Issue 
A&A
Volume 578, June 2015



Article Number  A97  
Number of page(s)  15  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201424030  
Published online  11 June 2015 
Gvirial: Gravitybased structure analysis of molecular clouds ^{⋆,}^{⋆⋆}
^{1}
MaxPlanck Institut für Radioastronomie,
Auf dem Hügel 69,
53121
Bonn,
Germany
email:
gxli@mpifrbonn.mpg.de
^{2}
University of Toledo, Ritter Astrophysical Observatory, Department
of Physics and Astronomy, Toledo
OH
43606,
USA
^{3}
MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStraße 1, 85740
Garching bei München,
Germany
Received: 18 April 2014
Accepted: 24 February 2015
We present the Gvirial method which aims to quantify (1) the importance of gravity in molecular clouds in the positionpositionvelocity (PPV) space; and (2) properties of the gas condensations in molecular clouds. Different from previous approaches that calculate the virial parameter for different regions, our new method takes gravitational interactions between all the voxels in 3D PPV data cubes into account, and generates maps of the importance of gravity. This map can be combined with the original data cube to derive relations such as the massradius relation. Our method is important for several reasons. First, it offers the ability to quantify the centrally condensed structures in the 3D PPV data cubes, and enables us to compare them in an uniform framework. Second, it allows us to understand the importance of gravity at different locations in the data cube, and provides a global picture of gravity in clouds. Third, it offers a robust approach to decomposing the data into different regions which are gravitationally coherent. To demonstrate the application of our method we identified regions from the Perseus and Ophiuchus molecular clouds, and analyzed their properties. We found an increase in the importance of gravity towards the centers of the individual molecular condensations. We also quantified the properties of the regions in terms of massradius and massvelocity relations. Through evaluating the virial parameters based on the Gvirial, we found that all our regions are almost gravitationally bound. Clusterforming regions appear are more centrally condensed.
Key words: gravitation / ISM: structure / ISM: kinematics and dynamics / stars: formation / methods: numerical / ISM: clouds
Appendices and movie are available in electronic form at http://www.aanda.org
Available at http://gxli.github.io/Gvirial/
© ESO, 2015
1. Introduction
Star formation takes place in the dense and shielded parts of the interstellar medium. Observations show that the interstellar medium exhibits complicated, irregular, and filamentary structures (Schneider & Elmegreen 1979; Williams et al. 2000; Goldsmith et al. 2008; Men’shchikov et al. 2010). Theoretically, such structures are created by various physical processes such as turbulence, gravity, magnetic field, and radiation.
To understand star formation it is necessary to understand how various physical processes affect it. Gravity is a longrange interaction, and plays important roles in most astrophysical processes at a multiple of physical scales. This is particularly true for turbulent molecular clouds (e.g., Heyer et al. 2009; Kauffmann et al. 2013). However, characterizing the role of gravity on a cloud at various physical scales is not straightforward.
In observations, the structure of molecular gas can be traced by spectral lines in the 3D positionpositionvelocity (PPV) space where molecular gas exhibits complicated structures. However, given the rich information obtained observationally, few constraints on the role of gravity in the clouds have been obtained so far. One major difficulty is to properly quantify the irregular structure of the gas. The virial parameter α_{vir} = 5σ^{2}R/GM^{1} is commonly used to quantify the importance of gravity, and to calculate it we must define a region in which σ and M can be evaluated^{2}. As a result, the virial parameter is only suitable for the cases where the structures are well defined. The morphology of the molecular interstellar medium is generally so complicated that in many cases it is difficult to separate individual objects from a continuous distribution of material. This has been further complicated by the fact that we observe the structures of the sky plane, and the structures identified from the observations are biased by various projection effects (Pichardo et al. 2000; Dib et al. 2006; Shetty et al. 2010; Beaumont et al. 2013).
Another difficulty with the virial parameter is that it is localized. To define the virial parameter we need to define the region, and by evaluating the virial parameter of that region, we automatically neglect the gravitational interaction between the region and its surroundings. As a result, the virial parameter can be used to quantify the importance of gravity for the individual structures, but it cannot be used to understand the importance of gravity for a cloud as a whole. Gravity is a longrange interaction, and to understand its importance, we need to understand how it works not it one region on one particular physical scale, but on a multiple of physical scales.
One case where such an understanding is required is the clusterforming region. In observations, such regions are found to be centrally condensed in both the positionposition space and the PPV space, and physically we expect that gravity is important at the centers of the clusterforming regions and should become less important if we move from the centers to the outskirts. This stratification, which is of crucial importance for understanding stellar clustering, awaits to be quantified.
In this paper we introduce a new method called Gvirial to quantify the importance of gravity in a variety of situations where mass is traced in the 3D PPV space. Instead of dividing the molecular cloud into regions based on isointensity contours in the 3D data cube (e.g., Williams et al. 1994), the method takes the gravitational interactions between all the voxels in the 3D PPV data cube into account, and generates maps of the importance of gravity in 3D. As a result, it provides constraints on the global importance of gravity. This is complementary to methods that calculate the selfgravity such as the virial parameter. Such a map of the importance of gravity also enables us to identify gravitationally coherent regions in molecular clouds and to quantify the structures of the regions.
In this paper, we introduce the Gvirial method (Sects. 2–5), and explore its usage in quantifying the importance of gravity and the structures of the molecular condensations seen in simulations (Sect. 5) and observations (Sect. 6). We conclude in Sect. 7.
2. Problem formulation
Observationally, molecular gas can be mapped by rotational transitions of molecules such as CO. Through proper modeling of the emission, it is possible to construct the distribution of the molecular gas in the form of a 3D PPV data cube which covers a continuous (x,y,v) space (1)where m represents the mass, x and y represent the spatial dimensions, and v represents the velocity dimension. Here has a dimension of ML^{3}T, and represents the amount of mass per unit area per unit velocity. The distribution of in the v dimension represents the distribution of gas at different velocities along the same line of sight.
We start with a mass distribution in a 3D PPV data cube (Fig. 1). We aim to understand how a particle at voxel i is bound by mass from all the voxels j (not excluding i). To achieve this, we split our task into two subtasks. First, we need to estimate the gravitational boundedness of voxel i by another voxel j in the 3D PPV data cube. Second, we need to estimate the gravitational boundedness of a voxel i based on the information we have about all the other voxels j.
2.1. Boundedness of a voxel pair in a 3D data cube
Fig. 1 Setup of the problem. We consider a mass distribution in a 3D PPV data cube. This is represented as the black grid. x–y represents the spatial dimensions and v represents the velocity dimension. The mass distribution is represented in blue. For each voxel i, its Gvirial can be determined by taking its interactions with all the voxels j (not excluding i) into account. See Sect. 2 for details. 

Open with DEXTER 
We start by defining the gravitational boundedness between a pair of voxels (i,j). We are interested in the extent to which a particle i is bound to another voxel j. For our purpose, i can be treated as a massless test particle, and the boundedness of particle i is determined by the mass of the voxel j.
The available measurable quantities in 3D PPV data cube include the mass m_{j}, the spatial separation δr_{ij} and the velocity separation δv_{ij}. We expect the boundedness to increase with m_{j} and decrease with δr_{ij} since gravitational attraction increases with the mass and decreases with the distance.
The remaining question is how to make use of the information contained in the velocity direction. We expect that voxels with large velocity differences are not likely to be bound to each other, for two reasons. First, a larger velocity separation implies higher kinetic energy, which consequently decreases the likelihood of it being gravitationally bound. Second, it has been found that the molecular gas follows the Larson’s relation (Larson 1981)^{3} where the spatial length and velocity dispersion are related by (2)As a result, a larger velocity separation implies a larger spatial separation, which implies a lower gravitationally boundedness.
Here, we define the gravitational boundedness contributed from voxel j to voxel i as (3)The boundedness I_{j → i} increases with mass and decreases with spatial and velocity separation.
Apparent similarities exist between Eq. (3) and the virial parameter. The virial parameter is defined as (Bertoldi & McKee 1992)^{4}(4)where m, r, and σ_{v} are the mass, radius, and velocity dispersion of the clumps. The virial parameter is a measurement of the gravitational boundedness of an object. The quantity I_{j → i} defined by Eq. (3) has a dimension of G Mass/Radius Velocity^{2}, which is the same as that of the virial parameter. Therefore, Eq. (3) can be viewed as a generalization of the virial parameter to a voxel pair, and a larger I_{j → i} is related to a larger chance for voxel i to be bounded by voxel j.
2.2. The Gvirial
We define the Gvirial of a voxel i as the sum of the gravitational boundedness contributed from all the voxels j (not excluding i). To be more precise, (5)where I_{j → i} comes from Eq. (3), and in the second step we take into account that the Gvirial is measured in the 3D PPV space. The physical meaning of Eq. (5) can be understood as follows. For one voxel i, its Gvirial is determined by summing up its boundedness with all the other voxels j.
In Eq. (5), if j is close to i in both the spatial and the velocity direction, it contributes more to the Gvirial. The contribution is proportional to the mass m_{j}, and inversely proportional to δr_{ij} and .
For a continuous distribution of material, Eq. (5) can be written as (6)where the integration is carried out over the whole data cube. Here has a dimension of ML^{2}V^{1} where M is mass, L is size, and V is velocity.
If the separation between i and j becomes small, the denominator of Eq. (6) approaches zero and the integrand can become large. If the density distribution is smooth, the term is still finite and does not induce a singularity. However, the term is singular. In our calculations, we change it to (7)which means we suppress the contribution to the Gvirial when the velocity separation is smaller than c_{0}, and in our case, c_{0} is comparable to the sound speed. We introduce this cutoff in order to make the integrand convergent. Nevertheless, there is a physical reason behind this. In a molecular cloud, the velocity dispersion that we observe comes from two parts: one from the thermal motion, and the other from the largescale streaming motion of the gas. The first part is almost scaleindependent, and the second part increases with the physical size. We are mainly interested in the balance between gravity and the largescale streaming motion, and as a result, it is reasonable to discard the contributions where δ_{v} ≲ c_{0}. In our calculations, c_{0} is chosen to be 1kms^{1}, which is comparable to the sound speed. The effect of changing c_{0} influences the absolute values of the Gvirial; however, the relative values of the Gvirial stay unchanged (Appendix B). Therefore, the Gvirial is a relative measure of the gravitational boundedness. In the case where the Gvirial map of several observations needs to be compared, a single value of c_{0} needs to be chosen in advance.
The reasons for the name Gvirial in Eq. (5) include the following: first, we named it Gvirial to emphasize its connection with the commonly used virial parameter; second, we added the letter G to emphasize that our virial parameter is a generalized version of the virial parameter and that it is global. Different from the case of Bertoldi & McKee (1992) and Goodman et al. (2009) where the virial parameter is used to quantify selfgravity, our Gvirial takes all the gravitational interactions between gas in the data cube into account. More clarifications concerning the concept of the Gvirial can be found in Appendix A.
3. Separating components with different velocities
One difficulty that we need to deal with is the lineofsight confusion. When calculating the Gvirial, different components along the same line of sight are distinguished through their velocities. If two components happen to have the same velocity, the Gvirial will be overestimated, since physically unassociated components will be treated as one single component in the calculations.
We argue that in many cases, the separation in the velocity axis provides information so that we can separate different components based on the velocity difference or at least alleviate the problem. We consider two cases; the first is the lineofsight confusion in a given molecular cloud, and the second is the lineofsight confusion in our Milky Way disk.
In a molecular cloud, the lineofsight confusion problem can be alleviated if the molecular gas follows the Larson’s relation (Larson 1981; RomanDuval et al. 2011) because the boundedness is proportional to (Eq. (3)) where δ_{r} and δ_{v} are the separations in position and velocity direction, respectively. We consider two points along the same line of sight separated by δ_{z}. According to the Larson’s relation, the velocity dispersion (which is a width of the statistical distribution velocity separation between the two points) is . Therefore, the boundedness related to δ_{z} by . In other words, the contribution is not likely to be large since the velocity is likely to be different in an averaged sense.
In the case of the Milky Way disk, different spiral arms usually have different velocities, they will be separated easily in our calculations. We note that in Eq. (3) the interaction is proportional to δr^{1} and δv^{2}. Therefore, a small separation in velocity will lead to a much bigger decrease in interaction and finally make the contribution to the total Gvirial negligible. Since the method uses velocity difference to deal with lineofsight confusion, it is more accurate for the cases where the velocity increases with distance (e.g., including systematic expansion and contraction where velocity increases with distance).
4. Numerical procedure
In the case of a 3D PPV data cube, for a voxel (x,y,v), the Gvirial is defined as (see Eqs. (6) and (7)) (8)where the integration is carried out over the whole data cube. With an observationally constructed distribution of mass in the 3D PPV data cube, we can easily construct the Gvirial using Eq. (8).
Equation (8) takes the form of a convolution, and can be conveniently calculated in Fourier space. In real space, the kernel is (9)and in Fourier space it can be shown that^{5}(10)Equation (10) enables us to calculate the Gvirial map efficiently. In our calculations, we first make a 3D fast Fourier transform (FFT) to the observationally constructed mass distribution and obtain . Then we calculate the Gvirial in the Fourier space (11)In the last step the Gvirial map is obtained by a inverse FFT of α_{k}.
We found that the absolute values of the Gvirial are dependent on c_{0} (see Appendix B); however, the relative values of the Gvirial are insensitive to c_{0}. As a result, the Gvirial is a relative measurement of gravitational boundedness rather than an absolute one. In order for the Gvirial maps from different regions to be comparable, a unique value of c_{0} has to be chosen; c_{0} also needs to be larger than the velocity resolution of the observations.
5. Quantifying the structure of molecular gas with Gvirial
We present applications of the Gvirial to quantify gravity and the structure of molecular gas. To begin with, we apply the method to the models where both the 3D density and 3D velocity structure are available. We focus on two aspects: first, we present the use of Gvirial in quantifying the importance of gravity in the 3D PPV data cube; second, we quantify the structures of molecular condensations bases on the Gvirial.
5.1. Defining Gvirial in the PPP space
To properly evaluate the accuracy of the Gvirial method, we first define the Gvirial^{model} in the PPP space where all the position and velocity information are available, and then compare it with the observationally reconstructed values of the Gvirial in the PPV space. Following Eqs. (3) and (5), we define the model Gvirial as (12)where (13)and (14)Here the voxel pair (i,j) has positions ((x_{i},y_{i},z_{i}),(x_{j},y_{j},z_{j})) and velocities . The parameter δ_{x} is the spatial resolution of the simulation. Writing Eq. (12) in its integral form, we have (15)where p stand for (x,y,z) and p′ stand for (x′,y′,z′). Here we choose c_{model} = 3 × c_{0} since the denominator of Eq. (15) is composed of a sum of the velocity difference from the three dimensions. Since the Gvirial is a relative measure of the gravitational boundedness (Appendix B), our results are not affected by this.
5.2. Quantifying the importance of gravity
We study how the importance of gravity traced in the Gvirial inferred from the simulated observations is related to the Gvirial theoretically defined in the models.
We consider two models. The first (hydrodynamic simulation model) was taken from a numerical simulation of turbulent gas. The simulation used here was carried out under periodical boundary conditions, and turbulence is injected through compressive forcing mode (Federrath et al. 2008), and no selfgravity is included. The details of the simulations are described in Federrath et al. (2010), RomanDuval et al. (2011)^{6}. Since selfgravity is not included in the simulation, the lineofsight confusion effect is probably overestimated because, as was found in Beaumont et al. (2013), in the simulations without gravity structures will overlap more in the PPV space. In this work, we make use of the density and velocity cubes from the simulations.
We make a cutout of a size of 4.8pc × 4.8pc × 4.8 pc from the snapshot at t = 5 T where T is the crossing timescale of the simulation. The computation of Eq. (15) is computationally expensive (o(N^{2})). To save computational time, we choose a portion of the simulation with a relatively large gas condensation and relatively complicated internal structures (Fig. 2).
Fig. 2 a) Map of the peak brightness temperature of the simulated ^{13}CO(1–0) emission along the velocity axis. b) Positionvelocity map of peak brightness temperatures of ^{13}CO(1–0) along the Yaxis. c) Positionvelocity map of peak brightness temperatures of ^{13}CO(1–0)) along the Xaxis. 

Open with DEXTER 
The second model (the analytical model) is constructed to resemble a typical molecular condensation (e.g., McKee & Tan 2003). It is defined within r_{in}<r<r_{out} where r is the radius, r_{in} = 1 pc, and r_{out} = 10pc. The density structure is (16)where r_{0} = 1pc; ρ_{0} = 10^{3} × m_{H2}, where m_{H2} is the mass of the H_{2} molecule; and γ_{ρ} are chosen to vary from − 1.8 to − 2.2. The velocity structure is parametrized as (17)where r_{0} = 1 pc, γ_{v} is chosen to vary from 0.35 to 0.6, and v_{0} = 1 kms^{1}. It can be viewed as an expanding sphere. At a given radius, the magnitude of velocity is distributed uniformly between 0.9v(r) and 1.1v(r), and the direction of the velocity is distributed uniformly in 3D. The model is sampled through a total of 3 × 10^{5} particles, and then turned into a 3D PPV data cube.
We chose the two models because they represent two typical situations. In the first case (the hydrodynamic simulation model), the model exhibits a high degree of physical complexity. In the second case (the analytical model), the density distribution is centrally condensed, and the model has welldefined massradius and velocityradius structures. We will study the behavior of the Gvirial in both cases.
The models are turned into a 3D PPV data cube using a simple radiative transfer model (Sect. 3.3 of RomanDuval et al. 2011). It is assumed that the ^{13}CO(1–0) emission is optically thin and n(^{13}CO)/n(H_{2}) = 1.7 × 10^{6} (Langer & Penzias 1990; Blake et al. 1987). The excitation temperature is assumed to be 10 K. In reality, the ^{13}CO(1–0) emission can be optically thick and both the abundance and excitation conditions can vary, and these can lead to inaccuracies. However, in testing our method, we are interested in the cases where mass can be reliably traced in the PPV data cube, and observationally this can be achieved by modeling several transitions of CO or using more reliable tracers.
In our calculation of the model Gvirial (), we choose c_{model} = 1 km s^{1}, and in our calculation of the Gvirial reconstructed from the simulated map (), we choose c_{0} = 0.3 km s^{1}. We make this choice because in Eq. (15) all three velocity components are considered, and in Eq. (6) only one velocity component is considered. Since Gvirial is a relative measure of the gravitational boundedness (Appendix B), the conclusions are not dependent on this.
Fig. 3 Comparison between the Gvirial_{model} calculated from the model and the Gvirial_{reconstructed}, which is reconstructed from the simulated observations in the PPV space. Panel a) shows the result for the hydrodynamic simulation; and panel b) shows the result for the analytical model with γ_{ρ} = −2.2 and γ_{v} = 0.5. The colors stand for the amount of mass in a given interval. 

Open with DEXTER 
In Fig. 3 we compare the Gvirial_{model} calculated from the model with the Gvirial_{reconstructed} reconstructed from the simulated observations in the PPV space. The first thing to be noticed is that the absolute values of the Gvirial differ because the Gvirial is only a relative measure of the gravitational boundedness, rather than an absolute one.
In general, a larger Gvirial_{model} is related to a larger Gvirial_{reconstructed}. The reconstructed Gvirial exhibits a higher uncertainty in the hydrodynamic simulation model since its flow has complicated structures.
Fig. 4 Comparison between between Gvirial_{model} and Gvirial_{reconstructed} in the PPV space for the hydrodynamical model described in Sect. 5.2. In the model, turbulence is injected through compressible forcing, and no selfgravity is included (Federrath et al. 2008). The Gvirial_{model} in the PPV space is defined as the massweighted average of the Gvirial_{model} in the PPP space, which is defined in 12. Here, the grayscale image is the simulated ^{13}CO(1–0) emission from the model. The red contours stand for the Gvirial_{model}, and the contour levels are (6, 5, 10). The blue contours stand for the Gvirial_{reconstructed}, and the contour levels are (3, 6). It worth noting that the Gvirial_{model} and the Gvirial_{reconstructed} agrees better at regions where mass is traced. This can be seen in the v = −0.2 km s^{1} channel. They agree better at the regions where mass is traced (e.g., the red arrow), but does not agree well at the region without mass (e.g., the blue arrow). 

Open with DEXTER 
To look further into the relation between Gvirial_{model} and Gvirial_{reconstructed}, we convert the Gvirial_{model} from the PPP space into the PPV space, and study the connection between the two (Fig. 4). At a given velocity interval, the Gvirial_{model} in the PPV space is defined as the massweighted average of the Gvirial_{model} in the PPP space, and the average is carried over the entire line of sight. There are some significant differences between the iso Gvirial_{model} contours (isosurfaces where the Gvirial takes constant values) and iso Gvirial_{reconstructed} contours: the iso Gvirial_{model} exhibit more structures and iso Gvirial_{reconstructed} is smoother because the Gvirial_{model} is unaffected by the lineofsight confusion. Thus, the Gvirial map constructed in the PPV space is better suited to studying the effect of gravity at large scale, but is not suitable for the study of the importance of gravity at small scale (e.g., individual cores). Although the iso Gvirial_{model} and the iso Gvirial_{reconstructed} does not agree completely, they agree better at positions where emission is enhanced and mass is traced, and tends to differ more where there is little emission. Since we are interested in the gravitational boundedness of the gas, this disagreement at regions where little mass is present is not as important as it might appear.
Fig. 5 Comparison between the structure of the model and the structure reconstructed through the Gvirial for different sets of model parameters. The upper panels show the results where the analytical model is constructed with different γ_{ρ}, and the lower panels show the results where the analytical model is constructed with different γ_{v}. The left panels are the massradius relations and the right panels are the velocity dispersionradius relations. Here the radius is scaled up by a factor of 1.4 and the velocity dispersion is scaled up by a factor of 1.45 for an easy comparison. 

Open with DEXTER 
Since the hydrodynamic simulation model exhibit complicated structures, the lineofsight confusion has a larger impact on the results than the case of the analytical model. However, in the gravityfree hydrodynamic simulation model the confusion probably has been overestimated. According to Beaumont et al. (2013), structures tend to overlap more without selfgravity.
5.3. Quantifying the internal structures
For a given region, its Gvirial map is centrally concentrated, and if we divide the region based on different isosurfaces of Gvirial values, we get a set of regions that nest inside one another. We propose to quantify the structures of the gas condensations in the PPV space with the massradius relation and velocity dispersionradius relation derived with the help of the isosurfaces of the Gvirial.
Inside a closed region where the Gvirial of the voxels is larger than a given threshold, we can evaluate parameters such as mass, radius, and velocity dispersion. Using a map of mass distribution m(x,y,v) and a map of the Gvirial α_{Gvirial}(x,y,v), the corresponding parameters are evaluated by taking all the voxels inside a given contour of the α_{Gvirial}(x,y,v) map into account. The mass of a region is defined as (18)where R denotes a coherent region that satisfies α_{Gvirial}(x,y,v)>α_{min} and the central velocity of a region is defined as (19)The velocity dispersion is defined as (20)The radius of the region is defined by diagonalizing the tensor of second moments of the position coordinates weighted by the intensity (Goodman et al. 2009), and is defined as (21)where σ_{min} and σ_{max} are the dispersions along the major and minor axes.
This set of definitions enables us to quantify the structure of molecular condensations observed in the PPV space in terms of massradius and velocity dispersionradius relations. If the molecular gas has a relatively diffuse morphology, we expect to see a steeper dependence of the enclosed mass M with the radius r; if the molecular gas is centrally condensed, we expect to see a shallower Mr dependence. Similar arguments can be applied to the dependence of velocity dispersion on radius σ_{v}r.
If the molecular gas is centrally condensed, the Mr and σ_{v}r relations of the gas condensations are well defined in the PPP (positionpositionposition) space. It is expected that we can reconstruct these relations observationally with the help of the Gvirial. If a molecular gas condensation has a diffuse and irregular morphology, these Gvirialderived relations can still represent the morphology of the gas condensations. The difference between different regions can be represented in the Mr and σ_{v}r planes.
To demonstrate the diagnostic power of the Mr and σ_{v}r relations, we generated a set of models with different density and velocity structures, and compare them with the Mr and σ_{v}r relations reconstructed with the Gvirial (Fig. 5). The differences in structure among different regions can be accurately reconstructed with the help of the Gvirial. Furthermore, the slope of the Mr and σ_{v}r relations are accurately reconstructed with the Gvirial^{7}.
We also plot the dependence of Gvirial as a function of radius for various models (Fig. 6). For all the models, we found an increase of Gvirial towards the centers of the regions. Larger Gvirial are found for the models with more condensed structure and lower velocity dispersion.
5.4. Short summary
In this section we demonstrated the usage of Gvirial to quantify the structure of molecular gas condensations. We summarize our findings and remind the reader of the caveats.
We found that the Gvirial can be used to quantify the importance of gravity and to quantify the structures of the gas condensations.

1.
The Gvirial measured from the PPV data cube is positivelyrelated to the Gvirial found in the model(PPP space), and a larger Gvirial is related to alarger probability of being gravitationally bound by the ambientgas.

2.
The Gvirial map can be used to derive massradius and velocity dispersionradius relations. The Gvirialderived relations are good reconstructions of the structures of the models when the models are symmetric, and can also provide good representations of the structures if the models are irregular.
Several caveats need to be noted. First, the Gvirial is a relative measure of the gravitational boundedness. A larger Gvirial means a larger change of being gravitationally bound. The absolute values of the Gvirial cannot be directly used to tell if a region is gravitationally bound or not. The absolute values of the Gvirial are dependent on c_{0}. Therefore, in order for the values of Gvirial from several maps to be comparable, a unique value of c_{0} needs to be chosen in advance. This value should be comparable to the sound speed of the medium, and in this work we have chosen it to be c_{0} = 1 km s^{1}. We also note that the Gvirial is reconstructed from the PPV space where the projection effect tends to smear out structures; it is more suitable for the study of the gravitational boundedness of the structures larger than the individual gas condensations.
Second, the Gvirial method can be used to derive the massradius and velocity dispersionradius relations for molecular condensations. However, the results need to be properly interpreted. If the structures of the gas condensations are regular and are close to being spherically symmetric, the Gvirialderived massradius and velocity dispersionradius relations should be good reconstructions of the real 3D structure. If the structure of the gas condensations are irregular, the Gvirial can be used to derive these relations. However, in this case, the Gvirialderived relations should be viewed as representations of the compactness of the structures in the PPV space, and are not necessarily related the compactness of the structures in the PPP space.
Fig. 6 Gvirial as a function of radii for models with different γ_{ρ} and γ_{v}. 

Open with DEXTER 
Finally, we would like to emphasize that the Gvirial methods works in the PPV space, and like other methods, its accuracy is limited by lineofsight contamination. The method relies on using the velocity difference to alleviate the lineofsight confusion. This requires different gas components to have different velocities, which is satisfied in many cases. For a single object, this requires the velocity separation to increase with radius. Whereas this generally holds for individual molecular clouds and clumps, it does not hold for some cores or disks where the velocity decreases with radius (e.g., collapsing cores).
6. Applications to molecular clouds
In this section we present applications of our method to several molecular clouds. Our method requires a map of the distribution of molecular gas in the 3D PPV space. Observationally, this can be conveniently achieved using rotational transitions of the CO molecule.
In the simplest case, this can be achieved by observing the ^{13}CO(1–0) transition alone. Assuming an excitation temperature of 10 K, a ^{12}CO abundance of X(^{12}CO/H_{2}) = 8 × 10^{5}, and a ^{12}CO to ^{13}CO ratio of R(^{12}CO/^{13}CO) = 45, the column density can be estimated as (Simon et al. 2001; RomanDuval et al. 2010) (22)where T_{mb} is measured in K and δ_{v} is the velocity resolution of the data cube measured in kms^{1}. Changing the excitation temperature to 20 or 30 K decreases the derived mass by 40% and 92%, respectively (Simon et al. 2001). For an observed ^{13}CO(1–0) data cube T_{mb}(x,y,v), the mass distribution m(x,y,v) can be obtained as (23)where δ_{x} is the voxel size of the spatial dimension in cm and δ_{v} is the channel width in kms^{1}. The values of m and in Eq. (6) are related by .
We apply our Gvirial method to the publicly available data from the COMPLETE survey (Ridge et al. 2006). The observations have a spatial resolution of 46″ and a velocity resolution of 0.067 km s^{1}, the mean RMS per channel is ~0.33 K in terms of , and the beam efficiency is ~0.5.
With a distance of 250 ± 50 pc and a total mass of 10^{4}M_{⊙} (Enoch et al. 2006), the Perseus molecular cloud is among the beststudied molecular clouds in the Milky Way. It is composed of several distinct regions: B1, B3, NGC 1333, and IC 348. The Ophiuchus molecular cloud has a distance of 125 pc and a mass of 7 × 10^{3}M_{⊙} (de Geus et al. 1989) and contains clouds L1688 and L1689. In particular, L1688 and NGC 1333 are clusterbearing regions. IC 348 also hosts a star cluster; however, it is generally considered older than the other regions in Perseus (Gutermuth et al. 2009).
In this section, we provide a study of the structure of the Perseus and Ophiuchus molecular clouds with our Gvirial method. We present maps of the Gvirial, together with an analysis of the structure of the regions in the clouds based on the Gvirial method.
Fig. 7 Channel maps of the ^{13}CO(1–0) emission and the corresponding Gvirial. The upper panel shows the results from the Perseus molecular cloud and the lower panel shows the results from the Ophiuchus molecular cloud. Grayscale images stand for ^{13}CO(1–0) emission and the red contours represents the Gvirial. Contour levels start from 1.2 in steps of 0.8. 

Open with DEXTER 
6.1. Maps of the Gvirial
In Fig. 7 we channel maps of the Gvirial in the 3D PPV space. Threedimensional renderings of the maps are presented in Appendix C. For both clouds, the ^{13}CO(1–0) maps contain complicated and filamentary structure, whereas the Gvirial maps are smooth and contain fewer structures. Interestingly, higher values of Gvirial are reached only at clusterbearing regions such as NGC 1333, IC 348, and L1688, and this highlights the importance of gravity in such regions. Different regions are indicated in Fig. 8.
6.2. Identification of regions
The Gvirial offers a new way to divide the molecular cloud into regions. Previously, this has been done either by visual inspection, or with contourbased algorithms such as clumpfind (Williams et al. 1994). It is worth mentioning that other regionfinding algorithms are available, such as dendrogram (Rosolowsky et al. 2008) and dochamp (Whiting 2012).
Many of the algorithms (e.g., clumpfind and dendrogram) are contourbased, and they tend to assign voxels to regions based on isocolumndensity contours in the 2D case or on isointensity contours in the case of a 3D PPV data cube. Molecular clouds are characterized by a set of complicated hierarchical structures, and a naive application of the clumpfind tends to produce hundreds of clumps for one single molecular cloud. It has also been pointed out recently that contourbased methods suffer from superposition and confusion when the volume filling factor of emitting material is large (Beaumont et al. 2013).
In our analysis we are interested in finding gravitationally coherent regions. This can be achieved by applying the dendrogram^{8} to the Gvirial map, which is a measurement of the gravitational boundedness. Figure 8 shows the regions identified by applying the dendrogram algorithm to our Gvirial map. The identified regions correspond to the “leaves” of the dendrogram. In the dendrogram calculations, the minimum difference between different contours is set to 0.4, and the regions correspond to the leaves of the dendrogram. Each identified region consists of a continuous list of voxels in the 3D PPV space. In Fig. 8 the projected boundaries of the regions are plotted^{9}.
Compared to clumpfind, a combination of the Gvirial method and the Dendrogram method tends to identify regions that are coherent while the clumpfind tends to break up those regions. The reason is that ourGvirial maps are much smoother than the original map^{10}. Since the Gvirial maps are much smoother, the results obtained with the Gvirial maps are much less sensitive to the technical parameters (e.g., the minimum separation of the contours) used in the computations compared to the case with the intensity maps. Therefore, the Gvirial offers a more robust definition of regions. With Gvirial we can study the structure of molecular gas in terms of these gravitationally coherent regions. In the 2D case, our method is similar to the gridcore code which is based on 2D projected gravitational potential (Gong & Ostriker 2011).
6.3. Internal structure of the regions
Fig. 8 Regions identified by applying the dendrogram algorithm to our Gvirial map. The results from the Perseus molecular cloud (upper panel) and Ophiuchus molecular cloud (lower panel) are presented. The grayscale images correspond to the velocityintegrated ^{13}CO(1–0) emission and the contours correspond to the projected boundaries of the identified regions. These regions correspond to the “leaves” of the dendrogram. The conventional names of the regions are labeled. 

Open with DEXTER 
In this section we study the internal structure of the molecular gas in the individual regions. Previously, the structure of molecular gas was studied in terms of the clumps, and quantified using parameters that are evaluated for the whole clumps. While these methods do provide constraints on the role of gravity inside the regions, the role of gravity on scales larger than the individual gas condensations is neglected. The spatial structure of molecular gas larger than the clump scale can be quantified using the dendrogram algorithm. However, the dendrogram method is contourbased, and only the connection between the adjacent regions in the tree diagram are preserved. This makes it difficult to use dendrogram to quantify the structure of molecular gas on the large scale.
Fig. 9 Dependence of velocity and velocity dispersion as a function of radius. Here the solid lines represent the velocities of the regions, and the bars represent the velocity dispersions of the regions at given radii. The vertical extent of the errorbars are the 3D velocity dispersion . 

Open with DEXTER 
Fig. 10 Amount of molecular gas enclosed in regions with different thresholds of the Gvirial. The xaxis is the threshold of Gvirial which we take to define the region, and the yaxis is the amount of gas that is enclosed in the region. 

Open with DEXTER 
Fig. 11 Gvirial as a function of radius for different regions. 

Open with DEXTER 
Fig. 12 Enclosed mass as a function of radius for different regions. The scaling relation from Larson (1981) and Kauffmann et al. (2010b) are added. For all the curves, the Gvirial decreases with increasing radii. 

Open with DEXTER 
Here we provide an analysis of the structure of the individual regions with the Gvirial method. As we have shown in Sect. 5.3, the intrinsic massradius and velocityradius relations in the model can be reconstructed with our Gvirial method to a good accuracy if the velocity dispersion increases with the size.
In Fig. 9 we plot the dependence of velocity v_{0} and velocity dispersion σ_{v} on the radii of the regions. The inner parts of the regions are clearly at the center of the outer part of the regions in velocity space. This means the central part of the regions remains quiescent with respect to the outer parts, and this is consistent with the findings by Kirk et al. (2010), Walsh et al. (2004) and André et al. (2007).
We assess the importance of gravity in these regions with the Gvirial method. In our method, a larger Gvirial is related to a larger importance of gravity, and as a result the importance of gravity at a given region can be quantified by measuring the amount of gas at different thresholds of the Gvirial parameter. In Fig. 10, we plot the dependence of gas mass on the Gvirial threshold for different regions, and in Fig. 11 we plot the dependence of the Gvirial on the radius. For all the regions, the Gvirial increases towards the centers. For clusterbearing regions such as NGC 1333 in Perseus and L1688 in Ophiuchus, much higher Gvirial values have been reached at their centers. This implies that gravity is more important for the clustered mode of star formation.
With our method it is straightforward to derive relations such as the massradius and velocitysize relation. Recently, there has been growing interest in quantifying cloud structure in terms of various masssize relations (e.g., Kauffmann et al. 2010a,b). In Fig. 12 we plot the masssize relation obtained using our Gvirial method (see Sect. 5). Our masssize relation is different from that of Kauffmann et al. (2010b) since we use the Gvirial contours to define our regions, and mass is evaluated within a given region where the Gvirial is larger than the threshold. In Kauffmann et al. (2010b) the mass in evaluated within a region where the column density is larger than a given threshold. In our case, along one single line of sight, only the gas that stays within a given surface of a constant value of the Gvirial is taken into account while in Kauffmann et al. (2010b) all the mass along the line of sight is taken into account. Similar to Kauffmann et al. (2010b), we also found that clusterbearing regions such as L1688 and NGC 1333 are more massive in terms of molecular gas than the regions devoid of clusters at a give radius. In Fig. 12 we also plot the scaling relations proposed in Larson (1981, m(r) = 460 M_{⊙}(r/ pc)^{1.9}, and Kauffmann et al. (2010b, m(r) = 400 M_{⊙}(r/ pc)^{1.7} , and found that both provide approximate descriptions of the structure of the regions.
Molecular clouds are believed to be turbulencedominated, and this is mainly inferred from the fact that the clouds obey the velocitylinewidth relation (Larson 1981). In Fig. 13 we plot the 3D velocity dispersion of our regions as a function of radius. Here the 3D velocity dispersion is times the velocity dispersion evaluated in Eq. (20). The scaling relation from Larson (1981) is also plotted (L/ pc ~ 1.01 × (σ_{v}/ kms^{1})^{0.38}). In our case, the velocitylinewidth relation is evaluated for centrally condensed objects, and a smaller physical scale is related to the inner part of a gas condensation, whereas in Larson (1981) a smaller physical scale is related to a smaller size of a subregion in a cloud^{11}. For our centrally condensed objects, the scaling relation from Larson (1981) seems to be valid. We also note that clusterbearing regions such as NGC 1333 and L1688 have much larger velocity dispersions at large radii.
Heyer et al. (2009) found a dependence of the scaling coefficient σ_{v}/r^{1 / 2} of Larson’s relation on the column density, and attributed this to the fact that the clouds are in selfgravitational equilibrium. We plot σ_{v}/r^{1 / 2} as a function of column density Σ ≡ M/πr^{2}, where M is the mass and r is the radius of our regions (Fig. 14). Since we decompose one region into a set of nested subregions based on the Gvirial, one single region will appear as a curve in Fig. 14. All our regions are quite close to being gravitationally bound, and the scatter around the yaxis is much smaller than what is found for Milky Way molecular clouds as studied in Heyer et al. (2009). This difference arises mainly from the fact that the Gvirial method tends to identify regions that are gravitationally coherent.
Fig. 13 3D velocity dispersion as a function of radius for different regions. The velocitylinewidth relation from Larson (1981) is also plotted. 

Open with DEXTER 
Fig. 14 σ_{v}/r^{1 / 2} as a function of column density σ for the regions. In order to be consistent with Heyer et al. (2009), σ_{v} is the 1D velocity dispersion as defined in Eq. (20) and Σ ≡ M/πr^{2} where M is the mass r is the radius. The solid straight line shows the boundary below which the structures are gravitationally bound. It is defined as σ_{v}/r^{1 / 2} = (πG/ 5)^{1 / 2}Σ^{1 / 2}. The gray crosses come from the catalogue of giant molecular clouds studied in RomanDuval et al. (2010). 

Open with DEXTER 
7. Conclusion
7.1. Results and perspectives
In this paper we proposed a general method (Gvirial) for studying the structure molecular clouds. Different from the previous methods which decompose molecular clouds based on contours in the 2D plane or 3D PPV data cubes, in this method a map is generated by taking all the gravitational interactions between all the voxels in the 3D data cubes into account.
The generated 3D Gvirial maps have a dimensionless unit that is the same as that of the virial parameter, and a larger Gvirial is related to a larger chance of being gravitationally bound. Therefore, the method provides a global picture of gravity in the PPV space. Using a hydrodynamical simulation model and a simple sphericalsymmetric model, we demonstrate the connection between Gvirial calculated in the observed PPV data cube and the Gvirial calculated from the model. We found that the two are positively related. A larger Gvirial is therefore linked to a larger chance of being gravitationally bound. Different from the virial parameter which quantifies selfgravity, the Gvirial quantifies the effect of global gravity on the gas where all the interactions between all the particles in the 3D data cubes have been taken into account.
A map of this kind enables us to to identify regions in molecular clouds in terms of gravitationally coherent regions. Compared with previous ways of defining regions based on emission, the regions identified from the Gvirial maps more coherent and are less dependent on the technical parameters. Another advantage of such a definition is that the regions identified are coherent under gravity and are likely to collapse on their own.
We also demonstrated how to use the Gvirial maps to quantify the structures of the identified gas condensations in terms of massradius and velocity dispersionradius relations. We found that both the massradius relation and the velocitydispersion relation of the model can be reconstructed with the help of the Gvirial with a good fidelity when the model is symmetric. If the model is not symmetric, the Gvirial map can used to quantify the structures of the gas condensations, and it allows us to compare different structures.
As examples, we analyzed the ^{13}CO(1–0) emission from the Perseus and Ophiuchus molecular clouds, and found that both can be decomposed into several regions which are gravitationally coherent (Fig. 8). Moreover, the clusterbearing regions show higher values of Gvirial at the centers which implies that gravity plays a more prominent role in these parts (Fig. 11). We carried out an analysis of a total of five regions identified in the Perseus and Ophiuchus molecular cloud, and derived masssize relations and velocity dispersionsize relations for the regions. We also found that clusterbearing regions are more massive at a given radius than those which do not bear a cluster (Fig. 12), and the clusterbearing regions have a higher velocity dispersion at the outer parts (Fig. 13).
The method is general and can be applied to a variety of objects observed in 3D PPV space where gravity is supposed to play a role, and from the observations the method offers ways to quantify and compare these structures. We leave these possibilities for further explorations.
The importance of gravity in molecular clouds remains unclear. The effect of gravity as compared to kinetic motion is usually quantified using the virial parameter. The major uncertainty of the virial parameter comes from the definition of boundaries the regions. With the Gvirial method, we can eliminate this uncertainty by focusing on the regions that are gravitationally coherent. As shown in Fig. 14, the Gvirialdefined regions are much closer to being gravitationally bound than the giant molecular clouds studied in Heyer et al. (2009). A study of a larger sample of molecular clouds can potentially tell if gravity is important throughout the Milky Way molecular clouds, and we leave this for a further study.
7.2. Caveats
Like all the other methods that quantify the ISM structures, our method is influenced by the lineofsight confusion. One advantage of our method is that the lineofsight confusion can be reduced by taking the velocity information into account. For a single object, if the velocity dispersion increases with radius, the Gvirial method can take this into account and reduce the lineofsight confusion effect. If different objects have different velocities, they will be separated easily. If different objects with comparable mass velocity are found on the same line of sight, the Gvirial method becomes inaccurate.
The Gvirial is a relative measure of the gravitational boundedness. As a result, it is not possible to tell whether a region is gravitationally bound based on the Gvirial. However, a larger Gvirial value means a region is more likely to be gravitationally bound. It is also possible to derive the virial parameter afterwards based on the isoGvirial contours.
Here σ is the velocity dispersion and M is the mass (Bertoldi & McKee 1992). The surface terms and the magnetic terms are neglected. See BallesterosParedes (2006) for a thorough discussion.
Such regions have been defined with the clumpfind (Williams et al. 1994) and the dendrogram (Rosolowsky et al. 2008) programs in the past.
In Larson (1981) the index is 0.38. In RomanDuval et al. (2011) the index is found to be 0.51.
Here our virial parameter is E_{p}/ 2E_{k}, and in Bertoldi & McKee (1992) the virial parameter is 2E_{k}/E_{p}.
The simulations are available at http://starformat.obspm.fr/starformat/documentation.jsp
To make the Gvirialderived relations overlap with the model values, we scaled the estimated radius by a factor of 1.4 and the estimated velocity dispersion by a factor of 1.45. This adjustment is made in order to make it easier to compare different curves. In practice, the radii of the estimated regions are based on the tensor of second moments of the position coordinates, and the radii do not necessarily coincide with the radii defined in the model. A similar argument also applies to the estimation of the velocity dispersion.
Avaliable at https://dendrograms.readthedocs.org/en/latest/
This is similar to the case of Smith et al. (2009) where they identified structures form a simulation in the 3D positionpositionposition space based on gravitational potential.
The physical scale here can have different meanings (see Lazarian & Pogosyan 2004; Heyer & Brunt 2004; RomanDuval et al. 2011).
Available at http://ytproject.org/
Acknowledgments
GuangXing Li is supported for this research through a stipend from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. GuangXing Li thanks Dr. KeJia Lee for helping with the calculations and thanks Dr. Arnaud Belloche for discussions. We thank Eve Ostriker and Hao Gong for email exchanges. This study makes use of data from the COMPLETE survey. We also thank the yt and astrodendro teams for making their codes available, and would like to thank Thomas Robitaille for email exchanges. Dr. James Urquhart and Rosie Chen are acknowledged for careful readings of the draft and for their helpful comments. We thank the anonymous referee for several thorough and careful reviews of the paper and his/her insightful comments, and thank Malcolm Walmsley for his efforts.
References
 André, P., Belloche, A., Motte, F., & Peretto, N. 2007, A&A, 472, 519 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 BallesterosParedes, J. 2006, MNRAS, 372, 443 [NASA ADS] [CrossRef] [Google Scholar]
 Beaumont, C. N., Offner, S. S. R., Shetty, R., Glover, S. C. O., & Goodman, A. A. 2013, ApJ, 777, 173 [NASA ADS] [CrossRef] [Google Scholar]
 Bertoldi, F., & McKee, C. F. 1992, ApJ, 395, 140 [NASA ADS] [CrossRef] [Google Scholar]
 Blake, G. A., Sutton, E. C., Masson, C. R., & Phillips, T. G. 1987, ApJ, 315, 621 [NASA ADS] [CrossRef] [Google Scholar]
 de Geus, E. J., de Zeeuw, P. T., & Lub, J. 1989, A&A, 216, 44 [NASA ADS] [Google Scholar]
 Dib, S., Bell, E., & Burkert, A. 2006, ApJ, 638, 797 [NASA ADS] [CrossRef] [Google Scholar]
 Enoch, M. L., Young, K. E., Glenn, J., et al. 2006, ApJ, 638, 293 [NASA ADS] [CrossRef] [Google Scholar]
 Federrath, C., Klessen, R. S., & Schmidt, W. 2008, ApJ, 688, L79 [NASA ADS] [CrossRef] [Google Scholar]
 Federrath, C., RomanDuval, J., Klessen, R. S., Schmidt, W., & Mac Low, M.M. 2010, A&A, 512, A81 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Goldsmith, P. F., Heyer, M., Narayanan, G., et al. 2008, ApJ, 680, 428 [NASA ADS] [CrossRef] [Google Scholar]
 Gong, H., & Ostriker, E. C. 2011, ApJ, 729, 120 [NASA ADS] [CrossRef] [Google Scholar]
 Goodman, A. A., Rosolowsky, E. W., Borkin, M. A., et al. 2009, Nature, 457, 63 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Gutermuth, R. A., Megeath, S. T., Myers, P. C., et al. 2009, ApJS, 184, 18 [NASA ADS] [CrossRef] [Google Scholar]
 Heyer, M. H., & Brunt, C. M. 2004, ApJ, 615, L45 [NASA ADS] [CrossRef] [Google Scholar]
 Heyer, M., Krawczyk, C., Duval, J., & Jackson, J. M. 2009, ApJ, 699, 1092 [NASA ADS] [CrossRef] [Google Scholar]
 Kauffmann, J., Pillai, T., Shetty, R., Myers, P. C., & Goodman, A. A. 2010a, ApJ, 712, 1137 [NASA ADS] [CrossRef] [Google Scholar]
 Kauffmann, J., Pillai, T., Shetty, R., Myers, P. C., & Goodman, A. A. 2010b, ApJ, 716, 433 [NASA ADS] [CrossRef] [Google Scholar]
 Kauffmann, J., Pillai, T., & Goldsmith, P. F. 2013, ApJ, 779, 185 [NASA ADS] [CrossRef] [Google Scholar]
 Kirk, H., Pineda, J. E., Johnstone, D., & Goodman, A. 2010, ApJ, 723, 457 [NASA ADS] [CrossRef] [Google Scholar]
 Langer, W. D., & Penzias, A. A. 1990, ApJ, 357, 477 [NASA ADS] [CrossRef] [Google Scholar]
 Larson, R. B. 1981, MNRAS, 194, 809 [NASA ADS] [CrossRef] [Google Scholar]
 Lazarian, A., & Pogosyan, D. 2004, ApJ, 616, 943 [NASA ADS] [CrossRef] [Google Scholar]
 McKee, C. F., & Tan, J. C. 2003, ApJ, 585, 850 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Men’shchikov, A., André, P., Didelon, P., et al. 2010, A&A, 518, L103 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Pichardo, B., VázquezSemadeni, E., Gazol, A., Passot, T., & BallesterosParedes, J. 2000, ApJ, 532, 353 [NASA ADS] [CrossRef] [Google Scholar]
 Ridge, N. A., Di Francesco, J., Kirk, H., et al. 2006, AJ, 131, 2921 [NASA ADS] [CrossRef] [Google Scholar]
 RomanDuval, J., Jackson, J. M., Heyer, M., Rathborne, J., & Simon, R. 2010, ApJ, 723, 492 [NASA ADS] [CrossRef] [Google Scholar]
 RomanDuval, J., Federrath, C., Brunt, C., et al. 2011, ApJ, 740, 120 [NASA ADS] [CrossRef] [Google Scholar]
 Rosolowsky, E. W., Pineda, J. E., Kauffmann, J., & Goodman, A. A. 2008, ApJ, 679, 1338 [NASA ADS] [CrossRef] [Google Scholar]
 Schneider, S., & Elmegreen, B. G. 1979, ApJS, 41, 87 [NASA ADS] [CrossRef] [Google Scholar]
 Shetty, R., Collins, D. C., Kauffmann, J., et al. 2010, ApJ, 712, 1049 [NASA ADS] [CrossRef] [Google Scholar]
 Simon, R., Jackson, J. M., Clemens, D. P., Bania, T. M., & Heyer, M. H. 2001, ApJ, 551, 747 [NASA ADS] [CrossRef] [Google Scholar]
 Smith, R. J., Clark, P. C., & Bonnell, I. A. 2009, MNRAS, 396, 830 [NASA ADS] [CrossRef] [Google Scholar]
 Turk, M. J., Smith, B. D., Oishi, J. S., et al. 2011, ApJS, 192, 9 [NASA ADS] [CrossRef] [Google Scholar]
 Walsh, A. J., Myers, P. C., & Burton, M. G. 2004, ApJ, 614, 194 [NASA ADS] [CrossRef] [Google Scholar]
 Whiting, M. T. 2012, MNRAS, 421, 3242 [NASA ADS] [CrossRef] [Google Scholar]
 Williams, J. P., de Geus, E. J., & Blitz, L. 1994, ApJ, 428, 693 [NASA ADS] [CrossRef] [Google Scholar]
 Williams, J. P., Blitz, L., & McKee, C. F. 2000, Protostars and Planets IV, 97 [Google Scholar]
Online material
Movie 1 (Access here)
Appendix A: Relation between Gvirial and the virial parameter
The relative importance between gravitational and kinetic energy is usually characterized with the virial parameter, which is introduced in Bertoldi & McKee (1992). To evaluate the virial parameter, it is necessary to define a structure on which the virial parameter is calculated.
One difference between Gvirial and the virial parameter is that to evaluate the Gvirial no such boundary is needed. Therefore the Gvirial can be viewed as a generalization of the virial parameter to a continuous distribution of mass.
To illustrate the physical meaning of the Gvirial, we define a quantity called particle virial, pvirial, which is the ratio between the gravitational energy of a particle to its kinetic energy α_{p−virial} = E_{p}/ 2E_{k}. The potential energy E_{p} is determined by m × φ where m is the mass of the particle and φ is the gravitation potential, and E_{k} is determined as 1 / 2 × m(v − v_{c})^{2}. To define the kinetic energy, the velocity of the center of mass v_{c} is needed. Therefore pvirial is only suitable in the cases where a center of mass can be easily found. In the case of a molecular cloud, this is not straightforward because inside a molecular cloud the condensations can move at different velocities. The velocity of the center of mass depends on which clumps are included in the calculation, which is not unique. This difficulty is illustrated in Fig. A.1.
Fig. A.1 The difficulty of finding a proper center of mass. We consider three clumps p1, p2, p3 moving at velocities v1, v2, v3. If we are interested in whether i is gravitationally bound, since the majority of attraction comes from p1, physically the center of mass should be the center of mass of p1 instead of the center of mass of the three clumps p1, p2, p3, since p2 and p3 do not contribute much gravitational attraction to i compared to p1. Therefore to estimate the gravitational boundedness, a naive calculation of the center of mass where all the mass in the whole cloud is included is not appropriate. In the case of a complicated distribution of gas, finding the center of mass is not straightforward. 

Open with DEXTER 
One major advantage of the Gvirial over the pvirial is that no center of mass is needed. The Gvirial is the sum of the gravitational boundedness, and to evaluate it we do not need to introduce the center of mass. Therefore, the Gvirial can provide estimates of the importance of gravity when the geometry of the object is complicated and the center of mass is not well defined.
Appendix B: Dependence on of Gvirial the smoothing velocity c_{0}
The only free parameter in our method is the smoothing velocity c_{0} in Eq. (6). We added this parameter to avoid the divergence of Eq. (6) when the separation of velocity is zero.
Physically, if the velocity separation is zero, and if two gas particles spatially coincide with each other, the gravitational interaction will cause them to collide. However, the velocity separation cannot be zero since the gas also has velocity dispersions that have either thermal or nonthermal origin (Larson 1981). A physically lower limit of c_{0} should be the sound speed.
In reality, this is also affected by the superposition along the line of sight, since different gases that are physically unassociated can stay along the same line of sight and therefore appear to be gravitationally bound. Therefore, it is difficult to find a unique value of c_{0} based on first principles. In our case, we choose c_{0} to be 1 km s^{1}, which is larger than and still comparable to the sound speed.
Here we investigate how our results can be affected by the parameter c_{0}. First, we made a Gvirial map by assuming c_{0} = 1 km s^{1} and then we made another map by assuming c_{0} = 2 km s^{1}. Finally, we made a voxelbyvoxel comparison of the maps (Fig. B.1). We found the major effect of changing c_{0} on the absolute values of Gvirial: if we change c_{0} to 2 km s^{1} the Gvirial decreases by a factor of ~2. However, there is good correspondence between the old and new Gvirial values. Therefore, changing c_{0} affects the absolute values of the Gvirial; however, the relative values are unaffected. Therefore, all our figures should be unaffected by a change of c_{0} and the Gvirial axes of Figs. 10 and 11 will change accordingly.
In the case where the Gvirial is applied to different data cubes, in order for the results to be comparable, a unique choice of c_{0} is necessary.
Fig. B.1 Comparison of Gvirial map of the Ophiuchus molecular cloud under different c_{0}. The horizontal axis is the Gvirial calculated assuming c_{0} = 1 km s^{1} and the vertical axis is the Gvirial calculated assuming c_{0} = 2 km s^{1}. The grayscale image stands for the number of voxels that fall into each bin. 

Open with DEXTER 
Appendix C: 3D renderings of Gvirial of the Perseus and Ophiuchus molecular clouds
Fig. C.1 Volume rending representations of the ^{13}CO(1–0) emission and the corresponding Gvirial map. The upper panel shows the result from the Perseus molecular cloud and the lower panel shows the result from the Ophiuchus molecular cloud. Blue stands for ^{13}CO(1–0) emission and red and orange stand for Gvirial. Both clouds are projected along the velocity direction. For the Perseus molecular cloud, the isosurfaces of ^{13}CO(1–0) emission start from 0.3 K and increase in steps of 0.64 K. The contours of the Gvirial starts from 1.2 and increase in steps of 0.8. For the Ophiuchus molecular cloud, the contours of ^{13}CO(1–0) emission start from 0.3 K and increase in steps of 1.24 K. The contours of the Gvirial starts from 1.2 and increase in steps of 0.8. 

Open with DEXTER 
In order to provide intuitive representations of our Gvirial maps, we present volume renderings of the 3D PPV data cubes with yt (Turk et al. 2011)^{12} in Fig. 7. A movie can be found in the electronic edition of the journal.
Appendix D: Comparison with other methods
The previous methods to quantify the structure of molecular condensations such as Clumpfind and Dendrogram focus on the structure traced by the intensity map. The Clumpfind method tends to produces isolated structures. The Dendrogram produces hierarchical representations of the nested isosurfaces in 3D molecular line data cubes, and offers multiscale decompositions. It can be applied to a much larger variety of situations. Both methods work in 2D (positionposition) and 3D (PPV) space and both methods decompose the data to some extent.
The Gvirial method presented in this paper is also based on a map of intensity in 3D PPV space. However, its output is neither a list of structures (as in the case of the Clumpfind) nor a tree representation of a hierarchy of structures (as in the gas of the Dendrogram). Instead, it produces a map of the importance of gravity in the 3D PPV. The output is also a map 3D PPV space. The Gvirial map itself does not provide a decomposition of the data. To decompose the cloud into regions and to analyze their properties, other methods are needed.
In this work, we use the Dendrogram to identify gravitationally coherent regions from the Gvirial map, and use the isoGvirial contours to quantify the properties of the regions in the Mr and σ_{v}r plane. The Gvirial method is not a replacement of the other methods, but it provides a new map on which those methods could be applied.
All Figures
Fig. 1 Setup of the problem. We consider a mass distribution in a 3D PPV data cube. This is represented as the black grid. x–y represents the spatial dimensions and v represents the velocity dimension. The mass distribution is represented in blue. For each voxel i, its Gvirial can be determined by taking its interactions with all the voxels j (not excluding i) into account. See Sect. 2 for details. 

Open with DEXTER  
In the text 
Fig. 2 a) Map of the peak brightness temperature of the simulated ^{13}CO(1–0) emission along the velocity axis. b) Positionvelocity map of peak brightness temperatures of ^{13}CO(1–0) along the Yaxis. c) Positionvelocity map of peak brightness temperatures of ^{13}CO(1–0)) along the Xaxis. 

Open with DEXTER  
In the text 
Fig. 3 Comparison between the Gvirial_{model} calculated from the model and the Gvirial_{reconstructed}, which is reconstructed from the simulated observations in the PPV space. Panel a) shows the result for the hydrodynamic simulation; and panel b) shows the result for the analytical model with γ_{ρ} = −2.2 and γ_{v} = 0.5. The colors stand for the amount of mass in a given interval. 

Open with DEXTER  
In the text 
Fig. 4 Comparison between between Gvirial_{model} and Gvirial_{reconstructed} in the PPV space for the hydrodynamical model described in Sect. 5.2. In the model, turbulence is injected through compressible forcing, and no selfgravity is included (Federrath et al. 2008). The Gvirial_{model} in the PPV space is defined as the massweighted average of the Gvirial_{model} in the PPP space, which is defined in 12. Here, the grayscale image is the simulated ^{13}CO(1–0) emission from the model. The red contours stand for the Gvirial_{model}, and the contour levels are (6, 5, 10). The blue contours stand for the Gvirial_{reconstructed}, and the contour levels are (3, 6). It worth noting that the Gvirial_{model} and the Gvirial_{reconstructed} agrees better at regions where mass is traced. This can be seen in the v = −0.2 km s^{1} channel. They agree better at the regions where mass is traced (e.g., the red arrow), but does not agree well at the region without mass (e.g., the blue arrow). 

Open with DEXTER  
In the text 
Fig. 5 Comparison between the structure of the model and the structure reconstructed through the Gvirial for different sets of model parameters. The upper panels show the results where the analytical model is constructed with different γ_{ρ}, and the lower panels show the results where the analytical model is constructed with different γ_{v}. The left panels are the massradius relations and the right panels are the velocity dispersionradius relations. Here the radius is scaled up by a factor of 1.4 and the velocity dispersion is scaled up by a factor of 1.45 for an easy comparison. 

Open with DEXTER  
In the text 
Fig. 6 Gvirial as a function of radii for models with different γ_{ρ} and γ_{v}. 

Open with DEXTER  
In the text 
Fig. 7 Channel maps of the ^{13}CO(1–0) emission and the corresponding Gvirial. The upper panel shows the results from the Perseus molecular cloud and the lower panel shows the results from the Ophiuchus molecular cloud. Grayscale images stand for ^{13}CO(1–0) emission and the red contours represents the Gvirial. Contour levels start from 1.2 in steps of 0.8. 

Open with DEXTER  
In the text 
Fig. 8 Regions identified by applying the dendrogram algorithm to our Gvirial map. The results from the Perseus molecular cloud (upper panel) and Ophiuchus molecular cloud (lower panel) are presented. The grayscale images correspond to the velocityintegrated ^{13}CO(1–0) emission and the contours correspond to the projected boundaries of the identified regions. These regions correspond to the “leaves” of the dendrogram. The conventional names of the regions are labeled. 

Open with DEXTER  
In the text 
Fig. 9 Dependence of velocity and velocity dispersion as a function of radius. Here the solid lines represent the velocities of the regions, and the bars represent the velocity dispersions of the regions at given radii. The vertical extent of the errorbars are the 3D velocity dispersion . 

Open with DEXTER  
In the text 
Fig. 10 Amount of molecular gas enclosed in regions with different thresholds of the Gvirial. The xaxis is the threshold of Gvirial which we take to define the region, and the yaxis is the amount of gas that is enclosed in the region. 

Open with DEXTER  
In the text 
Fig. 11 Gvirial as a function of radius for different regions. 

Open with DEXTER  
In the text 
Fig. 12 Enclosed mass as a function of radius for different regions. The scaling relation from Larson (1981) and Kauffmann et al. (2010b) are added. For all the curves, the Gvirial decreases with increasing radii. 

Open with DEXTER  
In the text 
Fig. 13 3D velocity dispersion as a function of radius for different regions. The velocitylinewidth relation from Larson (1981) is also plotted. 

Open with DEXTER  
In the text 
Fig. 14 σ_{v}/r^{1 / 2} as a function of column density σ for the regions. In order to be consistent with Heyer et al. (2009), σ_{v} is the 1D velocity dispersion as defined in Eq. (20) and Σ ≡ M/πr^{2} where M is the mass r is the radius. The solid straight line shows the boundary below which the structures are gravitationally bound. It is defined as σ_{v}/r^{1 / 2} = (πG/ 5)^{1 / 2}Σ^{1 / 2}. The gray crosses come from the catalogue of giant molecular clouds studied in RomanDuval et al. (2010). 

Open with DEXTER  
In the text 
Fig. A.1 The difficulty of finding a proper center of mass. We consider three clumps p1, p2, p3 moving at velocities v1, v2, v3. If we are interested in whether i is gravitationally bound, since the majority of attraction comes from p1, physically the center of mass should be the center of mass of p1 instead of the center of mass of the three clumps p1, p2, p3, since p2 and p3 do not contribute much gravitational attraction to i compared to p1. Therefore to estimate the gravitational boundedness, a naive calculation of the center of mass where all the mass in the whole cloud is included is not appropriate. In the case of a complicated distribution of gas, finding the center of mass is not straightforward. 

Open with DEXTER  
In the text 
Fig. B.1 Comparison of Gvirial map of the Ophiuchus molecular cloud under different c_{0}. The horizontal axis is the Gvirial calculated assuming c_{0} = 1 km s^{1} and the vertical axis is the Gvirial calculated assuming c_{0} = 2 km s^{1}. The grayscale image stands for the number of voxels that fall into each bin. 

Open with DEXTER  
In the text 
Fig. C.1 Volume rending representations of the ^{13}CO(1–0) emission and the corresponding Gvirial map. The upper panel shows the result from the Perseus molecular cloud and the lower panel shows the result from the Ophiuchus molecular cloud. Blue stands for ^{13}CO(1–0) emission and red and orange stand for Gvirial. Both clouds are projected along the velocity direction. For the Perseus molecular cloud, the isosurfaces of ^{13}CO(1–0) emission start from 0.3 K and increase in steps of 0.64 K. The contours of the Gvirial starts from 1.2 and increase in steps of 0.8. For the Ophiuchus molecular cloud, the contours of ^{13}CO(1–0) emission start from 0.3 K and increase in steps of 1.24 K. The contours of the Gvirial starts from 1.2 and increase in steps of 0.8. 

Open with DEXTER  
In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.