resolutions of 250 × 250 pixels (top left), 500 × 500 pixels (top right), and 1000 × 1000 pixels (bottom left). The Δvariance spectra for all three cases are compared in the bottomright subpanel. All display a selfsimilar regime with an exponent of the power law of α = β − 2 (i.e., Eq. (4)). All maps are normalized by their own mean value, and the vertical offset between the three Δvariance functions simply reflects the effect of this different normalization. 

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Fig. 5 fBm images with values of β ranging from2 to 4 in steps of 0.5. All maps have a resolution of 1000 × 1000 pixels. The Δvariance figures for all cases are compared in the bottomright subpanel. All display a selfsimilar regime with an exponent of the power law of α = β − 2 (i.e., Eq. (4)). All maps are normalized by their own mean value. The random number series generating the phases has been kept the same in all maps. 

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6.2 Fractal Brownian motion maps with additional structure
We now explore the effect of discrete structures on the Δvariance spectrum. The structures we superimpose on top of fBm images are generalized 2D Gaussian functions that are given by: (7)
where N_{G} is the added column density of the 2D Gaussian structure, N_{peak} is its peak value, x_{0} and y_{0} are the coordinates of the peak, and the terms a, b, and c are given by: (8)
In Eq. (8), σ_{1} and σ_{2} are the standard deviations of the 2D Gaussian along the major and minor axes, respectively, and θ is the angle between the Gaussian function major axis and the xaxis, defined in the counterclockwise direction. We generated a very large number of synthetic maps on which we superimposed one or several 2D Gaussian structures over fBm images in a controlled manner. All fBm images have a value of β = 2.4, similar to the value found in the Polaris cloud, and a resolution of 1000 × 1000 pixels. For individual structures, we varied the aspect ratio of the 2D Gaussian, , over a range of 1 to 10. The peak value of the 2D Gaussians is expressed in terms of the mean value of the fBm image, and the column density contrast between the peak of the 2D Gaussian and the mean value of the fBm, δ_{c}, is varied between 1 and 10. We also explored the effect of varying the absolute size of the Gaussian function with respect to the image size, as well as the effect of including multiple Gaussian functions in the fBm images.
Figure 6 displays five realizations of an fBm with superimposed Gaussian structures. The 2D Gaussians all have δ_{c} = 5, σ_{1} = 50 pixels, and an aspect ratio that is varied between 1 and 10. The Δvariance functions of these maps are displayed in the bottomright panel of Fig. 6 and are compared to the Δvariance function of a pure fBm image with β = 2.4. The inclusion of an additional structure in the fBm image increases the value of on all spatial scales. The increment of the Δvariance function with respect to the Δvariance of the pure fBm reaches a maximum on a scale that is on the order of the equivalent diameter of the injected structure. Figure 6 shows that as the aspect ratio is reduced, the position of the point where the deviation from the fBm is maximized in the Δvariance function moves to larger spatial scales. If we approximate the surface of the 2D Gaussian by the area that lies within [2σ_{1}, 2σ_{2}] (i.e., where most of the signal lies), the equivalent diameter is then given by . The measured positions of the points of maximumdeviation in the Δvariance functions in Fig. 6 do indeed confirm that the position of maximum deviation is well approximated by D_{eq}. A deviation from this value (by up to ≈30%) can be observed for smaller structures (i.e., in this case, the most elongated) as they are less well resolved on the grid.
We explored the effect of varying the contrast between the injected 2D Gaussian structure and the underlying fBm image. Figure 7 displays five realizations where the value of is varied between 1 and 5. For all cases, the other parameters are fixed to f = 5 and σ_{1} = 50 pixels. All images have a resolution of 1000 × 1000 pixels, and the underlying fBm has an exponent of β = 2.4. The lowerright panel in Fig. 7 displays the corresponding Δvariance functions, which, here once again, are compared to the Δvariance of the fBm image. Figure 7 shows that higher contrasts (δ_{c}) between the selfsimilar fBm and the injected structure lead to higher values of the on spatial scales equal to D_{eq}.
In Fig. 8, we investigate the effect of the surface area by increasing the number of structures that are superimposed onto the underlying fBm image. In this figure, one or several similar structures (all with δ_{c} = 3, f = 5, and σ_{1} = 50 pixels) are superimposed onto the fBm images (all with 1000 × 1000 pixels resolution and β = 2.4). The lowerright panel in Fig. 8 shows that increasing the surface area of the injected structures has a significant impact (i.e., linear with the number) on the increase of the on spatial scales that are on the order of the size of the structures. We also note an increase in the width of the excess of the Δvariance spectrum up to scales of 250 pixels (i.e., larger than the sizes of the individual structures themselves) when there are several structures. This is the direct signature of the 250 pixel separation between the structures. We also explore the effect of changing the size of the structure with respect to the image size while fixing the aspect ratio and column density contrast. Figure 9 displays five realizations with f = 5, δ_{c} = 3 but where σ_{1} is varied between 150 and 16.67 pixels. The Δvariance functions for these realizations are displayed in the lowerright panel of Fig. 9. Here as well, the increment in with respect to the underlying fBm is maximized on scales that are on the order of the equivalent diameter of the structure, .
Given the expression of the Δvariance in Eq. (3), one thus expects that the amplitude of the maximum deviation of in the presence of structures from the of an fBm (defined hereafter as ) and which occurs on spatial scales that are equal to the equivalent diameter of the structures, to scale with , where A is the total area covered by the structures. We verify whether this scaling holds for all cases displayed in Figs. 6–9. We calculate the area as being A = N_{s}π2σ_{1}σ_{2}, where N_{s} is the number of structures present on the map. Figure 10 displays the value of as a function of . A linear scaling between these quantities is found, even though we observe a small deviation from linearity for smaller values of . This is due to the fact that when structures are small, there are larger uncertainties associated with the determination of their surface.
Fig. 6 2D Gaussian structures injected on top of an fBm image with β = 2.4. The 2D Gaussian functions have an aspect ratio (f = σ_{1}∕σ_{2}) that is varied in the range [1–10], and all have a value of δ_{c} = 5 and a fixed size of σ_{1} = 50 pixels. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function of the underlying fBm image. 

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Fig. 7 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions have an aspect ratio f = 5, a value of σ_{1} = 50 pixels, and a column density contrast between the peak of the 2D Gaussian and the mean value of the fBm, δ_{c}, that is varied between 1 and 5. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

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Fig. 8 One or several similar 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions have an aspect ratio f = 5, a value σ_{1} = 50 pixels, and δ_{c} = 3. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

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6.3 Toward more realistic configurations
In principle, the generation of realistic column density maps could rely on numerical simulations of turbulent and selfgravitating molecular clouds. However, the parameter space can be very large, namely, models with or without gravity, with various magnetic field strengths, and with various driving schemes, Mach numbers, and turbulence driving scales. When gravity is included, the extracted information will also unavoidably depend on the time evolution of the simulated clouds. This remains a valuable approach that has in fact been explored to a certain extent (e.g., Ossenkopf et al. 2001; Ossenkopf 2002) and deserves to be explored further with more refined models. In this work, we prefer to generate models whose parameters can be easily controlled and for which we can easily understand and disentangle the effects on the Δvariance function. Asin Sect. 6.2, we superimposed 2D Gaussians on top of predefined fBm images. However, instead of including individual structures or structures that are set apart from each other, we now include 2D Gaussians with specific distribution functions that characterize their properties. The parameters we varied are the number of 2D Gaussian structures, N_{s}, the distribution function of the size of the major axis , the distribution function of the aspect ratios , and the distribution function of the structures column density contrast . Each structure is assigned a randomly drawn orientation on the map, and the structures are allowed to overlap. Keeping in mind that clumps, cores, and filaments such as those found in the column density map of CygnusX North may have a more complex internal structure than 2D Gaussian functions, we aim to understand which combination of the parameters leads to Δvariance functions that are similar to that of the CygnusX North region. More broadly, our aim is also to understand the sensitivity of the Δvariance to the choice of the distribution functions that characterize the statistical properties of the structures.
The distribution of sizes and aspect ratios of cores and clumps in molecular clouds is likely to depend on the density tracer as well as on the clump identification algorithm. To illustrate this, in Fig. 11 we compare the size (i.e., major axis; L_{1}, left panel) and aspect ratio (f, middle panel) distributions of structures found in the Herschel infrared Galactic Plane survey (HiGAL; Molinari et al. 2016; Elia et al. 2017) and in the Five College Radio Astronomy Observatory (FCRAO) CO survey of the outer Galaxy (HCS; Heyer et al. 1998; Dib et al. 2009). Structures in the HiGAL survey are extracted from 250 μm emission maps (78 952 objects in total), whereas the HCS survey is based on the (1–0) transition in ^{12}CO molecular line observations, in which 10 156 discrete structures were identified. The clouds and clumps reported in the HiGAL survey are ostensibly more roundish than the ones detected in molecular line observations^{4}. The distribution functions in Fig. 11 are normalized and are thus transformed into probability distribution functions.
For the aspect ratio distributions of the HiGAL and FCRAO HCS clouds and clumps, (Fig. 11, middle panel), we find that these distributions are best approximated by the following function: (9)
Fitting the distributions of aspect ratios for the HiGAL (for f > 1) and HCS (for f > 2.5) clouds yields [η = −1.15 ± 0.04, A_{f} = 0.75 ± 0.13] and [η = −0.37 ± 0.01, A_{f} = 0.41 ± 0.10], respectively. The results of the corresponding fits are shown in Fig. 11 (dashed red lines, middle panel). For any other chosen value of η, the corresponding value of A_{f} can be calculated by requiring that , and where f_{min} and f_{max} are the lower and upper limits on f, respectively.In the same vain, we fitted the normalized distribution function of the size of the major axis, . Here again, we find that the data is best fitted with a function that is given by: (10)
Using Eq. (10) the fit to the data of the HiGAL clouds for values of L_{1} in the range 1.5 pc ≥ L_{1} ≥ 5 pc and for the HCS clouds using values of L_{1} in the range 4 pc ≥ L_{1} ≥ 70 pc yields values of the parameters ξ and of [] and [], respectively. The fit functions are displayed with the dashed red lines in Fig. 11 (lefthand panel). For any other value of ξ, the corresponding values of can be obtained by requiring that , where L_{1,min} and L_{1,max} are the lower and upper limits on L_{1}.
The distribution of column density contrasts of dense structures in nearby molecular clouds is not yet fully established. Recent work by Arzoumanian et al. (2019) derived the contrast between the average column density on filament crests and their local background for filaments detected in a number of nearby molecular clouds. Roy et al. (2019) constructed the distribution function of the contrast between filaments and their local background and found that it scales as for δ_{c} > 1. The exact scaling found in Arzoumanian et al. (2019) and Roy et al. (2019) may not apply directly to our synthetic models since we define the contrast as being the one between the peak column density of the structure and the mean value over the entire map. With this in mind, we parameterized the distribution of column density contrasts as being a power law of the form: (11)
where A_{c} is a normalization coefficient that is given by , and δ_{c,min} and δ_{c,max} are the lower and upper limits on δ_{c}. We took δ_{c,min} = 1 in all cases and varied δ_{c,max} between 3 and 10. This is consistent with the range of values found by Arzoumanian et al. (2019) for filaments and with the range of column densities that are present in the high density tail of the NPDF of the CygnusX North cloud. We chose three values of ψ, of 2, 2.5, and 3, which, as an extrapolation of the results presented in Arzoumanian et al. (2019), should cover both variations due to differences in the clouds environmental conditions and variations due to temporal evolution (i.e., selfgravitating structures will have, statistically, higher column density contrasts at time goes by).
As stated above, we would like to understand the sensitivity of the Δvariance spectrum inrelation to the underlying distribution functions of the different parameters, and, as a byproduct, understand which particular set of parameters can help generate a Δvariance spectrum that resembles the one found in CygnusX North. In principle, the parameter space is relatively large with four free parameters to probe (N_{s}, η, ξ, and ψ), and even larger if the lower and upper limits on f, L_{1}, and δ_{c} are also varied. As a first step, we explore below models of synthetic clouds whose properties are inspired from the HiGAL and the HCS samples. In a second step, we expand this “forward modeling” approach and present a broader parameter study where we vary, in a more systematic fashion, the parameters of the distributions functions.
For any given choice of (N_{s}, η, ξ, ψ), and owing to the fact that the orientations and positions of the injected structures are random, it is important that, for each choice of the parameters, a statistically significant number of realizations is performed in order to capture the mean behavior and standard deviation around the mean of the Δvariance spectrum. We chose to perform 25 realizations with any given set of the parameters. Furthermore, and owing to computational limitations, we performed the synthetic calculations on maps with 1000 × 1000 pixels, whereas the map of CygnusX North has 5740 × 5740 pixels. We assigned the same physical size to each pixel in the synthetic maps as in the observations (≈ 0.025 pc), and we matched the mean A_{V} in the synthetic maps to the mean A_{V} of the CygnusX North cloud. Because we are comparing the results from the synthetic maps to the observations of the CygnusX North region, all synthetic maps (i.e., both the reference fBm and the fBm plus structures maps) were convolved with a beam similar to that of the observations. The beam is represented by a Gaussian function whose FWHM = 18.2′′. We compared the Δvariance spectra of the synthetic maps and the observations on scales that are larger than the beam size, namely scales that are ≥ 0.15 pc.
Fig. 9 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions all have an aspect ratio f = 0.2, a column density contrast between the peak of the 2D Gaussian and the mean value of the fBm, δ_{c} = 3, and a value of σ_{1} that is varied between 150 and 16.67 pixels. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

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Fig. 10 Maximum deviation of the Δvariance function in the presence of structures from that of a pure fBm as a function of the quantity , where A is the area covered by the discrete structure(s) and δ_{c} is the column density contrast between the peak of the structure and the mean value of the underlying fBm. The dashed line has a slope of one. 

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Fig. 11 Fractional probability distribution functions of the size of the major axis (left) and aspect ratio of clouds (middle) found in the 250 μm maps of the HiGAL survey (Molinari et al. 2016; Elia et al. 2017; full line) and in the ^{12}CO FCRAO HCS survey (Heyer et al. 2001; Dib et al. 2009; dashed line). The distribution of column density contrast is assumed to be a power law. The distributions of L_{1} and f are fitted with parameterized functions. The values of the parameters of the fit are reported in the main text. 

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6.3.1 HiGAL and HCSlike clumps
As illustrative examples, we first explored the resulting Δvariance spectra corresponding to population of clumps and cores similar to the ones found in the HiGAL (submm) and HCS (CO) surveys. We recall that the values of ξ and η are [ − 0.04, −0.37] and [−0.89,−1.15] for the HiGAL clouds and the CO clouds, respectively, which implies that the CO structures in the HCS surveyare both larger and more elongated than structures detected in the HiGAL survey. The lower and upper limits on the aspect ratios for the submmlike and COlike clouds are taken to be (f_{min} = 1, f_{max} = 4) and (f_{min} = 3, f_{max} = 12), as per the observational constraints (Fig. 11, middle panel). For the lower limits on the major axis size, we extrapolated the major axis size distributions for both the submmlike and COlike clouds down to the resolution limit such that L_{1,min} = 0.025 pc^{5}. For the upper limit on L_{1}, we adopted a common value of L_{1,max} = 5 pc for both the HiGALlike and HCSlike clouds. Adopting a larger value of L_{1,max} (up to ≈70 pc) for the HCSlike clouds would be excessive on the grounds that CygnusX North does not contain any such large structures, and this is further motivated by the fact that we are considering a region that is 5.74 smaller, in each direction, than the real map. In the absence of additional information from the surveys, we imposed, in both cases, a value of ψ =2.5 and lower and upper bounds on δ_{c} of 1 and 3, respectively.
The top panel in Fig. 12 displays three examples of the HiGALlike maps generated with N_{s} = 200 (left), 300 (middle), and 400 structures (right) out of a total of 25 realizations performed for each case. The size of the major axis, aspect ratio,and column density contrast for each individual structure are randomly sampled from the corresponding distribution functions, and the structures are assigned random positions and orientations and overlaid on top of an fBm image with β =2.4. The bottom panel in Fig. 12 displays the corresponding Δvariance spectra, which are calculated, in each case, as the mean spectrum from the 25 realizations (blue triangles). The Δvariance spectra for the synthetic maps are compared to the spectrum of the underlying fBm (open diamonds) and to that of the CygnusX North cloud (filled stars). As observed earlier in the case where individual structures are injected (in Sect. 6.2), the Δvariance spectrum in the presence of structures shows a departure from that of the underlying fBm, and, in the case of an entire population of structures, the point of maximum departure from the underlying fBm case corresponds to the characteristic scale of the ensemble of structures that are injected onto the map. Figure 12 shows that a better agreement is obtained for N_{s} = 300. We followed the same procedure and generated structures similar to those found in the HCS survey. Three examples of such maps with N_{s} = 50, 100, and 200 are displayed in Fig. 13 (top panel). The HCSlike structures have a shallower spectrum of major axis sizes, and the corresponding Δvariance spectrum peaks at higher spatial scales than their HiGALlike counterparts (Fig. 13, bottom panel). Figures 12 and 13 show that while the HiGALlike structures provide a better match to the observations of CygnusX, neither of these two cloud samples fit the observations of CygnusX North well. However, it is useful to compare these two cases to CygnusX in order to highlight how structures with fundamentally different statistical properties impact the Δvariance spectrum. These cases also provide a starting point for a more detailed exploration of the parameter space.
Fig. 12 Top: examples of 200 (left), 300 (middle), and 400 (right) 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian structures are randomly sampled using the distribution functions of the major axis size and aspect ratio distributions of the HiGAL clumps. The column density contrasts are randomly sampled from a distribution with ψ = 2.5. All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum of the synthetic models for cases with N_{s} = 200 (left), N_{s} = 300 (middle), and N_{s} = 400 (right) injected structures. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to those of the CygnusX North region and an fBm with β = 2.4. 

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6.3.2 Parameter study
In this section, we perform a broader parameter study and investigate how the Δvariance spectrum is affected by variations in the distribution functions of the structure sizes, aspect ratios, and column density contrasts. It is nearly impossible to cover the entire parameter space for the four parameters (N_{s}, η, ξ, ψ). We therefore fixed N_{s} = 300, and adopted, in the first instance, a value of ψ = 2.5. We varied the shapes of the major axis and aspect ratio distribution functions and considered values of ξ = [−0.8, −0.6, −0.4, −0.2] and η = [−1.15, −0.75, −0.35]. Furthermore, we fixed here the lower and upper bounds of the structures sizes (i.e., size of the major axis) to L_{1,min} = 0.025 pc and L_{1,max} = 5 pc, respectively. The lower and upper bounds on the aspect ratios are fixed in all cases to the values of f_{min} = 3 and f_{max} = 12, and the lower and upper limits on the column density contrasts are δ_{c,min} = 1 and δ_{c,max} = 3. Examples ofmaps generated with each permutation of these parameters are displayed in Fig. 14. The Δvariance spectra for all of these cases are displayed in Fig. 15. Here again, we performed 25 realizations with each set of parameters and computed the mean value and standard deviation on each spatial scale. What Fig. 15 reveals is that the Δvariance spectrum is more sensitive to the shape of the distribution of the major axis sizes, characterized here by the parameter ξ, than to the distribution of aspect ratios (parameter η). Shallower distributions of the major axis (ξ = −0.2) lead to an over abundance of larger structures on the map and to a noticeable mismatch of the Δvariance for those cases with the Δvariance spectrum of the CygnusX North region, irrespective of the value of η. In contrast, steeper distribution functions of the major axis (e.g., ξ = −0.8) lead to significantly less variance than what is observed in CygnusX North on scales ≳ 0.2 pc. For intermediate values of ξ in the range [−0.6,−0.4], there is a good agreement between the Δvariance spectrum of the synthetic models and the spectrum of CygnusX North. Models with ξ = −0.4 present the best fit, but one can reasonably argue that cases with ξ = −0.6 could still be considered a good fit to the data if higher values of N_{s} were employed. From the grid of models shown in Fig. 15, the case with ξ = −0.4 and η = −0.35 represents the best fit to the observations. This corresponds to a mean value of the size of the major axis of pc and to a mean value of the aspect ratio of . This implies a mean value of the minor axis pc, and, taking σ_{1} = L_{1}∕3 and σ_{2} = L_{2}∕3, this yields a value of the effective size pc, which is very close to the mid position of the plateau in the Δvariance spectrum of the CygnusX North cloud.
Figure 16 shows the individual Δvariance spectra for the 25 individual realizations (i.e., gray lines) when structures are randomly drawn from the distribution functions with the best fitting set of parameters, namely N_{s} = 300, ξ = −0.4, η = −0.35, and ψ = 2.5. We adopted these values of N_{s}, ξ, and η and further investigated the effect of the remaining parameters.
In order to further explore the effect of the aspect ratio, f, we performed additional tests in which we varied its lower and upper bounds. In addition to the fiducial case in which f_{min} = 3 and f_{max} = 12, we considered models with (f_{min} = 1, f_{max} = 6) and (f_{min} = 1, f_{max} = 12). All other parameters were fixed to those of the best fitting model in Fig. 15, namely ξ = −0.4, η = −0.35, ψ =2.5, and (L_{1,min} = 0.025 pc, L_{1,max} = 5 pc), (δ_{c,min} = 1, δ_{c,max} = 3). Figure 17 (top panel) displays examples of the maps for each one of the considered cases. The calculations of the Δvariance spectra in those cases (Fig. 17, bottom panel) show that the existence of more roundish structures (i.e., larger structures for the same major axis size) results in Δvariance spectra thatpeak at higher spatial scales, and those cases present a poor fit to the observations of CygnusX North.
We now explore the effect of varying the distribution function of the column density contrast. In addition to the fiducial case with ψ =2.5, we constructed synthetic maps with ψ = 2 and ψ =3. We also generated additional maps in which the value of ψ is fixed to 2.5 and varied the values of the lower and upper limits on the column density contrast, δ_{c}. We considered cases with (δ_{c,min} = 1, δ_{c,max} = 3; fiducial case shown earlier) and cases with (δ_{c,min} = 1, δ_{c,max} = 5) and (δ_{c,min} = 1, δ_{c,max} = 10). The remaining parameters were fixed to their fiducial values, namely ξ = −0.4, η = −0.35, (L_{1,min} = 0.025 pc, L_{1,max} = 5 pc), and (f_{min} = 3, f_{max} = 12). Figure 18 (top panel) displays selected realizations of maps generated with various values of ψ, and Fig. 19 (top panel) displays examples of maps generated with different values of δ_{c,max} and with a fixed value of ψ = 2.5. The Δvariance spectra for various cases of ψ are shown in Fig. 18 (bottom panel) and are always calculated as being the mean values from 25 realizations. Overall, the Δvariance spectrum is less impacted by variations in ψ, even though values of ψ ≥ 2.5 lead to a better agreement with the observations. On the other hand, allowing for higher values of the maximum column density contrast, δ_{c,max}, has an impact on the amplitude of the deviation from the Δvariance of the underlying fBm, but it has no effect on the position of the point of maximum deviation (Fig. 19, bottom panel), in agreement with our findings in Sect. 6.2. We find that a value of δ_{c,max} = 3 fits the observations better.
In summary, we are able to show that it is possible to reproduce Δvariance spectra that resemble that of the CygnusX North region under reasonable assumptions of the size distributions of structures, their aspect ratios, and column density contrasts. Broadly speaking, reproducing the Δvariance spectrum of the CygnusX North region requires a size distribution that is steeper than the size distribution of structures detected in CO surveys, such as the HCS survey, and shallower than the one inferred from the HiGAL submm survey. We also show that the observations are best fitted when structures are allowed to have aspect ratios that are predominantly ≳ 3.
Fig. 13 Top: examples of 50 (left), 100 (middle), and 200 (right) 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian structures are randomly sampled using the distribution functions of the major axis size and aspect ratio distributions of the HCS clouds. The column density contrasts are randomly sampled from a distribution with ψ = 2.5. All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum of the synthetic models for cases with N_{s} = 50 (left), N_{s} = 100 (middle), and N_{s} = 200 (right) injected structures. Each synthetic Δvariance spectrum is the average over 25 realizations and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to those of the CygnusX North region and an fBm with β = 2.4. 

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Fig. 14 Examples of synthetic maps generated by overlaying structures (2D Gaussians, N_{s} = 300) on top of an fBm image with β = 2.4. The properties of the structures are randomly sampled from distribution functions of the aspect ratio (Eq. (9)), size of the major axis (Eq. (10)), and column density contrast (Eq. (11)). All models shown here share the same values of δ_{c} = 2.5, (L_{1,min} = 0.025 pc, L_{1,max} = 5 pc), (f_{min} = 3, f_{max} = 12), and (δ_{c,min} = 1, δ_{c,max} = 3). All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. 

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6.3.3 Contribution of compact sources to the Δvariance plateau in CygnusX
In this section, we examine the contribution of compact sources to the observed plateau in the Δvariance spectrum of CygnusX North. In the previous section, we generated populations of structures whose properties were sampled from parameterized distribution functions and overlaid these structures on an fBm image. Here, we follow a different approach and extract the compact sources from the map of CygnusX North before reinjecting them onto the fBm image. To this purpose, we used a newly developed clump finding algorithm. The details of the code will be presented in a forthcoming paper (Bontemps et al., in prep.). Here we simply summarize its basic concepts. This code uses secondorder spatial derivatives in order to recognize high curvature peaks where 2D Gaussian fits are applied, after subtracting a local background. It uses an improved determination of the background emission, thanks to a recently developed minimization, to interpolate an empty space (the footprint of a detected source) in a 2D map. Applying this code to the map of CygnusX North, we were able to detect a total of 1242 compact sources. The mean values of σ_{1} and aspect ratio (f) for this sample of compact sources are 0.23 pc and 1.32, respectively. This implies a mean effective size for the compact sources of pc, which corresponds, roughly, to where the Δvariance of Cygnus has its peak. Because the synthetic maps we are using have 1000 × 1000 pixels and are thus ≈ 33 times smaller than the map of CygnusX North, we injected a total of (1242∕33) ≈ 38 compact sources onto each synthetic map. Unlike synthetic maps generated earlier, only the underlying fBm is convolved with the Gaussian beam since the compact sources extracted from the CygnusX North map are already affected by beam smearing.
We generated 50 synthetic maps such that each core is statistically selected at least once and, for each map, the 38 structures that are injected are randomly sampled from the list of structures that are extracted from the observational map and are assigned random positions and orientations on the map. Figure 20 (lefthand panel) displays one of the realizations of the synthetic maps using this approach, and the mean Δvariance spectrum calculated from the 50 realizations is shown in the righthand panel of Fig. 20. The exponent of the fBm in this case is also taken to be β = 2.4. We also generated other models with different values of the fBm exponent, in the range β = [2, 3] (figures not shown for redundancy). The Δvariance spectrum of the models (i.e., Fig. 20) exhibits a peak at ≈ 0.6−0.8 pc, which is at the lower end of the plateau found in the observations. However, there is no agreement between the models and the observations, neither in terms of the width of the Δvariance spectrum nor its amplitude. What Fig. 20 reveals is that the compact sources taken alone, despite having an important contribution to the signal at ≈0.6−0.8 pc, cannot explain the full extent of the plateau that is observed in the Δvariance spectrum of CygnusX North; it further reveals that there is a need to consider a distribution of structures that includes both larger and more elongated objects in order to explain the observations. A broader distribution of sizes is required in order to reproduce the broad Δvariance spectrum in CygnusX North as well as to adjust the amplitude of the spectrum to the observed values since more extended structures can provide the intermediate column densities between the compact sources and the underlying fBm structure.
Fig. 15 Δvariance spectra related to the models presented in Fig. 14. Each synthetic spectrum (blue triangles) is the average over 25 realizations of the maps with the same set of parameters. In all models, N_{s} = 300 and ψ = 2.5. The full line is the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and that of an fBm image with β = 2.4. In the grid of models, the best fit to the data of Cygnus North is for the case with ξ = −0.4 and η = −0.35. 

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Fig. 16 Same as the figure with ξ = −0.4, η = −0.35, ψ = 2.5, and N_{s}= 300 displayed in Fig. 15 but additionally showing the Δvariance spectra of individual realizations with this set of parameters. 

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7 Discussion
The present study relies on the comparison of observations with synthetic maps that are generated using 2D fBm images with superimposed structures. As stated above, the structures are injected at random positions and are not necessarily associated with existing higher column density regions in the fBm images. Additionally, the random position and orientation that are assigned to each structure imply that the structures are not spatially correlated, and this may affect the signal on spatial scales on the order of the structures’ effective separation. Furthermore, fBm images, albeit a good proxy for selfsimilar structures, are known to differ from real molecular clouds in terms of their multifractal nature, or, more specifically, their lack thereof (e.g., see details in Elia et al. 2018). There is, however, no reason to believe that any of these assumptions or simplifications are critical to the analysis. Our results demonstrate that the Δvariance spectrum of a complex region such as CygnusX North can be reproduced reasonably well using realistic distribution functions of the characteristics of these structures (size, contrast, aspect ratio). Nonetheless, it is important to stress that a more physical model is still needed in order to tie the existence of these structures to the physical conditions that prevail in the gas and to the initial conditions of the gas when the molecular cloud has started to assemble. While some refinements can be made to the empirical models presented in this work, it is probably safe to state that numerical models that incorporate most or all of the necessary physics – and that preferably simulate galactic scales larger than the clouds themselves while resolving the internal structure of the clouds – constitute the next step for comparing models to the observations. Complex features, such as striations that are observed in molecular clouds (e.g., Heyer et al. 2016; Tritsis et al. 2018), can naturally emerge selfconsistently in numerical models and are harder to implement in empirical models.
On the observational side, we recall that the Herschel maps presented in this work have been resampled to a higher resolution by a factor of two in order to match the resolution of the 250 μm maps. This was done using the method detailed in Palmeirim et al. (2013). This approach has the advantage of increasing the dynamical range of the maps, but it may have introduced additional signal on small scales; this, in turn, may have contributed to worsening the agreement between the synthetic models and the observations of CygnusX on these scales. While revisiting this correction is well beyond the scope of this paper, this effect is possibly what is causing the Δvariance spectrum in CygnusX North to fall less sharply at smaller spatial scales than what is expected from the effects of beam smearing (i.e., the slope of the Δvariance spectrum in the first bin is shallower than the slope in the second bin of the spectrum). Another possible issue relates to the existence of an underlying selfsimilar regime in CygnusX. The simple experiments presented in Figs. 6–9 when a single (or a few similar) structure(s) is (are) superimposed onto the fBm image show that the selfsimilarity in the Δvariance spectrum is preserved on scales that are either smaller or larger than the effective diameter of the structure(s). In the case of multiple structures with different sizes, contrasts, and elongations, the underlying selfsimilarity is perturbed on a larger range of spatial scales. Thus, the identification of a selfsimilar regime in CygnusX North, if it exists, would in principle require higher resolution observations in order to probe the shape of the Δvariance at smaller spatial scales and/or a larger map, possibly connecting to the H I gas at the outer edges of the cloud in order to probe the shape of the spectrum at larger spatial scales.
8 Conclusions
The internal structure of molecular clouds holds important clues regarding the physical processes that lead to their formation and their subsequent dynamical evolution. While the overall morphology of a molecular cloud can be linked to its star formation activity (Dib & Henning 2019) and thus provide hints about the cloud’s assembly mechanism, the internal structure of the cloud also holds important information about the fragmentation process and the competition between different physical processes that redistribute matter within the cloud. Using the Δvariance spectrum, we have characterized the structure of the CygnusX North and Polaris Flare molecular clouds. These two clouds represent two extremes in terms of their star formation activity in the Milky Way. In Polaris, the structure of the cloud as revealed by the Δvariance is selfsimilar over more than one order of magnitude in spatial scales. In contrast, the Δvariance spectrum of CygnusX North exhibits an excess (compared to Polaris) and a plateau in the range of physical scales of ≈ 0.5−1.2 pc. The departure from selfsimilarity in a region such as CygnusX North is due to the existence of overdense structures, including compact sources (i.e., hubs and ridges), and more elongated clumps and filaments. In such a region, these structures may arise as a result of largescale compressions (i.e., converging flows) before being dominated by their own selfgravity. They are also likely to be affected by the mechanical and radiative feedback from massive stars that form in the cloud.
In order to explain the observations of CygnusX North, we built synthetic maps in which we overlaid a population of discrete structures (i.e., 2D Gaussians) on top of an fBm image. The properties of these structures, such as their major axis sizes, aspect ratios, and column density contrasts, are randomly drawn from parameterized probability distribution functions of these quantities. We show that the inclusion of discrete structures “on top” of a selfsimilar image increases the Δvariance, and this increment has its maximum on spatial scales that are equal to the effective size of the injected structures (or to an effective mean size of the structures if they have a spectrum of sizes and elongations). Using this forward modeling approach, we are able to show that, under very plausible assumptions, it is possible to reproduce a Δvariance spectrum that resembles that of the CygnusX North region. We also used a “reverse engineering” approach in which we extracted the compact structures in the CygnusX North cloud and reinjected them onto an fBm map. The calculated Δvariance spectrum using this approach deviates from the observations and is an indication that the range of characteristic scales (≈ 0.5−1.2 pc) observed in CygnusX North is not only due to the existence of compact sources, but is a signature of the whole population of structures that exist in the cloud, including more extended and more elongated structures such as ridges and hubs. Such structures are required in order to broaden the peak of the Δvariance spectrum and also because they provide the required intermediate column densities that reduce the contrast between the compact sources and the potentially underlying fBm, bringing the amplitude of the Δvariance in line with the observations. At present, it is relatively difficult to ascertain which physical process leads to the formation of structures with scales in the range 0.5−1.2 pc. While gravity is the suspected culprit because its effect can precede from an evolutionary point of view over those of stellar feedback, ananalysis of the pillars and globules in the Cygnus OB2 association has shown that these structures have typical sizes of ≈ 0.6 pc (Schneider et al. 2016a). This indicates that feedback may be responsible, at least partially, for generating the peak in the Δvariance spectrum that is observed in the entire CygnusX North region. Further work should shed more light on the possible correlation between the shape of the Δvariance spectrum and star formation activity, such as the surface density of star formation and the intensity of the radiation field in different parts of the cloud. Independently, the application of the Δvariance method to numerical simulations of selfgravitation clouds with and without feedback effects will also help explain the dominant physical processes that can generate a structure similar to the one observed in the CygnusX North molecular cloud.
Fig. 17 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the lower and upper bounds of the aspect ratios, f_{min} and f_{max}, respectively. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrumof the synthetic models for the three cases with the considered sets of f_{min} and f_{max}. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to that of an fBm image with β = 2.4. 

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Fig. 18 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the exponent ψ. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum spectra of the synthetic models for the three cases with ψ = 2, 2.5, and 3. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and that of an fBm image with β = 2.4. 

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Fig. 19 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the lower and upper bounds of column density contrast δ_{c,min} and δ_{c,max}, respectively. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectra of the synthetic models for the three cases with the considered sets of δ_{c,min} and δ_{c,max}. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to that of an fBm image with β = 2.4. 

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Fig. 20 Left: synthetic column density map generated by reinjecting the compact source extracted from the CygnusX North map on top of an fBm image (convolved with a beam with FWHM = 18.2′′) with β = 2.4. Right: Δvariance spectrum of the synthetic models using this approach. The Δvariance spectrum of the models is the average over 50 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to the case of an fBm image with β = 2.4. 

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Acknowledgements
We thank the referee for a careful reading of the paper and for useful suggestions. We would also like to thank Shantanu Basu, Bruce Elmegreen, Jan Palouš, João Alves, Nanda Kumar, and Alessio Traficante for their feedback and useful discussions on an early version of the paper. S.D., S.B., and N.S. acknowledge support from the french ANR and the german DFG through the project “GENESIS” (ANR16CE92003501/DFG1591/21). Some of the calculations were performed on Copenhagen’s University DCSC cluster which is supported by a research grant (VKR023406) from the Villum Foundation. This research has made use of data from the Herschel Gould Belt survey project^{6}. The HGBS is a Herschel Key Project jointly carried out by SPIRE Specialist Astronomy Group 3 (SAG3), scientists of several institutes in the PACS Consortium (CEA Saclay, INAFIAPS Rome and INAFArcetri, KU Leuven, MPIA Heidelberg), and scientists of the Herschel Science Center (HSC).
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The IDL package for calculating the Δvariance can be found at https://hera.ph1.unikoeln.de/~ossk/Myself/deltavariance.html
By construction, the mean value of the fBm is zero. We applied an arbitrary offset to the maps in order to insure that all values were positive. The maps were then normalized by their mean value. The addition of a constant offset for the whole map does not alter the shape of the Δvariance spectrum since the relative differences between pixels remain the same.
The HiGAL sources were extracted with CuTEx (Molinari et al. 2016), which is designed to identify relatively roundish sources. In principle, during the detection step, it keeps only structures with both minor and major axes ranging from 1 to 3 instrumental point spread functions. Subsequently, starting from this initial guess, the 2D Gaussian fit that is used to determine the flux as well as the final estimate of the two axes has an additional tolerance to adjust itself on the source profile, so that one can find one (or both) of the two axes shorter than 1 RMS or longer than 3 RMS. Usually, no large differences are found between the two axes, so that the ratio is never larger than ≈4. On the contrary, algorithms used to extract sources from CO surveys, such as the one used in Heyer et al. (2001) that is based on a friendoffriend approach in positionpositionvelocity space, do not have any constraint on source size. If the CO emission is kinematically connected over an elongated area, such a structure might be identified as a single source with a large aspect ratio.
In CygnusX North, the real lower limit on the sizes of the structures and the true shape of the size distribution are very uncertain in the regime of small sizes. The choice of L_{1,min} = 0.025 pc basically means no lower limit (as it is the smallest resolved structure), while all other higher values would be questionable.
All Figures
Fig. 1 Column density maps of the CygnusX North cloud (left) and the Polaris Flare cloud (right). Column densities are displayed in units of the visual extinction using the conversion cm^{−2}. 

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In the text 
Fig. 2 Column density distribution function in the CygnusX North cloud (left panel) and the Polaris Flare cloud (right panel). The full red lines in both panels show a fit by a lognormal function in the low column density regime (≲ 5 × 10^{21} cm^{−2} in CygnusX North and ≲1 × 10^{21} cm^{2} in Polaris). The dashed line is a fit to the powerlaw regime that is observed in both regions in the intermediate column density regime, while the tripledot dashed red line in the case of the CygnusX North region is a fit to the shallower power law in the high column density regime. 

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In the text 
Fig. 3 Deltavariance functions calculated for the CygnusX North cloud (left) and the Polaris Flare cloud (right). The vertical dashed black lines in both panels mark the position of the spatial resolution for each of these two regions. The vertical dashed red lines in the case of Polaris mark the spatial range over which the powerlaw fit is performed. We do not attemptany fit in CygnusX North because the underlying selfsimilar regime is heavily perturbed by the presence of structures (see Sect. 6.3). 

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In the text 
Fig. 4 fBm images with β = 2.5 and with resolutions of 250 × 250 pixels (top left), 500 × 500 pixels (top right), and 1000 × 1000 pixels (bottom left). The Δvariance spectra for all three cases are compared in the bottomright subpanel. All display a selfsimilar regime with an exponent of the power law of α = β − 2 (i.e., Eq. (4)). All maps are normalized by their own mean value, and the vertical offset between the three Δvariance functions simply reflects the effect of this different normalization. 

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In the text 
Fig. 5 fBm images with values of β ranging from2 to 4 in steps of 0.5. All maps have a resolution of 1000 × 1000 pixels. The Δvariance figures for all cases are compared in the bottomright subpanel. All display a selfsimilar regime with an exponent of the power law of α = β − 2 (i.e., Eq. (4)). All maps are normalized by their own mean value. The random number series generating the phases has been kept the same in all maps. 

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In the text 
Fig. 6 2D Gaussian structures injected on top of an fBm image with β = 2.4. The 2D Gaussian functions have an aspect ratio (f = σ_{1}∕σ_{2}) that is varied in the range [1–10], and all have a value of δ_{c} = 5 and a fixed size of σ_{1} = 50 pixels. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function of the underlying fBm image. 

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In the text 
Fig. 7 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions have an aspect ratio f = 5, a value of σ_{1} = 50 pixels, and a column density contrast between the peak of the 2D Gaussian and the mean value of the fBm, δ_{c}, that is varied between 1 and 5. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

Open with DEXTER  
In the text 
Fig. 8 One or several similar 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions have an aspect ratio f = 5, a value σ_{1} = 50 pixels, and δ_{c} = 3. All maps are normalized to their mean value. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

Open with DEXTER  
In the text 
Fig. 9 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian functions all have an aspect ratio f = 0.2, a column density contrast between the peak of the 2D Gaussian and the mean value of the fBm, δ_{c} = 3, and a value of σ_{1} that is varied between 150 and 16.67 pixels. Bottomright figure: corresponding Δvariance functions calculated for each case, and these are compared to the Δvariance function calculated for the underlying fBm image. 

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In the text 
Fig. 10 Maximum deviation of the Δvariance function in the presence of structures from that of a pure fBm as a function of the quantity , where A is the area covered by the discrete structure(s) and δ_{c} is the column density contrast between the peak of the structure and the mean value of the underlying fBm. The dashed line has a slope of one. 

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In the text 
Fig. 11 Fractional probability distribution functions of the size of the major axis (left) and aspect ratio of clouds (middle) found in the 250 μm maps of the HiGAL survey (Molinari et al. 2016; Elia et al. 2017; full line) and in the ^{12}CO FCRAO HCS survey (Heyer et al. 2001; Dib et al. 2009; dashed line). The distribution of column density contrast is assumed to be a power law. The distributions of L_{1} and f are fitted with parameterized functions. The values of the parameters of the fit are reported in the main text. 

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In the text 
Fig. 12 Top: examples of 200 (left), 300 (middle), and 400 (right) 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian structures are randomly sampled using the distribution functions of the major axis size and aspect ratio distributions of the HiGAL clumps. The column density contrasts are randomly sampled from a distribution with ψ = 2.5. All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum of the synthetic models for cases with N_{s} = 200 (left), N_{s} = 300 (middle), and N_{s} = 400 (right) injected structures. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to those of the CygnusX North region and an fBm with β = 2.4. 

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In the text 
Fig. 13 Top: examples of 50 (left), 100 (middle), and 200 (right) 2D Gaussian structures injected on top of an fBm image with β = 2.4. The Gaussian structures are randomly sampled using the distribution functions of the major axis size and aspect ratio distributions of the HCS clouds. The column density contrasts are randomly sampled from a distribution with ψ = 2.5. All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum of the synthetic models for cases with N_{s} = 50 (left), N_{s} = 100 (middle), and N_{s} = 200 (right) injected structures. Each synthetic Δvariance spectrum is the average over 25 realizations and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to those of the CygnusX North region and an fBm with β = 2.4. 

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In the text 
Fig. 14 Examples of synthetic maps generated by overlaying structures (2D Gaussians, N_{s} = 300) on top of an fBm image with β = 2.4. The properties of the structures are randomly sampled from distribution functions of the aspect ratio (Eq. (9)), size of the major axis (Eq. (10)), and column density contrast (Eq. (11)). All models shown here share the same values of δ_{c} = 2.5, (L_{1,min} = 0.025 pc, L_{1,max} = 5 pc), (f_{min} = 3, f_{max} = 12), and (δ_{c,min} = 1, δ_{c,max} = 3). All syntheticmaps are convolved with a beam whose FWHM = 18.2′′. 

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In the text 
Fig. 15 Δvariance spectra related to the models presented in Fig. 14. Each synthetic spectrum (blue triangles) is the average over 25 realizations of the maps with the same set of parameters. In all models, N_{s} = 300 and ψ = 2.5. The full line is the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and that of an fBm image with β = 2.4. In the grid of models, the best fit to the data of Cygnus North is for the case with ξ = −0.4 and η = −0.35. 

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In the text 
Fig. 16 Same as the figure with ξ = −0.4, η = −0.35, ψ = 2.5, and N_{s}= 300 displayed in Fig. 15 but additionally showing the Δvariance spectra of individual realizations with this set of parameters. 

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In the text 
Fig. 17 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the lower and upper bounds of the aspect ratios, f_{min} and f_{max}, respectively. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrumof the synthetic models for the three cases with the considered sets of f_{min} and f_{max}. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to that of an fBm image with β = 2.4. 

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In the text 
Fig. 18 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the exponent ψ. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectrum spectra of the synthetic models for the three cases with ψ = 2, 2.5, and 3. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and that of an fBm image with β = 2.4. 

Open with DEXTER  
In the text 
Fig. 19 Top: examples of 2D Gaussian structures injected on top of an fBm image with β = 2.4. The maps only differ in the values of the lower and upper bounds of column density contrast δ_{c,min} and δ_{c,max}, respectively. All other parameters have the same values (see text for details). All synthetic maps are convolved with a beam whose FWHM = 18.2′′. Bottom: Δvariance spectra of the synthetic models for the three cases with the considered sets of δ_{c,min} and δ_{c,max}. Each synthetic spectrum is the average over 25 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to that of an fBm image with β = 2.4. 

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In the text 
Fig. 20 Left: synthetic column density map generated by reinjecting the compact source extracted from the CygnusX North map on top of an fBm image (convolved with a beam with FWHM = 18.2′′) with β = 2.4. Right: Δvariance spectrum of the synthetic models using this approach. The Δvariance spectrum of the models is the average over 50 realizations, and the full lines represent the 1σ dispersion around the mean. The synthetic Δvariance spectra are compared to that of the CygnusX North region and to the case of an fBm image with β = 2.4. 

Open with DEXTER  
In the text 
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