3.2.1. Dendrogram

The issues of identifying structures in a PPV cube using the dendrogram algorithm are discussed in Zhou et al. (Reference Zhou2024a). Consequently, we employed the same approach as outlined in Zhou et al. (Reference Zhou2024a) and (Reference Zhou2024b) to identify dense gas structures. We conducted a direct identification of hierarchical (sub-)structures based on the 2D integrated intensity (Moment 0) map of CO (2 $-$ 1) emission. Subsequently, we extracted the average spectrum of each identified structure to delve into its velocity components and gas kinematics. All the retained structures on the strictly masked Moment 0 map are reliable; we therefore only require the smallest area of the identified structure larger than 1 beam and do not set other parameters in the algorithm to reduce the dependence of the identification on the parameter settings. Dendrogram algorithm decomposes the intensity data (Moment 0 map) into hierarchical structures. As illustrated in Fig. 1 of Zhou et al. (Reference Zhou2024a), all identified structures can be divided into three categories, that is, i-leaves (isolated leaves), c-leaves (clustered leaves), and branches. In this work, we merged c-leaves and i-leaves. Finally, there are two kinds of structures, that is, leaf and branch structures. In Figs. 2 and 3, the structures identified by the dendrogram algorithm exhibit a strong correspondence with the background integrated intensity maps.

Figure 2.

Dendrogram structures. (a) Leaves; (b) branches. Red and orange eclipses represent type1 and type3 structures, respectively.

Figure 3.

Same as Fig. 2, but for NGC5236.

The algorithm characterises the morphology of each structure by approximating it as an ellipse. Within the dendrogram, the rms sizes (second moments) of the intensity distribution along the two spatial dimensions define the long and short axes of the ellipse, denoted as a and b. As described in Zhou et al. (Reference Zhou2024a), a smaller ellipse is obtained with a and b, necessitating a multiplication by a factor of 2 to appropriately enlarge the ellipse. Then the effective physical radius of an ellipse is $R\rm_{eff}$ = $\sqrt{2a \times 2b}*D$ , where D is the distance of the galaxy. For a structure with the area A, the total integrated intensity of the structure is $I_{\rm CO}$ , then the mass of the structure can be calculated by:

(1) \begin{equation}M = \alpha^{2-1}_{\rm CO} \times I_{\rm CO} \times A,\end{equation}

here $\alpha^{2-1}_{\rm CO} \approx 6.7\,\rm M_{\odot} \rm pc^{-2} (\rm K*km s^{-1})^{-1}$ (Leroy et al. Reference Leroy2022).

The large-scale velocity gradients due to the galaxy rotation will contribute to the non-thermal velocity dispersion. Before extracting the average spectra of the identified structures, we subtracted the large-scale velocity gradients in PPV data cube based on the constructed gas dynamical model in Section 3.3.3. According to their averaged spectra, the structures with absorption features were eliminated first. Following Zhou et al. (Reference Zhou2024b), we only consider type1 (single velocity component) and type3 (blended velocity components) structures in this work.

As detailed in Zhou et al. (Reference Zhou2024a), two screening criteria were applied for structure refinement: (1) Removal of repetitive branch structures; and (2) Exclusion of branch structures exhibiting complex morphology. Finally, there are 853 and 2581 retained structures for NGC4321 and NGC5236, respectively, as listed in Table 1 and marked in Figs. 2 and 3 by ellipses. NGC5236 is $\sim$ 3 times nearer than NGC4321, the total number of the identified structures in NGC5236 is also $\sim$ 3 times more than that in NGC4321.

Table 1.

Dendrogram structures in NGC 4321 and NGC 5236.

3.2.2. Scaling relations

The scaling relations provide insights into the physical states of the structures. Although the identified structures in NGC5236 are much more than that in NGC4321, the scaling relations in Fig. 4 are comparable. Due to the further distance, some faint structures were filtered out in NGC4321, which can be seen in NGC5236. In Fig. 4, the faint or low-density structures produce dispersive tails, similar to the Fig. 14 of Zhou et al. (Reference Zhou2024a) for the G333 molecular cloud complex in the Milky Way.

Figure 4.

Scaling relations of leaf and branch structures of NGC 4321 and NGC 5236. (a)-(c) and (a1)-(c1) $\sigma-N*R$ ; (d)-(f) and (d1)-(f1) $\sigma-N$ ; (g)-(i) and (g1)-(i1) $\sigma-R$ . $\sigma$ , R and N are the velocity dispersion, effective radius, and column density of each structure, respectively. Here, “P” represents the Pearson coefficient. Dashed vertical line marks the boundary of two fittings.

In Fig. 4(g)–(i), it appears that only branch structures demonstrate a discernible correlation between velocity dispersion and scale. Clumps or cores characterised by high column density exhibit greater velocity dispersion relative to those with lower column density, a phenomenon attributed to gas motions associated with gravitational collapse (Ballesteros-Paredes et al. Reference Ballesteros-Paredes, Hartmann, Vázquez-Semadeni, Heitsch and Zamora-Avilés2011; Traficante et al. Reference Traficante2018; Ballesteros-Paredes et al. Reference Ballesteros-Paredes, Vázquez-Semadeni, Palau and Klessen2018; Li et al. Reference Li2023; Zhou et al. Reference Zhou2024a). As depicted in Fig. 4(d)–(f), the positive correlations observed between velocity dispersion and column density suggest a gravitational origin for the velocity dispersion. In the context of pure free-fall, an anticipated $\sigma-N*R$ relation would exhibit a slope of 0.5. For a more convenient comparison with $\sigma-R$ and $\sigma-N$ relations, we transform the Heyer relation, $\sigma/R^{0.5} \propto N^{0.5}$ , to the form $\sigma \propto (R*N)^{0.5}$ (Eq. 3 in Ballesteros-Paredes et al. Reference Ballesteros-Paredes, Hartmann, Vázquez-Semadeni, Heitsch and Zamora-Avilés2011). Both relations should ideally have a slope of 0.5. However, in Fig. 4(a)–(c), the slopes of the $\sigma-N*R$ relations are notably shallower than 0.5, suggesting a deceleration from the expected behaviour of pure free-fall gravitational collapse.

3.3.1. Identification of filaments

The FILFINDER algorithm (Koch & Rosolowsky Reference Koch and Rosolowsky2015) was used to identify and characterise filaments in the galaxy. Following the method described in Zhou et al. (Reference Zhou2022) and (Reference Zhou2023b), we used the Moment 0 maps of CO (2 $-$ 1) emission from the data products of the PHANGS-ALMA survey to identify filaments. Fig. 1(b) and (e) display the skeletons of the identified filaments superimposed on the Moment 0 maps. Notably, these skeletons exhibit a high degree of agreement with the gas structures traced by CO (2 $-$ 1) emission. In this work, we only consider the filaments along the spiral arms (the main filaments), where have strong CO (2 $-$ 1) emission, which allows us to study large-scale continuous gas motions in the galaxy.

Almost all algorithms have the issue of parameter settings; thus, we try not to discuss the identified filament itself in this work, such as its length and width. Actually, we mainly regard the FILFINDER algorithm as a tool to draw lines, these lines can make connections between discrete dense structures or put dense structures in gas networks. Then by extracting the velocity information along these lines, we can study how the surrounding gas converges to dense gas structures (gravitational centres). In Zhou et al. (Reference Zhou2022) and (Reference Zhou2023b), the filaments traverse multiple local hub-filament structures, characterised by fluctuations in velocity and density along the filaments. These local dense structures, acting as gravitational centres, facilitate the accretion of surrounding diffuse gas, culminating in the formation of local hub-filament structures. In the PPV space, a hub-filament structure can be represented as a funnel structure, as demonstrated in Fig. 9 of Zhou et al. (Reference Zhou2023b). The gradient of the funnel profile serves as an indicator of the gravitational field’s strength. Consequently, examining the local velocity gradients along the filaments provides insights into the strength of local gravitational fields.

In Fig. 1, although the velocity fluctuations at small scales can be seen everywhere, they are hidden in global large-scale velocity gradients due to the galaxy rotation. To derive the local velocity fluctuations, we must first subtract the large-scale velocity gradients. We take two methods to solve this issue.

3.3.2. Linear fitting

Fig. 1(c2)&(c) and (f2)&(f) show the intensity-weighted velocity (Moment 1) and integrated intensity (Moment 0) of CO (2 $-$ 1) line emission along the main filaments with intense velocity and density fluctuations. A segment of NGC5236’s main filament was used to demonstrate the velocity gradient fitting in Fig. 5. Here, we eliminated the large-scale velocity gradient by employing a simple linear function for modelling. Subsequently, we subtracted this model from the velocity field obtained from the Moment 1 map along the main filament. We divided the main filaments into many segments to ensure that each segment is as straight as possible, thus increasing the accuracy of the linear fit, as shown in Fig. 5(a). The fitted straight line can be the baseline, and then we subtracted the baseline for each segment to leave only the residual local velocity fluctuations. The good correspondence between velocity gradients and intensity peaks in Fig. 5(b) indicates that the surrounding gases are converging to the centre of the gravitational potential well. For each segment of the main filament, we first estimated the global velocity gradients between velocity peaks and valleys at two sides of the intensity peaks (representing gravitational centres) and ignored local velocity fluctuations. Then we also derived additional velocity gradients over smaller distances around the local intensity peaks, as illustrated in Fig. 5(b). Generally, large-scale velocity gradients are associated with multiple intensity peaks. Given that NGC4321 and NGC5236 are good face-on galaxies, here we ignored the projection effect in velocity gradient fitting.

Figure 5.

A segment of NGC5236’s main filament, used to demonstrate the velocity gradient fitting. (a) Linear fitting of the large-scale velocity gradient due to the bulk motion; (b) velocity field of the segment in panel (a) after subtracting the fitted large-scale velocity gradient. Local velocity gradients are fitted in the ranges defined by the red vertical dashed lines in panel (c), and straight lines show the linear fitting results. The colour-coding of straight lines are meaningless, but it is convenient for the reader; (c) blue and red dotted lines show the normalised velocity and intensity, respectively. Orange box marks a gravitational coupling structure.

3.3.3. Gas dynamical model

As a comparison, we also removed the bulk motion or the galaxy rotation by creating gas dynamical models using the Kinematic Molecular Simulation (KinMS) package of Davis et al. (Reference Davis2013) based on the Moment-1 map, as shown in Fig. 6(b). Then we used the Moment 1 map to subtract the created model and obtained the velocity residual map. Finally, we extracted the velocity fluctuations from the velocity residual map rather than the Moment 1 map. The subsequent velocity gradient fitting is the same with Section 3.3.2.

Figure 6.

(a) The velocity field (Moment 1) of NGC 4321 in observation; (b) gas dynamical model created by the Kinematic Molecular Simulation (KinMS) package; (c) velocity fluctuations after removing the created model.

3.3.4. Statistical analysis of the fitted velocity gradients

In Sections 3.3.2 and 3.3.3, the galaxy centres were excluded in the fitting. As shown in Fig. 7, two methods give consistent fitting results. In Fig. 8, we can find the same gas kinematic modes presented in Zhou et al. (Reference Zhou2023b) and (Reference Zhou2024b). The variations in velocity gradients at both small and large scales (with a boundary around 500 pc) align with expectations from gravitational free-fall, with central masses of $\sim$ 10 $^{5}$ –10 $^{6.5}$ M $_\odot$ and $\sim$ 10 $^{6}$ –10 $^{7.5}$ M $_\odot$ . This implies that sustaining velocity gradients on larger scales requires correspondingly larger masses, and larger masses imply larger scales, suggesting that the larger-scale inflow is driven by the larger-scale structure which may arise from the gravitational clustering of smaller-scale structures, in harmony with the presence of hierarchical or multi-scale hub-filament structures within the galaxy and the gas inflows from large to small scales. In the orange box marked in Fig. 5(b) and (c), multiple peaks are coupled together to form a gravitational potential well on larger scale, and each peak itself is also a local gravitational potential well.

Figure 7.

(a) The correlation between the fitted velocity gradient and the scale; (b) The mass distribution of the identified structures.

Figure 8.

Statistical analysis of all fitted velocity gradients. (a) Velocity gradient vs. the length. The colour lines show the freefall velocity gradients for comparison. For the freefall model, red, magenta, green, cyan, blue and black lines denote masses of 10 $^{5}$ M $_\odot$ , 10 $^{5.5}$ M $_\odot$ , 10 $^{6}$ M $_\odot$ , 10 $^{6.5}$ M $_\odot$ , 10 $^{7}$ M $_\odot$ , and 10 $^{7.5}$ M $_\odot$ , respectively. Panels (b), (c), and (d): Zoomed maps with lengths $\lt$ 500 pc (small scale), $\sim$ 500–1 500 pc (medium scale), and $\gt$ 1 500 pc (large scale) in panel (a).

In Fig. 7(a), almost all measured velocity gradients can be fitted in the mass range $\sim$ 10 $^{5}$ –10 $^{7}$ M $_\odot$ , which is consistent with the mass distribution of the identified structures shown in Fig. 7(b), indicating that local dense structures and their complex as gravitational centres will accrete the surrounding diffuse gas and then produce the observed velocity gradients at different scales.

3.3.5. Deviation of the free-fall model

We can do a deeper analysis for the fitted velocity gradients with a simple model. Assuming free-fall,

(2) \begin{equation}\nabla v = -\frac{d}{dR}\sqrt{\frac{2GM}{R}}=\sqrt{\frac{GM}{2R^3}},\end{equation}

in this equation, velocity gradient is more sensitive to scale. Thus, we only fit the correlation between velocity gradient and scale in Fig. 7(a). In equation (2), $\nabla v \propto R^{-1.5}$ , however, the linear fitting in Fig. 7(a) gives $\nabla v \propto R^{-0.8}$ , thus $\sim$ 2 times smaller than the slope of the free-fall model, also indicating the slowing down of a pure free-fall gravitational collapse presented in Section 3.2.2.