RJ-plots: An improved method to classify structures objectively

An automated technique to identify and classify the morphology of 2D structures called J-plots is presented in Jaffa et al. (2018). This technique constructs two quantities, called J-values, by first calculating the principal moments of inertia of a structure, I₁ and I₂, and comparing it to an equivalent circle of equal area and total weight¹¹1Jaffa et al. (2018) refer to this quantity as the mass of the structure as they are concerned with structures identified in column density images. We adopt total weight due to its generality., $a$ and $W$. The total weight of a structure is the integral of the weighting of each pixel (e.g. column density or intensity); the exact quantity is unimportant. The equivalent circle has a moment of inertia of:

The J-values may thus be defined as:

By construction J₂ $\leq$ J₁. These two quantities are plotted against each other to produce a J-plot which may be used for classification (an example is the left panel of figure 2). A structure which has uniform surface density and is perfectly circular has J₁=J₂=0 and lies at the origin of a J-plot; a centrally concentrated quasi-circular structure has J₁ $>$ 0, J₂ $>$ 0 and lies in the upper-right quadrant of a J-plot (a J-core); a centrally under-dense quasi-circular structure has J₁ $<$ 0, J₂ $<$ 0 and lies in the bottom-left quadrant of a J-plot (a J-bubble); and an elongated structure has J₁ $>$ 0, J₂ $<$ 0 and lies in the bottom-right quadrant of a J-plot (a J-filament).

This method provides a simple and efficient method for the classification of structures in simulations, real observations and synthetic observations (Clarke et al., 2018; Petkova et al., 2021; Neralwar et al., 2022). However, due to the construction that J₂ $\leq$ J₁, only half of the J-space may be occupied; importantly while the entire bottom-right quadrant (elongated structures) may be filled, only half of the upper-right and bottom-left (centrally concentrated and rarefied structures respectively) may be. This leads to a bias towards identifying structures as filaments and against identifying ring-like or concentrated structures.

To help compensate for the above-mentioned biases, we develop and present an improved version of J-plots called RJ-plots. This paper is structured in the following manner: in section 2 we describe the improvements made to J-plots as well as test and calibrate the RJ-plot technique against a number of basic structure shapes; in section 3 we show a selection of applications of RJ-plots using both observational data (section 3.1) and simulation data (section 3.2); and in section 4 we conclude.

To demonstrate how the characteristics of a structure influence its resulting J/RJ-values, it is constructive to use a set of readily parametrisable basic shapes. Here we use ellipsoidal Plummer-like density profiles to produce a range of shapes parametrized using their aspect ratio, $A$, and their central density $\rho_{c}$. Examples of these shapes can be seen in figure 1.

When constructing the shapes we use 1000x1000 pixel images to minimise grid artefacts from attempting to capture curved shapes with a Cartesian grid. The surface density of a shape is given as:

otherwise the density is set to zero. Here the aspect ratio A is define as $x_{0}/y_{0}$ and $y_{0}$ is set to 8 pixels, allowing aspect ratios up to 15 to be considered. One can readily see that increasing $\rho_{c}$ produces a centrally over-dense structure while setting $\rho_{c}$ to a negative value produces a centrally under-dense structure. Taking the maximum between the Plummer-like profile and zero avoids negative surface densities when $\rho_{c}$ $<$ 0. These shapes are not to be considered necessarily typical shapes identified in the ISM, but are readily parametrisable and understood, allowing for a method to investigate the sensitivities of using moment of inertia based classification schemes.

The left panel of figure 2 shows the J-plot resulting from considering shapes with 1 $\leq$ A $\leq$ 15 in steps of 0.5, and with -100 $\leq$ $\rho_{c}$ $\leq$ 900 in steps of 5. Each ‘track’ in the J-plot is the result of a shape with constant aspect ratio and varying central density. The results from shapes with A=1 lie along the J₂=J₁ line, with centrally over-dense structures ($\rho_{c}$ $>$ 0) in the upper-right quadrant and centrally under-dense structure ($\rho_{c}$ $<$ 0) in the bottom left quadrant. Increasing aspect ratio leads to shapes moving into the bottom-right quadrant, with the $\rho_{c}$=0 shapes moving approximately along the J₂=-J₁ line.

By considering these basic shapes the biases in J-plots are apparent. Even small amounts of elongation moves structures quickly into the bottom-right elongated structure quadrant, with an A=1.5 and $\rho_{c}$=0 structure having (J₁,J₂) $\sim$ (0.25,-0.25), leading to a strong preference to classify objects as filaments. Furthermore, with increasing aspect ratio it becomes increasingly difficult for structures which are centrally over/under-dense to be correctly classified as they move more deeply into the elongated structure quadrant. Moreover, as $|$J${}_{i}|$ $\leq$ 1, with increasing aspect ratio the available J-space to express differences in central density becomes increasingly constrained.

To address these limitations we propose that a more appropriate method for structure classification is to use rotated J-values, or RJ-values for short. As the two lines J₂=J₁ and J₂=-J₁ are seen to be important with shapes moving along them as their elongation and central density is modified, a rotation of 45 degrees anti-clockwise may be performed such that these lines become two orthogonal axes, i.e:

With this step the two sets of information are quasi-separated with R₁ encoding elongation and R₂ encoding central under/over-density. Due to $|$J${}_{i}|$ $\leq$ 1 there exist a maximum value of R₂ as a function R₁, i.e. R₂ = $\sqrt{2}-$ R₁, reducing the range of space to express differences in central density as elongation increases. This may be removed by normalising R₂ by this maximum value such that:

This results in a space, 0 $\leq$ R₁ $\leq$ $\sqrt{2}$ and -1 $\leq$ R₁ $\leq$ 1, in which any point may be occupied by a shape. Being rotated and re-scaled values of J-values, RJ-values do not contain additional information about a shape’s morphology but, as shown below, provide more explicit and easy to interpret results compared to J-values.

The RJ-plot resulting from considering the shapes described in section 2.1 can be seen in the right-panel of figure 2. One sees that each track resulting from shapes with a given aspect ratio leads to a parabola; thus the two RJ-values do not completely separate the elongation information from the central density. However, the parabolae are shallow, showing a near separation of information, and that there is considerable improvement compared to J-values. Note also that neither axis responds linearly to changes in aspect ratio or central density, but they do respond monotonically.

It is important to discuss how best to interpret the value of R₂. While the basic shapes considered have either a single high-density region at their centre or a completely hollow, enclosed interior, this dichotomy is not what R₂ is measuring. Rather, this quantity measures how weight is distributed with respect to the centre of weight of the object. Shapes where weight is predominately close to but not necessarily on top of the centre of weight of the object, e.g. a structure with multiple high-density clumps close to but not at its centre, will have high R₂ values, while structures where weight is predominately located far from the centre of weight, e.g. a highly curved filament where the centre of weight lies outside of the structure, will have low R₂ values. This point can be seen in figure 3 which shows how sectors of an annulus move in RJ-space as the angle of the sector varies. When the angle is low, the centre of weight remains inside of the sector and R₂ $\sim$ 0. As the angle increases the sector appears increasingly filamentary as R₁ increases until at $\theta$ $\sim$ 60$\degree$ the centre of weight begins to move outside of the structure and R₂ decreases. As the angle continues to increase, the centre of weight shifts until it reaches approximately (0,0), at which point R₂ reaches its minimum and R₁ begins to decrease. When the annulus is complete it reaches R₁=0 and an R₂ value determined by its thickness.

Considering the discrete classification of structures using the RJ-plot, one can readily split the space into four sections. To separate between elongated and quasi-circular structures we use the parabola corresponding to the A=2 shapes; the exact choice of aspect ratio is arbitrary as any value greater than 1 is by definition elongated. The resulting fit yields the equation:

which can be seen as the red line in figure 2. Thus, a structure where R₁ $>$ R${}_{1}^{E}$ is regarded as elongated and regarded as quasi-circular when the inverse is true. Further, the R₂=0 line may be used to separate structures which are centrally over-dense (R₂ $>$ 0) from those which are centrally under-dense (R₂ $<$ 0). Note that as shown in figure 3, structures with R₁ $>$ R${}_{1}^{E}$ and R₂ $<$ 0, are most likely curved filaments and partial shells rather than strongly elongated bubbles, while half-to-complete shells lie in the region R₁ $<$ R${}_{1}^{E}$ and R₂ $<$ 0.

The Structure, Excitation and Dynamics of the Inner Galactic InterStellar Medium (SEDIGISM) survey was performed using the Atacama Pathfinder Experiment (APEX) telescope from 2013-2016 (Schuller et al., 2017, 2021). The survey covered the range -60$\degree$ $\leq$ $l$ $\leq$ 18$\degree$ and $|b|$ $\leq$ 0.5$\degree$, using the molecular emission lines ¹³CO and C¹⁸O (J=2-1). Here we use the SEDIGISM catalogues from Duarte-Cabral et al. (2021), Urquhart et al. (2021) and Neralwar et al. (2022), which together provide a sample of 10,663 clouds with associated properties such as dense gas fraction, as well as their J-values and visually determined morphology classification.

To highlight a multi-scale approach to the use of RJ-plots we use simulation data from Clarke, Williams & Walch (2020) to investigate the morphology of sub-structures within a non-equilibrium, accreting filament. These simulations were performed using the moving-mesh code Arepo (Springel, 2010). The code solves the three-dimensional hydrodynamic equations including self-gravity with time-dependent coupled chemistry and thermodynamics. The initial set-up is a cylindrical, turbulent colliding flow; the set-up quickly forms a dense filament at the $z$-axis. In total 10 simulations were performed with identical properties, only the random seed used to generate the turbulent velocity field being changed. Accretion from the colliding flows drives internal turbulence and the filaments fragment to form numerous cores as well as filamentary sub-structures (see Clarke et al., 2017, 2018; Clarke, Williams & Walch, 2020, for the formation process and properties of these sub-structures). The simulations are stopped when $\sim$5 - 10$\%$ of the mass of the filament is contained in sink particles and at this point the simulations are analysed. This occurs about 0.5 - 0.6 Myr after the start of the simulation and the filaments have reached a mass between $\sim$150 M${}_{{}_{\odot}}$ and $\sim$250 M${}_{{}_{\odot}}$. Prior to analysis, the Arepo Voronoi-mesh is mapped to a fixed Cartesian grid with resolution of 0.01 pc. For each simulation we use two viewing angles, those aligned with the $x$- and $y$-axes, to produce a total of 20 column density plots. An example column density image is shown in figure 7. For full details on the simulations see Clarke et al. (2018) and Clarke, Williams & Walch (2020).

To identify structures on multiple scales we use a dendrogram approach, specifically the Python package astrodendro²²2http://www.dendrograms.org/. The dendrograms are constructed on the logarithmic column density as the images cover a large dynamic range. The parameters used are: min$\_$value = 21.5, min$\_$delta = 0.3, min$\_$npix = 9. This value of min$\_$value is adopted as it is sufficiently high as to only include the inner filament region of the map but sufficiently low as to ensure that there is only one single dendrogram trunk, i.e. a single structure on the largest scale. The value of min$\_$delta = 0.3 corresponds to roughly a doubling in linear space; increasing its value will decrease the number of structures found between the dendrogram trunk and its leaves, while decreasing will increase this number. We consider 0.3 to be a good compromise value and one which does not greatly affect the results. Setting min$\_$npix = 9 places a minimum diameter of a structure of $\sim$0.03 pc and ensures that any structure is at least marginally resolved. The left panel of figure 7 shows the dendrogram structures identified in this manner for simulation S1 as contours, the colour of the contour denoting a structure’s level in the dendrogram.

We calculate the RJ-values of each of the structures; this is done using the linear column density image of each structure. The RJ-plot generated from considering the structures in simulation S1 can be seen in the right panel of figure 7. There one can see that the majority of the structures are elongated, with few lying to the left of the red boundary line. Further, higher level dendrogram structures (i.e. smaller and more deeply embedded) are more likely to be concentrated, while lower level structures appear as curved elongated structures (R₁ $>$ R${}_{1}^{E}$, R₂ $<$ 0), showing a distinction between the larger-scale filaments and the smaller-scale cores.

Considering all 20 column density maps results in 396 total structures with masses ranging from 0.36 to 354.3 M${}_{{}_{\odot}}$ and equivalent radii from 0.02 - 0.88 pc. A structure’s equivalent radius is defined as $r=\sqrt{a/\pi}$, where $a$ is the structure’s area; this is only approximate due to the non-circular nature of the structures but may be used to estimate the size of a structure. Figure 8 shows the RJ-plots containing all 396 structures, colour-coded by the structure size and average surface density. One can see that the majority of structures are elongated ($\sim$80%) as well as the fact that there exists two trends in RJ-space. The first set of structures extends vertically between 0.5 $\lesssim$ R₁ $\lesssim$ 1.1 and have relatively large radii (r $\sim$ 0.4 pc) and moderate surface densities ($\Sigma\sim 500$ M/pc²). These structures lie in the same region as the sectors of an annulus, as considered in figure 3, and are regions of the filament which are highly curved. As an example see the southern section of the filament in figure 7. The second set of structures lie close to R₂ $\sim$ 0 but show a clear anti-correlation between R₁ and R₂, i.e. less elongated structures are more centrally concentrated. These structures are typically small (r $\lesssim$ 0.1 pc), have high surface densities ($\Sigma\gtrsim 1000$ M/pc²), and are associated with the small core structures in the filament.

These trends across scales can be seen in figure 9 which shows the cumulative distributions of sizes and average surface densities of structures split into two sets of binary classifications: elongated vs quasi-circular, and centrally over-dense vs. under-dense. Quasi-circular structures are both smaller and denser than elongated structures, and the same is true of centrally over-dense structures compared to centrally under-dense structures. In the right panel of figure 9 is shown the fraction of structures below a given equivalent radius which are quasi-circular compared to elongated. As one moves to smaller scales, quasi-circular structures become more common (reaching $\sim 40\%$ at 0.03 pc)³³3At this small scale, structures contain approximately 30 pixels. Such a reduced number of pixels may affect measurements of the central concentration due to the central area being poorly resolved; however, it ought to have minimal effect on the elongation as a range of aspect ratios can still be achieved (i.e. 1-30). but never become the dominant structure morphology. Thus, considering fragmenting filaments, structures predominately remain filamentary across all scales and core-like structures inherit their elongation from the parental filament.