The Milky Way atlas for linear filaments II. clump rotation versus filament orientation

The unbiased linear filament sample, the first large-scale linear filaments across the full Galactic plane (Wang et al., 2024), was combined with the SEDIGISM survey (Structure, Excitation and Dynamics of the Inner Galactic Interstellar Medium), resulting eight target filaments (F2, F3, F34, F35, F36, F39, F40, and F41). The filament sample includes filaments of varying distances, lengths, masses, luminosities, and linearities (Wang et al., 2024).

Data from the SEDIGISM survey, conducted with the Atacama Pathfinder Experiment 12 m submillimetre telescope (APEX, Güsten et al., 2006) were used. The survey overview papers (Schuller et al., 2017, 2021; Duarte-Cabral et al., 2021) provide a comprehensive account of the observations, data reduction, and data-quality checks. In total, the SEDIGISM survey observed 84 deg², covering from -60${}^{\circ}{}\leq{}l{}\leq{}$+18^∘, and $|{}b{}|{}\leq{}$0.5^∘, with a few extensions in $b$ towards some regions, as well as an additional field towards the W43 region (+29${}^{\circ}{}\leq{}l{}\leq{}$+31^∘). The ¹³CO (2-1) images used here are from the DR1 data set (Schuller et al., 2021), which has a typical 1$\sigma$ sensitivity of 0.8 $-$ 1.0 K (in T_mb) per 0.25 km s^-1 channel and an FWHM beam size of 28^′′.

The filaments were identified from Hi-GAL clumps (Elia et al., 2021) using the Minimum Spanning Tree (MST) algorithm developed by Wang et al. (2016). The specialized MST algorithm is applied to identify filaments in Position-Position-Velocity space, implying that clumps are only clustered as a filament when they are in close proximity with comparable velocities. In Figure 1, the clumps are marked with circles and linked by minimizing the total length of connecting edges. The edges represented by blue solid lines are straight lines joining the clumps, and the lengths of the edges indicate the distance between each pair of clumps. The fitted filaments are marked by end-to-end lines connecting two clumps at the filament tips, shown as the black solid lines in Figure 1. Details of the identified linear filaments can be found in Wang et al. (2024). Orientations ($\theta_{f}$) of the target filaments were determined by using the positions of the clumps at the two tips, with measurements taken clockwise from the East.

For most filaments, the distances between clumps are consistent with the findings reported by Elia et al. (2021). According to Wang et al. (2024), there is no evident selection bias in the distance distributions of the target filaments, supporting the assumption that all clumps within a target filament are equidistant. Consequently the related parameters of the clumps reported in Elia et al. (2021) are scaled to derive the velocity gradient, the specific angular momentum ($J/M$), and the ratio ($\beta$) between the rotational energy and gravitational energy. The scaled properties of clumps in target filaments are listed in Table 1. Clumps, which appear as localized increases in column density within clouds, are generally assumed to be spherical in shape with rigid-body rotation, producing a linear gradient, $\mathbf{\nabla}{}v_{\mathrm{LSR}}$. In the local standard of rest (LSR) velocity field, the linear gradient aligns perpendicular to the rotation axis.

The velocity gradients were measured following the method described in Goodman et al. (1993), which fits the function

Here, $v_{LSR}$ represents an intensity weighted average velocity along the line of sight and $v_{0}$ is velocities of clumps (see column (8) in Table 1). $\Delta{}l$ and $\Delta{}b$ are the offsets from the center positions (listed in column (2) and (3) of Table 1) of clumps in the Galactic longitude and Galactic latitude in radians, respectively, while $c_{1}$ and $c_{2}$ are the projections of the gradient per radian onto the $l$ and $b$ axes, respectively. Finally, the magnitude of the velocity gradient is defined by

where $D$ is the distance (column (4) in Table 1) of clumps. Its orientation (i.e., the orientation of the increasing velocity) measured east of north is given by

Based on the spherical shape of clumps, the velocity field of the HiGAL clumps within the target linear filaments was fitted using a least-squares method as outlined in Equation (1) from the ¹³CO (2-1) images. The magnitude of the gradient ($\cal{}G$), its direction ($\theta_{\cal{}G}$), and their errors were then calculated based on the resulted $c_{1}$ and $c_{2}$ of the least-squares fit. According to the confidence-level simulation results in Goodman et al. (1993), the velocity gradient of the target clumps can be reliably fitted when they contain at least nine spatial pixels in the ¹³CO (2-1) images. One clump in each of the two filaments (F35 and F39) was not well fitted. The derived $\cal{}G$ values distribute between 7.0 $\times$ 10^-2 km s^-1pc^-1 and 4.8 km s^-1pc^-1, listed in column (4) of Table 2. Its direction, $\theta_{\cal{}G}$, are presented in column (5) of Table 2. Figure 2 shows the distribution of the obtained $\cal{}G$ values compared with the clump radius ($R$) and mass ($M$). One could note that $\cal{}G$ tends to decrease as $R$ and $M$ increase.

If molecular clouds are rotating, their current angular momentum could provide insights into their evolution processes. The specific angular momentum ($J/M$), which represents angular momentum per unit mass, is commonly used to compare angular momenta in different parts with comparable masses. For a spherical clump with a power-law density distribution, $\rho~{}\propto~{}r^{-\cal{}A}$, $J/M$ is computed as in Xu et al. (2020b):

When the density of a clump is uniform, $\cal{}A$ is set to 0. Here, $\cal{}A$ = 1.6 (Bonnor, 1956) is adopted, resulting in $J/M$ being reduced by about 30% compared to a uniform density. The assumption of spherical geometry introduces a systematic difference of 20% in the results of specific angular momentum for elongated clumps when the rotation axis is considered parallel to the clump axis. The gradient direction, which in our assumption is a direction of clump rotation, is not always parallel to either of the clump axes. In fact, the angles appear to be random, suggesting that a spherical geometry is the most reasonable assumption for our analysis. The derived $J/M$ values are listed in column (6) of Table 2 and are plotted in Figure 3(a). The derived $J/M$ values range from 8.0 $\times$ 10^-4 pc kms^-1 to 0.1 pc kms^-1. We found a monotonic increase of $J/M$ as a function of clump size ($R$), following a power-law relation $J/M~{}\propto~{}R^{1.5\pm 0.2}$, which is consistent with  Goodman et al. (1993).

The ratio ($\beta$) between rotational energy (E_r) and gravitational energy (E_g) is used to quantify the dynamical role of rotation, defined by Xu et al. (2020b) as

In this equation,

The assumption of $\cal{}A$ = 1.6 (Bonnor, 1956) leads to a 70% reduction in E_r and a 31% increase in E_g, which cumulatively results in $\beta$ being two times lower than that of a uniform sphere. The derived $\beta$ values range from 1.0 $\times$ 10^-5 to 0.1, as given in column (7) of Table 2 and plotted in Figure 3(b). The lack of variation in $\beta$ implies that it may remain constant regardless of $R$, a conclusion consistent with the results from Goodman et al. (1993). As explored in previous studies  (e.g. Goodman et al., 1993; Curtis & Richer, 2011; Xu et al., 2020b), the small values of $\beta$ signifies that rotation alone provides insignificant support against gravitational collapse. To facilitate clear comparison, the obtained values of $J/M$ and $\beta$ in this work were analyzed alongside those from previous woks, as illustrated in Figure 3(c). This discussion can be found in Section 4.1.

To quantify the alignment between the velocity gradients of clumps and their natal filaments, the relative difference between the angles, $|{}\theta_{f}-\theta_{\cal{}G}{}|$, is derived through

where ‘MIN’ refers to the minimum angular difference between $\theta_{f}$ and $\theta_{\cal{}G}$. The derived $|{}\theta_{f}-\theta_{\cal{}G}{}|$ values range from 3.9 to 89.3, as presented in column (8) of Table 2.

To investigate the distribution of $|{}\theta_{f}-\theta_{\cal{}G}{}|$, Monte Carlo simulations in three-dimensional (3D) space were conducted, following the approach described by Stephens et al. (2017). In this method, two random unit vectors are generated within a unit sphere in 3D, and the angle ($\theta_{3D}$) between these two vectors is measured. A total of 10⁶ pairs of unit vectors are generated to produce 10⁶ angles of $\theta_{3D}$, constrained to a range of 0^∘ – 90^∘. If $\theta_{3D}$ exceeds 90^∘, the values of 180^∘ - $\theta_{3D}$ values are adopted (Equation 6). Angles within 0^∘ – 20^∘ are defined as parallel, 20^∘ – 70^∘ as random, and 70^∘ – 90^∘ as perpendicular. The angles of $\theta_{3D}$ are then projected onto a two-dimensional (2D) space. Figure 4 plots the cumulative distribution function of our $|{}\theta_{f}-\theta_{\cal{}G}{}|$ and the projected $\theta_{3D}$ (the three blue dashed lines), which were applied to illustrate what our sample might look like. The consistency between the two distributions is assessed using the p-values of the Anderson–Darling (AD) test. A p-value close to 1 suggests a likely consistency between the distributions, whereas a p-value close to 0 indicates their inconsistency. For our dataset, the AD test yields a p-value of 0.98, indicating that the observed distribution of angles bears more resemblance to a random distribution. This finding suggests a non-correlation between the gradient direction of a clump and the orientation of its natal filament. Furthermore, it indicates that the rotation axis of a clump does not depend on the orientation of its natal large-scale filament.

The derived $J/M$ and $\beta$ in this work were compared with those form earlier reports, which included clumps at various evolutionary stages (e.g. Phillips, 1999), ranging from starless and pre-stellar cores in dark clouds (Goodman et al., 1993; Caselli et al., 2002; Xu et al., 2020b) to dense cores/clumps in high-mass clouds (Pirogov et al., 2003; Li et al., 2012; Tatematsu et al., 2016; Xu et al., 2020b). Notably, the $J/M$ and $\beta$ values for 30 well-resolved cores reported by Xu et al. (2020a) were obtained at the Jeans scale the Orion Molecular Cloud (OMC) 2/3 using high-resolution ALMA (Atacama Large Millimeter/submillimeter Array) N₂H⁺ images. Additionally, Xu et al. (2020a) reported a random distribution for $|{}\theta_{f}-\theta_{\cal{}G}{}|$, consistent with the findings of this study. When considering the complete sample (the values of $J/M$ and $\beta$ from this study and previous studies), $J/M$ and $\beta$ were fitted as functions of clump size $R$, as presented in Figure 3(c). A power-law relation between $J/M$ and $R$ persists across all measurements, yielding a best-fit slope of 1.58, which closely aligns with the value of 1.5 found in this study. The values of $J/M$ for our clumps, represented by green circles in Figure 3(c), are lower than those reported in other studies. This is mainly due to a consistently higher density of our HiGAL clumps , which are distributed along condensed filaments. Other clumps shown in Figure 3(c), even those from high-mass star-formation regions, are often characterized by different observational methods, primarily using CO tracing lower densities. Although $\beta$ shows significant scatter, its small value suggests that rotational energy constitutes only a minor fraction of the gravitational energy, indicating that observed rotation cannot prevent the gravitational collapse of clumps in molecular clouds.

Modifications in the alignment between a clump and its natal filament can reveal their dynamical states, such as collapse, and the manner of gas transfer within filaments. Various angles (e.g., angle among outflow axes, velocity gradients, magnetic fields, disk axes, and filament orientation) between clump and its natal filament have been statistically analyzed to explore any existing trends of alignment. For example, Planck Collaboration et al. (2016) and Soler (2019) studied the relative angles between magnetic fields at local clumps and filament orientations. They found that magnetic fields predominantly align parallel to low column density filaments, whereas they tend to be perpendicular to high column density filaments. At smaller scales, Zhang et al. (2014) found that the magnetic fields at dense core scales align either parallel or perpendicular to parsec-scale magnetic fields. Furthermore, such bimodal distribution in the angles between the momentum (outflow axes) of clumps and filament orientation has also been detected in previous works (Anathpindika & Whitworth, 2008; Wang et al., 2011; Kong et al., 2019; Anathpindika & Di Francesco, 2022). Conversely, a random distribution of the relative angles between momentum and filament orientation/magnetic field has been reported in several recent studies (Stephens et al., 2017; Punanova et al., 2018; Xu et al., 2020a; Baug et al., 2020). Our finds are consistent with this random distribution, suggesting that the relative orientation between the rotational axis of clumps and filaments might not be as deterministic as previously thought. For the special location, whether located centrally or at one tip, the most massive clump (black solid circles in Figure 1) in each filament also has no obvious preference orientation, shown as Figure 5.

Compared to several previous studies (e.g. Arzoumanian et al., 2018; Zhang et al., 2020; Guo et al., 2022; Liu et al., 2023; Dewangan et al., 2024), our sample is robust because the orientation of the filaments considered here are closely related with the embedded clumps, characterized by the positions of the two tip clumps (see Section 2.2). In these works, the identification of filaments was carried out employing 2D identification schemes like FILFINDER (Koch & Rosolowsky, 2015), DisPerSE (Sousbie, 2011), and SExtractor (Bertin & Arnouts, 1996), which often have no direct relation with the embedded cores/clumps. Our experience suggests that these technical differences are unlikely to explain the observed diversity in the relative orientations between the rotational axes of clumps and filaments. A comprehensive study using high spectral and spatial resolution images of a larger sample is required to understand this inconsistency better.