Density Exponent Analysis -- A new vision towards gravitational collapse of molecular clouds

7 The evolution of molecular interstellar clouds, during which stars form, is a complex, multi-scale 8 process. The power-law density exponent describes the steepness of density proﬁles in the log-log 9 space, and it has been used to characterize the density structures of the clouds. Its eﬀectiveness 10 results from the widespread emergence of power-law-like density structures in complex systems that 11 have reached intermediate asymptotic states. However, its usage is usually limited to spherically 12 symmetric systems. Importing the Level-Set Method, we develop a new formalism that generates 13 robust maps of a generalized density exponent k ρ at every location for complex density distributions. 14 By applying it to a high ﬁdelity, high dynamical range map of the Perseus molecular cloud constructed 15 using data from the Herschel and Planck satellites, we ﬁnd that the density exponent exhibits a 16 surprisingly wide range of variation ( − 3 . 5 . k ρ . − 0 . 5). Regions at later stages of gravitational 17 collapse are associated with steeper density proﬁles. Inside a region, gas located in the vicinities of 18 dense structures has very steep density proﬁles with k ρ ≈ − 3, which form because of depletion. This 19 density exponent analysis reveals diverse density structures in a molecular cloud, forming a coherent 20 picture that gravitational collapse and accretion contribute to a continued steepening of the density 21 proﬁle. We expect our method to be eﬀective in studying other power-law-like density structures, 22 including the density structure of granular materials and the Large-Scale Structure of the Universe. 23


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Many astrophysical processes, including the gravitational collapse of molecular clouds, are complex and 25 multi-scaled. Residing in the Galactic disk, the clouds are the nurseries of stars. They are open systems 26 that interact with the environment constantly. Their collapses involve an interplay between turbulence 27 [1], gravity, magnetic field [2], ionization radiation, and Galactic shear, resulting in highly complex 28 density distributions. 29 A clear picture of how the collapse occurs is yet to be achieved. Modern, high spatial dynamical 30 range observations provide maps that contain an unprecedented amount of information [3]. Measures of the underlying structures is often overlooked. Besides, to derive these measures, one needs to specify 36 the boundary of a region in advance, which can be a challenging task, especially for data that contain 37 heterogeneous structures. 38 Power-law density structures such as ρ ∝ r kρ (k ρ < 0) are common in systems which have reached 39 intermediate asymptotic states, such that the behaviors are independent of the details of the initial 40 and/or boundary conditions [7,8]. One such example is the gravitational collapse of molecular cloud, 41 where stationary collapse leads to ρ ∼ r −2 [9,10,11,12,13], and accretion flow around dense objects 42 have ρ ∼ r −1.5 [14,15]. For these systems, measuring the power-law density exponent k ρ would enable 43 us to distinguish different structures, and the value of k ρ can be directly compared against models to 44 achieve understandings. Various attempts have been made to measure the density exponent. The most 45 obvious approach is to fit spherical models to observational data. However, as the majority of regions

Model:
ρ ≈ ρ 0 (r/r 0 ) k ρ Figure 1: Evaluation the Level-Set Density Exponent. We first divide a region using a set of isosurfaces. A typical region R 1 would be surrounded by an isosurface at ρ = ρ 1 . Inside this region, there exists one or a few subregions (R 2,i ) surrounded by isosurfaces at ρ = ρ 2 . The equations on the right hand side describe how the density exponent k ρ at voxels included in R 1 yet not included in R 2,i ) is computed. The spacing between the adjacent isosurfaces are exaggerated for a clearer view.
we study are non-spherical, this approach is limited in practice. Another way is to derive the power-law 47 density exponent using the density probability distribution function (PDF) [10,16,17]. Although the 48 procedure is straightforward, this statistical approach only allows for the derivation of an "effective" 49 density exponent, which contains no information on how gas organizes spatially.  2 Method 58 For a spherical system, the density structure can be described as ρ(r) where r is the radius. To measure the steepness of the density profile in a given location, we adopt a local model where ρ ∼ r kρ , and in the vicinity of r, the value of k ρ can be derived as k ρ (r) = log(ρ(r + δ r )) − log(ρ(r)) log(r + δ r ) − log(r) , where δ r << r.

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The goal is to measure the density exponent for clouds of arbitrary geometries. Assuming a 3D 60 density structure ρ(x, y, z), we divide the region using a set of densely-spaced iso-density contours, after 61 which each subregion should be surrounded by a contour at ρ = ρ 1 , one or a few contours at ρ = ρ 2 , and 62 it should contain values ranging from ρ 1 to ρ 2 . Assuming that the region R 1 has a volume of V 1 , inside 63 this region, there can be a few (n) subregions R 2,i surrounded by isosurfaces with ρ = ρ 2 (ρ 2 > ρ 1 ), and 64 these subregions have volumes of V 2,i = V 2,1 , . . . V 2,n . The size of the region can be approximated as 65 r 1 ∝ (V 1 ) 1/3 , and the effective size of all subregions altogether can be approximated as r 2 ∝ ( i V 2,i ) 1/3 .

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The Level-Set Density Exponent is The procedure is illustrated in Fig. 1, and the resulting map is called the Density Exponent Map.

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The advantage of the Density Exponent Analysis lies in its robustness and resolving power: the 69 method is directly applicable to maps that contain heterogeneous structures and can be used to dis-70 tinguish these structures. As an example (Fig. 2), we construct a model which contains two spherical 71 clumps of different density profiles. We derive its density PDF and produce a density exponent map. The 72 density PDF contains limited information since the spatial information is lost completely. In contrast,

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This corresponds to 40% of the gas contained in the Perseus region and the region is surrounded by 89 a diffuse envelope that contains gas that does not contribute directly to the star formation. We also 90 excluded unresolved regions -patches surround by contours whose sizes are smaller than 3 pixels (0.08 91 pc), from our analyses. To derive the density exponent map, the data is divided using 100 contour levels 92 equally spaced in log(ρ).

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In Fig. 3 we plot the density exponent distribution at the cloud center plane, which contains all the 94 line-of-sight density maximums. k ρ ranges from -3.5 to -0.5, forming a highly inhomogeneous pattern 95 which contain variations on different scales. 96 3.1 Inter-regional variations: Density-driven collapse 97 The Perseus clouds can be separated into a few pc-sized subregions. Each region has a corresponding 98 gravitational potential dip [19], and tehse regions can collapse to from star clusters or associations. We 99 first divide the cloud into these regions and evaluate parameters including the mass-weighed mean density 100 exponent and the mean densities. We also characterize these regions by deriving a quantify called the −3 ± 0.7 −2 ± 0.7 −1 ± 0.7 In the background, we plot the density exponent map of the IC348 region and the NGC1333 region, respectively. The centers of the circles represent to the location of dense cores, and the radii of the circles correspond to the accretion zone radii of the dense cores r acc = Gm core /σ 2 v . The colormap is the same as that used in Fig. 3.
scenario. Since all regions belong to the Perseus molecular cloud, we assume that they have almost the

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The map also allows us to identified a new region called "Perseus Cirrous" (Fig. 3) for the first time.

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Although overlooked by previous studies, the region stands out in our analyses due to its shallow density 118 profile (k ρ ≈ −1). The region occupies the shallow end of the density exponent parameter space, and its 119 structure should be representative of the structure gas at early stages of gravitational collapse.   profile of dispersion of 2 n pixels. The final 3D density structure is assumed to be the sum of these slabs.

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When combining these slabs, we aligned them such that the density maximums stay on the same plane.

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Due to the lack of information on the distribution of gas along the line of sight direction, the density 216 structure we constructed is not identical to but resembles the real one. We first test our method by 217 producing a 3D clump of where ρ ∼ r −2 , projected it to 2D, and verified that our reconstruction allows us 218 to recover the density exponent to an accuracy of 0.01. Then, using results from numerical simulations

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[24], we perform density exponent analysis on both the original data and the 3D data constructed from 220 a 2D projection. We limit ourselves to gas with n(H 2 ) 250 cm −3 and find that the reconstructed 221 cloud and the original cloud are similar in terms of k ρ (Fig. 5). The original cloud has a mass-weighted whose sizes are smaller than three pixels (56", 0.08 pc), are also indicated.

IC348
all gas dense gas unresolved k < -2.5 -2.5 < k < -1.5-1.5 < k < -0.5 k > -0.  Figure 6: Results from individual regions. Left panels: maps of the density exponent k ρ . Right panels: we plot the total amount of gas, the amount of dense gas, the amount of spatially-unresolved gas, as well as a distribution of gas in regions of different k ρ . Properties including total gas mass, mean density, dense gas fraction, and mass-weighted density exponent, are indicated in the right panels. Limited by the resolution, it is impossible to derive the density exponent for gas contained in contours whose sizes are smaller than 3 pixels. Mass contained in these contours are excluded from the analyses. The total amount of unresolved gas are indicated as "unresolved".