© The Authors 2025

1 Introduction

Open clusters are formed from interstellar gas in dense molecular clouds (e.g., Lada & Lada 2003). The spatial distribution of member stars in open clusters changes over time as clusters evolve. Quantifying the spatial distribution can thus provide valuable insights into the formation process and the subsequent dynamical evolution of open clusters.

According to the theory of hierarchical star formation of Kruijssen (2012), gravitationally bound young open clusters (with ages ≤100 Myr) are primarily formed in high-density regions (Treviño-Morales et al. 2019; Vázquez-Semadeni et al. 2017; Ward et al. 2020). Such clusters exhibit fractal substructures, as predicted by the conveyor belt mechanism (Arnold et al. 2017; Clarke 2010; Fujii et al. 2022). On the contrary, clusters formed in low-density (filamentary) regions are often characterized by filamentary substructures, and such clusters disperse rapidly after gas removal.

Internal two-body relaxation and the influence of the external Galactic tides play an important role in the evolution of older open clusters (with ages exceeding 100 Myr). Internal two-body relaxation shapes open clusters with a dense core, accompanied by a lower-density halo (Pang et al. 2022b). When Galactic tidal forces are substantial, elongated tidal-tail substructures tend to emerge over time (Pang et al. 2021a, 2022b; Tang et al. 2019; Röser & Schilbach 2019).

A natural first approach to studying the morphology of open clusters is through visual inspection, although this method lacks accuracy and reproducibility. Another popular method to quantify 2D morphology is calculating the radial density profile of star clusters and then fitting the profile to various models, such as the Elson-Fall-Freeman (EFF) model (Elson et al. 1987) or the King model (King 1962). The aforementioned pioneering works on the morphology of star clusters only used the 2D (projected) distribution of member stars.

With the release of Gaia data (Gaia Collaboration 2016), morphology research shifted from 2D spacial data to 3D spatial data, which is attributed to the significantly improved parallax measurement. Pang et al. (2021a) introduced a quantitative approach to determine the 3D morphology for the region within the tidal radius of the cluster, using the ellipsoid fitting, which is an advanced version for estimating the ellipticity of a 2D projected distribution (Chen et al. 2004; Tarricq et al. 2022). However, this method is sensitive to extended outliers and is therefore mostly useful for the dense, bound regions of open clusters. To better distinguish between the extended substructures outside tidal radius in star clusters into different types, visual inspection was used again in Pang et al. (2022b) to carry out the classification. Pang et al. (2022b) successfully classified open clusters into filamentary, fractal, halo, and tidal-tail types. However, only 53 out of 85 open clusters in their catalog have been classified. The complexity of the 3D structure of a star cluster cannot be easily analyzed by the human eye alone. A quantitative approach is required to objectively and accurately determine the 3D morphology of star clusters.

In this paper, we quantified the morphology of open clusters in the solar neighborhood by introducing a quantity that does not depend on the density profile. The fractal dimension (Mandelbrot & Wheeler 1983) is measured for characterizing shapes by quantifying the complexity. The fractal dimension has been widely applied in scientific research. In astronomy, for example, the fractal dimension can be used to study the large-scale distribution of galaxies (Elmegreen & Elmegreen 2001; Ribeiro & Miguelote 1998). Feitzinger & Galinski (1987) calculated the fractal dimension of star-forming sites in 19 spiral galaxies for classification. The concept of fractal dimension has also been used to study turbulence in Giant HII Regions (Caicedo-Ortiz et al. 2015), and the distribution of the interstellar medium (Sánchez et al. 2005). The fractal dimension was first used to examine the spatial distribution of young open cluster samples in the solar neighborhood by de La Fuente Marcos & de La Fuente Marcos (2006). In the latter study, each open cluster was considered as a point. Our study is the first to calculate the fractal dimension for individual open clusters in the solar neighborhood based on the 3D positions of member stars obtained from Gaia DR 3 data using the box-counting method (Grassberger 1983).

This paper is organized as follows. In Sect. 2, we introduce the two open cluster catalogs. The fractal dimension is described in Sect. 3, with Sect. 3.1 detailing its computation and Sect. 3.2 addressing its uncertainty. In Sect. 4, we analyze the fractal dimension in different regions of each cluster (Sect. 4.1) and discuss the fractal dimensions of different morphological types in Sect. 4.2. A classification based on fractal dimension is performed in Sect. 4.3. Sect. 5 provides discussions on the fractal dimension in relation to open cluster dynamical evolution (Sect. 5.1), star formation within open clusters (Sect. 5.2), and Galactic structures (Sect. 5.3). Finally, we provide a brief summary of our findings in Sect. 6.

2 Open clusters catalogs

Gaia DR 3 (Gaia Collaboration 2023) has offered unprecedented high-accuracy positions on the sky and parallax for more than 1.8 billion sources, and so has Gaia EDR3 (Gaia Collaboration 2021). In this study, we used the catalog based on Gaia DR 3 (Gaia Collaboration 2023): the Pang star cluster catalog (Pang et al. 2024), and compared the findings with those of the Hunt star cluster catalog (Hunt & Reffert 2024).

We investigated star clusters within 500 pc in the solar neighborhood from the catalog of Pang et al. (2024), based on (X, Y, Z) coordinates of the member stars of 93 star clusters. Li et al. (2021); Pang et al. (2022a, 2021a, 2024, 2022b, 2021b) identified member stars of a total 93 star clusters. Their study strictly required member stars to have measurement uncertainties below 10% in their parallax and photometry. The machine learning algorithm Stars’ Galactic Origin (Yuan et al. 2018, StarGO) was used to select star cluster members based on Gaia EDR 3 and DR 3 data (Gaia Collaboration 2021, 2023). The field star contamination rate was kept within a 5% threshold, corresponding to a membership probability of 95%. The heliocentric Cartesian coordinates of member stars in these open clusters were adjusted based on the distance correction using the Bayesian approach proposed in Carrera et al. (2019), which assumes a two-component prior. A normal distribution prior is assumed for the individual distances to the cluster stars, while an exponentially decreasing profile prior is used for the distances to field stars (Bailer-Jones 2015). The corrected distance for each star is derived from the mean of the posterior distribution. No distance correction was applied to cluster Group X, as it lacks a central concentration of stars and instead exhibits a two-piece fragmented spatial distribution. Group X is currently in the final stages of disruption (Tang et al. 2019).

For comparison with the Pang catalog clusters, we extended our investigation to open clusters from the all-sky cluster catalog of Hunt & Reffert (2024) with distances below 500 pc. We further selected open clusters with a median color-magnitude diagram class above 0.5 and a cluster significance test score greater than 5σ (see Hunt & Reffert 2023, for details). As for the Pang catalog clusters, only member stars with parallaxes and photometric measurements within a 10% uncertainty (Lindegren et al. 2018, Appendix C) were included for further analysis. We obtained a total of 127 open clusters. The distances of individual member stars were corrected using the same Bayesian approach (Carrera et al. 2019) that was applied to the catalog of Pang et al. (2024).

In total, the fractal dimension analysis includes 93 open clusters from Pang et al. (2024) (hereafter: the Pang catalog clusters) and 127 open clusters from Hunt & Reffert (2024) (hereafter: the Hunt catalog clusters) mainly within 500 pc in the solar neighborhood. The basic properties for these two samples of clusters are listed in Tables A.1 and A.2.

3 Fractal dimension

3.1 Computation of fractal dimension

We quantified the morphological complexity of open clusters using the fractal dimension, which serves as a measure of fractal structures. The fractal dimension offers a quantitative tool to study the spatial distribution of individual stars inside the cluster. A lower value of fractal dimension indicates a more clumpy (substructured) morphology, and vice versa.

In this work, we considered the box-counting dimension, also known as the Minkowski–Bouligand dimension (Peitgen et al. 2004), which can be obtained using the box-counting method (Grassberger 1983). The box-counting method covers the dataset with boxes of varying sizes and counts the boxes needed to cover the entire dataset. The 3D coordinates of the member stars are standardized so that they have zero mean and unit variance before the start of box-counting, which guarantees the coordinates become all dimensionless as well. Thus, the box length L below is also dimensionless.

The fractal dimension f_dim via the box-counting method is defined as $$f_{\text{dim}}=-\frac{d{\log }_{e}N(L)}{d{\log }_{e}L},$$(1)

where L is the length of the box and N(L) is the number of boxes counted to cover the spatial distribution of member stars corresponding to the given L. Given a finite dataset of discrete points, this derivative must clearly be estimated using an approximate prescription, such as taking a finite difference. Moreover, its value may change as a function of box size. In our case we estimated f_dim as the slope that is obtained by linear regression between log_e N(L) and − log_e L (see Fig. 1).

Fig. 1 shows that plateaus emerge when the number of boxes N approaches the number of the member stars (horizontal dashed lines in Fig. 1). To mitigate the bias on the plateaus and automate the computation process, we defined a fixed fitting range of [− log_e 2, log_e 2] (indicated by the vertical dashed lines in Fig. 1). This is a range in scales equal to 1/2 and 2 times the scale set by the standard deviation of the coordinates, which is roughly the half-mass radius.

Fig. 1 Box-counting loge N vs. − loge L of Pleiades and NGC 3532. The slopes of the red solid lines fitted by robust regression represent the estimated fractal dimension fdim. The intervals defined by the vertical dashed lines correspond to the fitting range [− loge 2, loge 2]. The horizontal dashed line in each panel indicates the natural logarithm of the total number of member stars in the respective star clusters.

3.2 Uncertainty estimation

Here we discuss the dependence of the uncertainty in the fractal dimension on (1) observational errors in the Gaia data, (2) the number of member stars in each cluster n_s, and (3) the choice of contamination rate.

Gaia’s astrometric uncertainties (RA, Dec, and parallax) cause uncertainties in the 3D positions of individual stars, which can affect the derived fractal dimensions. To mitigate this, we re-assigned the position of each member star in the Pang catalog, by drawing from a Gaussian distribution centered at the observed position, with the dispersion based on the observational uncertainty. The fractal dimension of the cluster was then recalculated with the newly assigned stellar positions. This process was repeated 1000 times for each cluster. The relative error of fractal dimension associated with astrometric uncertainty is only a few percent (2–6%), which is considered acceptable.

To estimate the effect of the member number, n_s, on the fractal dimension, we selected three example clusters with total member numbers ranging from 240 to above 2500, NGC 3532, NGC 6991, and the Pleiades, and randomly resampled different numbers of member stars (without replacement) to recalculate the value of f_dim. For different values of n_s ∈ [30, 50, 100, 200,…] (except for NGC 6991, which has approximately 240 stars, for which an interval of 20 was used), 100 iterations were performed. The corresponding mean fractal dimension and standard deviation of these 100 trails were computed and presented in Fig. 2a to c. The fractal dimension of each of these three clusters increases with growing member number n_s, while the relative error (standard deviation, blue error bars) decreases with n_s. The relative error varies from 19% (NGC 3532 with n_s = 30) to 0.4% (Pleiades with n_s = 1400).

In Fig. 3, we investigated how the contamination rate from the Pang catalog changes the fractal dimension in five example clusters, which have different morphologies (tidal-tail and halo-types). We observed a small decline of the mean fractal dimension (represented by blue crosses in Fig. 3) when the contamination rate increased from 1% to 5%. However, this might be induced by low-number statistics, as the member number significantly drops when the contamination rate is below 5%. On the other hand, the fractal dimension increases significantly as the contamination rate surpasses 15%. This is mainly attributed to the inclusion of more field stars, which generates artificial halolike substructures in the outskirts (Pang et al. 2022b), resulting in an artificial increase of the fractal dimension. The Pang catalog adopted a 5% contamination rate for membership. As can be seen in Fig. 3, the dispersion of the fractal dimension is the largest at 5%, implying that clusters of different morphologies can be effectively distinguished via fractal dimension. Therefore, members with a 5% contamination rate are appropriate for fractal dimension analysis.

4 Fractal dimension of open clusters in the solar neighborhood

In this section, we analyzed the results of the fractal dimension of open clusters in the solar neighborhood across different regions. We considered the entire cluster (the all-member region), the region inside the tidal radius (r ≤ r_t), and the region outside the tidal radius (r > r_t).

The tidal radii of the open clusters from both catalogs are computed as $$r_t={\left(\frac{G{M}_{\text{cl}}}{2{(A-B)}^2}\right)}^{\frac{1}{3}}$$(2)

(Pinfield et al. 1998), where G is the gravitational constant, M_cl is the total mass of the cluster (the sum of the masses of the individual cluster members), and parameters A and B are the Oort constants. Here we used A = 15.3 ± 0.4 km s⁻¹ kpc⁻¹ and B = −11.9 ± 0.4 km s⁻¹ kpc⁻¹ (Bovy 2017).

To avoid the bias of outliers, we set a threshold for the number of stars n_s > 30 used to compute the fractal dimension. Regions with fewer than 30 stars were not assessed in our procedure.

Fig. 4 displays the fractal dimension of open clusters as a function of distance. We did not observe any correlation between the fractal dimension and distance among the Pang catalog clusters (panels a to c). Therefore, the distance correction we applied allowed us to successfully recover the cluster morphologies, which would otherwise be distorted by the uncertainties of the parallax measurements.

A similar trend is observed in Hunt catalog clusters in the all-member region and r ≤ r_t regions (panels d to e). However, a weak correlation is observed for Hunt clusters in r > r_t region (panel f), and the fractal dimension decreases as the distance increases. As a significant number of the Hunt clusters are elongated along the line of sight, the distance correction method is not optimal for correcting this morphology (see Figure 4 in Pang et al. 2021a). This indicates that a small bias is introduced when we use the fractal dimension to quantify the extended regions of Hunt catalog clusters. We will address this effect in the analysis below.

4.1 Distribution of fractal dimension in different regions

The distribution of the fractal dimension of clusters from Pang et al. (2024) (for the all-member region) is presented in Fig. 5a.

The mean fractal dimension is f_dim = 1.46. The results for the regions inside and outside the tidal radius are represented with the red and blue histograms in Fig. 5b. The mean fractal dimension for r > r_t is around 1.53, with a prominent peak around f_dim ≈ 1.4 and a secondary peak at f_dim ≈ 1.7. The mean fractal dimension for r ≤ r_t increases to f_dim = 1.81, with two prominent peaks at 1.7 and 2.0.

In Figs. 5c and d, we display the distribution of the fractal dimension for three regions of clusters in the Hunt catalog clusters. The mean fractal dimension is approximately 1.65 for the all-member region while that for r > r_t and r ≤ r_t is 1.68 and 1.80 respectively. The distribution of fractal dimension is uni- modal for these three regions, while the all-member region is more symmetric. Both catalogs indicate that the fractal dimension for stars r > r_t is smaller than that for r ≤ r_t (Figs. 5b and d). However, the fractal dimension of Pang catalog clusters at r > r_t covers a larger range, from roughly 0.8 to 2.0, compared to 1.0 to 2.1 in Hunt catalog clusters. Unlike the Pang catalog clusters (Fig. 5 b), we found no significant difference between the fractal dimension distribution for r ≤ r_t (red histogram in Fig. 5d) and r > r_t (blue histogram in Fig. 5d) of the Hunt catalog clusters except the slight shift of their peaks.

Among the three regions, the fractal dimension patterns are the same for both catalog clusters. The mean value at r ≤ r_t is the highest, whereas the mean for r > r_t is lower, and the mean for the all-member region is the lowest. This indicates that the region inside the tidal radius is more uniform, while that outside the tidal radius exhibits more substructure. This is in agreement with the conclusion from Pang et al. (2022b), which classifies the substructures outside the tidal radius into the filamentary, fractal, halo, and tidal tails. The lowest mean value for the all-member region is induced by the fixed fitting range of fractal dimension (Fig. 1).

Fig. 2 Dependence of the fractal dimension fdim,test on the number of member stars, ns, for three example clusters: NGC 3532 (a), NGC 6991 (b), and Pleiades (c). The blue triangles and the error bars indicate the corresponding mean fractal dimension and the standard deviation in the corresponding ns group of 100 iterations.

Fig. 3 Fractal dimension vs. contamination rate for five example clusters, NGC 2516 (gray stars), NGC 3532 (red diamonds), NGC 6991 (orange triangles), Pleiades (pink circles), and Praesepe (brown squares). The blue crosses and the error bars are the mean fractal dimension and the standard deviation of five clusters at the same contamination rate.

Fig. 4 Relation between fractal dimension and the corrected distance of the Pang catalog clusters (left panels) and the Hunt catalog clusters (right panels) in the solar neighborhood for the all-member region (a and d), r ≤ rt (b and e), and r > rt (c and f). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value below 0.05 indicates a rejection of the null hypothesis.

4.2 Fractal dimension of morphological types

Pang et al. (2022b) visually categorized the morphology outside the tidal radius for 53 open clusters that show significant substructure (Col. 3 in Table A.1) into four types: f1: filamentary, f2: fractal, h: halo, and t: tidal-tail. To be explicit, f1-type clusters are younger than 100 Myr with unidirectional elongated substructures, while f2-type clusters are also younger than 100 Myr but have fractal substructures. For clusters older than 100 Myr, h-type clusters have a compact core with a low-density halo in the outskirts, whereas t-type clusters correspond to those with two tidal tails.

Fig. 6 presents the histogram of the fractal dimensions for the morphology-classified clusters from Pang et al. (2022b). The fractal dimension of f1 and f2 types clusters (all-member region, panels a and b) exhibits the largest scatter, as reflected by its greater standard deviation of 0.213 and 0.179. In comparison, the standard deviation for halo-type and tidal-tail-type clusters have smaller values of 0.046 and 0.167, respectively. The distributions of fractal dimension for the f2-type and t-type clusters (all-member region, panels b and d) are very similar. The Kolmogorov-Smirnov test on these two distributions results in a p-value of 0.80 (significantly larger than the threshold value of 0.05), which means that these two distributions are very similar. From the right panels in Fig. 6, we found that the mean fractal dimension values for the f1-type in r ≤ r_t and r > r_t are 1.85 and 1.66, respectively; for the f2-type, they are 1.82 and 1.62; for the h-type, 2.12 and 1.61; and for the t-type, 1.85 and 1.61. From a comparison between the fractal dimensions of these four morphological types in the inner and outer regions, it is evident that the fractal dimension for r ≤ r_t is consistently greater than that for r > r_t, indicating a more uniform distribution within the tidal radius. The 53 morphologically classified clusters follow the same trend that is observed in all 93 clusters from the Pang catalog (Fig. 5b). The difference between r > r_t and r ≤ r_t is most significant in halo- and tidal-tail-type clusters. These older clusters (h-type and t-type) have more unbound stars escaping in the outer halo-structure or tidal tails, respectively.

Fig. 5 Histograms of the fractal dimension for open clusters. (a) Histogram of fractal dimension for the Pang catalog clusters. (b) Overlapping histograms of fractal dimension for the Pang catalog clusters at r ≤ rt (red) and r > rt (blue). (c) Histogram of fractal dimension for the Hunt catalog clusters. (d) Overlapping histograms of fractal dimension for the Hunt catalog clusters at r ≤ rt (red) and r > rt (blue).

4.3 Classification based on fractal dimension

In this section, we aimed to quantitatively classify the morphology of open clusters via fractal dimension. The classification was based on the fractal dimension of three distinct regions as features: the all-member region ( f_dim,all), the region outside the tidal radius $r>r_t\left(f_{\dim ,\text{r}>{\text{r}}_{\text{t}}}\right)$, and the region inside the tidal radius $r\le r_t\left(f_{\dim ,\text{r}\le {\text{r}}_{\text{t}}}\right)$.

We adopted the Gaussian mixture model (GMM) for the classifying process. GMM is characterized by its soft classifying approach, allowing each cluster to be associated with k groups (we set k ∈ [2, 3, 4, 5] in GMM), and calculate the silhouette score corresponding to the k groups for individual clusters. The k-group with the highest mean silhouette score in the cluster sample is selected to proceed for further analysis. Finally, we applied the Bayesian Information Criterion (BIC) to select the optimal number of groups, k, while avoiding over-fitting. The model with a low BIC value is considered as the optimal result. This method is flexible and can accommodate groups of diverse sizes.

We present the average silhouette score in Table 1. The silhouette scores quantify the similarity of a data point to its assigned group compared to other groups, with values in the range of −1 to 1. An average silhouette score above 0.5 suggests strong alignment with the assigned group and therefore good classification. When a data point’s silhouette score approaches −1, the classification completely fails.

The classification results for clusters from the Pang catalog are presented in Table 1. The fractal dimension for the all member region and r ≤ r_t achieves a similar mean silhouette score > 0.5 and classifies the Pang catalog clusters into two groups. There is an age difference between these two groups. The number of clusters in each group (see Table 1) allows for a statistically significant comparison. The classification based on the fractal dimension for the r > r_t region is ineffective. We were unable to divide clusters into different groups. Thus, the fractal dimension for r > r_t alone does not appear to be a reliable feature for morphology classification.

As shown in Pang et al. (2022b), there are two kinds of clusters which both exhibit elongated substructures, but which have very different ages. The filamentary-type clusters are young (<100 Myr) and have inherited their filamentary structures from the filaments of molecular clouds. On the other hand, older clusters (>100 Myr), develop extended tidal tails and halos over time, due to the internal dynamical evolution and external Galactic tidal field. Therefore, the fractal dimension alone cannot be used to distinguish between different cluster shapes outside r_t among fractal, filamentary, halo, and tidal-tail substructures (Pang et al. 2022b).

Fig. 6 Histograms of the fractal dimension for the morphology- classified clusters. Left: histogram of fractal dimension for the allmember region in the 53 clusters from the Pang catalog, categorized by morphological type: (a) f1, (b) f2, (c) h, and (d) ${\overline{f}}_{\text{dim},\text{all}}$ represents the mean fractal dimension for each corresponding type. Right: overlapping histograms of fractal dimension for the 53 clusters from the Pang catalog at r ≤ rt (red) and at r > rt (blue), categorized by morphological type: (e) f1, (f) f2, (g) h, and (h) $\text{t}.{\overline{f}}_{\text{dim },\text{r}\le {\text{r}}_{\text{t}}}$ and ${\overline{f}}_{\text{dim, r}>{\text{r}}_{\text{t}}}$ represent the mean fractal dimension for each corresponding type of r ≤ rt (red) and r > rt (blue).

5 Discussion

5.1 Fractal dimension and open cluster dynamical evolution

An earlier study by Sánchez & Alfaro (2009) first attempted to identify the relation between the fractal dimension of open clusters and cluster age, based on 2D positions. However, due to low-number statistics (8 clusters), they were unable to draw a definitive conclusion. In Figs. 7a–c, we investigated the dependence of the fractal dimension on cluster age of the Pang catalog clusters. For the all-member region (panel a), the fractal dimension shows an anti-correlation with age, with the Spearman coefficient s = −0.38 and a p-value less than 0.01. This implies that the morphology of the star clusters becomes more clumpy as the clusters evolve. This is expected from the dynamical evolution of open clusters, which results in the emergence of tidal tails and halos (Tarricq et al. 2022), and therefore the fractal dimension decreases.

In Figs. 7a and d, we over-plotted simulation results from Ussipov et al. (2024), who simulated star clusters with varying Star Formation Efficiencies (SFEs) of 0.15 (purple circles), 0.17 (red triangles), and 0.20 (orange crosses), over ages ranging from 50 Myr to 1500 Myr. In their simulation, the fractal dimension evolves a little before 200 Myr. After 200 Myr, simulated star clusters start to develop tidal tails and lose members under the influence of the Galactic tidal field. Therefore, their fractal dimension declines significantly. The simulation results and our observations both cover a similar range of fractal dimensions and exhibit a generally negative trend. The fractal dimension for Pang clusters at the region r > r_t (Fig. 7b) also shows a declining trend with age (s = −0.25), similar to that of the all-member region.

On the other hand, for stars at r ≤ r_t in Pang clusters (panel c in Fig. 7) the fractal dimension and the age show no dependence, with the Spearman coefficient s being only −0.06, along with a p-value 0.57. In the first 10 Myr, a significant amount of crossing times have passed inside r ≤ r_t, gravitational interactions of member stars can effectively erase substructures. Therefore, further increases in the fractal dimension are asymptotically small. Given the scatter in the fractal dimension, we were unable to statistically detect the potentially slight relationship.

However, no correlation is observed between the fractal dimension and cluster age for the Hunt catalog clusters, as shown in Figs. 7d–f. The bias we observed, as well as the weak dependence of fractal dimension on distance for Hunt clusters at r > r_t (Fig. 4f), might dilute the true dynamical evolution signature reflected by fractal dimension.

Fig. 7 Relation between the fractal dimension and the cluster age in Pang catalog clusters (left panels) and in Hunt catalog clusters (right panels) in the solar neighborhood for the all-member region (a and d), r > rt (b and e), and r ≤ rt (c and f). The colored open symbols and curves in panels a and d represent the evolution of the fractal dimension taken from the simulations by Ussipov et al. (2024) with SFEs of 0.15 (purple circles), 0.17 (red triangles), and 0.20 (orange crosses). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value less than 0.05 means rejection of the null hypothesis.

5.2 Fractal dimension and star formation in open clusters

Star formation in the solar neighborhood has been found to be hierarchical (Pang et al. 2022b). We attempted to constrain the outcome of the star formation process in the solar neighborhood using the fractal dimension. Fig. 8 illustrates the relationship between fractal dimension and cluster observed total mass, M_cl, which is obtained by summing up the individual stellar masses provided by both catalogs. A pronounced correlation is observed between fractal dimension and cluster mass: f_dim for all three regions in the clusters of both catalogs increases with cluster mass. This trend is most prominent at r ≤ r_t.

These results align with the framework of hierarchical star formation. Through the conveyor belt mechanism, stars along filaments are transferred into dense hub regions via infalling flows, and these infalling stellar groups merge (Treviño-Morales et al. 2019; Vázquez-Semadeni et al. 2017). The stellar groups formed in the filaments have a smaller SFE, and therefore cannot remain bound after gas expulsion. These low-mass groups inherited their filamentary shape from filaments and are characterized by lower fractal dimensions. While they merge to form more massive, and more centrally concentrated structures, the fractal dimension increases (Schmeja & Klessen 2006). This correlation is particularly strong within r ≤ r_t, where mergers are taking place. The higher value of f_dim among more massive clusters supports the hypothesis that they are probably formed hierarchically.

Fig. 8 Relation between fractal dimension and the observed total cluster mass in Pang catalog clusters (left panels) and Hunt catalog clusters (right panels) in the solar neighborhood for the all-member regions (a and d), for r ≤ rt (b and e), and for r > rt (c and f). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value less than 0.05 means rejection of the null hypothesis.

5.3 Fractal dimension and galactic structures

The analysis of the transition in the fractal dimension has been proposed by Sánchez et al. (2010) to determine the typical size of star complexes in galaxies. On smaller scales in a galaxy, for example, star-forming regions, the fractal dimension is lower for the spatial distribution of star complexes (groups of star clusters and stellar groups), which resemble the fractal star formation cores induced by turbulence in giant molecular clouds. When we include stellar objects (from stars to star clusters) within a larger scale, the spatial distribution is consistent with a nearly uniform distribution. For example, the Galactic interstellar medium (ISM) has a universal fractal dimension 2.3 ± 0.3 (Elmegreen & Falgarone 1996), which represents a uniform distribution.

Below, we quantified the clustering strength of star clusters in the Milky Way via fractal dimension and aimed to search for the transition of fractal dimension, which is induced by Galactic structures where stars are being formed. We adopted the median position of all members as the location of each cluster and calculated the fractal dimension for the distribution of open clusters within different distances. Fig. 9a displays the fractal dimension for the spatial distribution of star clusters from both Pang (pink circles) and Hunt (orange triangles) catalogs in the solar neighborhood. The computation of fractal dimension was only carried out for distances beyond 200 pc since there are fewer than 10 clusters within 100 pc. The computation was carried out with an interval of 100 pc. At a distance of ~300 pc, we observed a transition of f_dim in both catalogs. This distance is close to the edge of the Local Bubble (O’Neill et al. 2024), which is a cavity characterized by high-temperature, low-density plasma, surrounded by a shell of dust and gas (Cox & Reynolds 1987). Almost all star formation in the solar vicinity occurs on the surface of the Local Bubble (Zucker et al. 2022). Beyond 300 pc, the fractal dimension increases dramatically. The plateau observed in the fractal dimension above 500 pc for the Pang catalog clusters is due to a very small number of clusters beyond 500 pc.

In our Galaxy, the star-forming regions are concentrated along the spiral arms. To probe the larger-scale star formation structures in the Galaxy, we made use of all the open clusters in the Hunt catalog (a total of 3457 open clusters). Fig. 9b presents the fractal dimension for the distribution of these clusters across the Galaxy, computed with a 100 pc interval, consistent with the calculation inside the solar neighborhood. At the distance of 600 pc, the fractal dimension of open clusters approaches the value of Galactic ISM f_dim ~ 2.3 (Elmegreen & Falgarone 1996). This is the location at the edge of the Local Arm (Reid et al. 2019), beyond which star formation and the spatial density of open clusters decline. We also noticed a rapid drop in the fractal dimension from 2.3 to 1.7–1.9 at some locations, reaching several local minima. This occurs when the distribution of clusters approaches spiral arms, where the concentration of open clusters is higher. A comparison with the location of Galactic spiral arms (Hao et al. 2021; Reid et al. 2019) indicates that the local minima of f_dim correspond to the locations of the Perseus Arm (at ∼1700 pc), the Scutum-Centaurus Arm (∼2500 pc), the Norma Arm (∼3500 pc), and the Outer Arm (∼4200 pc). Therefore, the transition of the fractal dimension for the spatial distribution of open clusters provides a unique perspective to trace Galactic structures. It successfully recovers the small-scale structure inside the Solar neighborhood (the Local Bubble and the Local Arm) to larger structures in the Galaxy (the different spiral arms).

6 Summary

We determined the fractal dimension for open clusters in the solar neighborhood using 3D position from Gaia DR 3. Our cluster sample includes 93 clusters from the catalog of Pang et al. (2024), and 127 clusters from the catalog of Hunt & Reffert (2024), mainly within 500 pc from the Sun. The box-counting method (Grassberger 1983) was applied to compute the fractal dimension of these clusters for three distinct regions: the allmember region; the region r < r_t and the region r > r_t, where r_t is the tidal radius. The open clusters can be effectively classified by their fractal dimension using the GMM algorithm. Based on the fractal dimension analysis of these open clusters, our results can be summarized as follows:

• The fractal dimension for the Pang catalog clusters in three distinct regions share a common peak around f_dim = 1.7. The fractal dimension is lower in the r > r_t region, while it is highest within r ≤ r_t;

• We analyzed the fractal dimensions of open clusters across the four morphological cluster types from (Pang et al. 2022b): filamentary (f1), fractal (f2), halo (h), and tidal-tail (t), and conclude:

  • For the fractal dimension of the all-member region, f1- type clusters exhibit the largest variability and the highest mean value, while h-type clusters show the opposite;

  • Comparing the fractal dimensions of these morphological types across different regions, we found that the morphology inside the tidal radius is more uniform, while the overall morphology is more clumpy.

• For the Hunt catalog clusters, the fractal dimension in all different regions exhibits a uni-modal distribution. The mean value of the fractal dimension, f_dim = 1.65 is larger than that of the Pang catalog f_dim = 1.46;

• The GMM was utilized to classify clusters based on either the fractal dimension of all-member or r ≤ r_t regions, which divides the Pang catalog clusters into two groups with observable age differences. Fractal dimension for stars r > r_t alone cannot properly classify cluster morphology;

• We investigated the dynamical evolution of open clusters, star formation in open clusters, and Galactic structures, using the fractal dimension:

  • For the Pang clusters, the fractal dimension tends to decrease with age in both the all-member region and the r > r_t region, which is consistent with the general evolutionary trend of the simulated clusters with varying SFEs described in Ussipov et al. (2024). In contrast, no correlation is observed for Hunt clusters across all three regions;

  • A strong correlation between the fractal dimension and cluster mass is observed in both catalogs across all three regions, with the relationship being more pronounced within r ≤ r_t. This trend is closely linked to the hierarchical star formation process, in which massive clusters are built from the mergers of small filamentary groups;

  • The fractal dimension for the distribution of open clusters in both catalogs within the solar neighborhood increases rapidly beyond the shell of the Local Bubble (approximately 300 pc). In the distribution of open clusters across the Galaxy (Hunt catalog, a total of 3457), local minima where f_dim < 1.9 likely indicate the location of different spiral arms: the Local Arm (∼600 pc), the Perseus Arm (∼1700 pc), the Scutum-Centaurus Arm (∼2500 pc), the Norma Arm (∼3500 pc), and the Outer Arm (∼4200 pc).

Our study provides a novel approach to quantitatively analyze the 3D morphology of open clusters in the solar neighborhood through the fractal dimension. Further research will be conducted using more precise 3D spatial data that will be released from Gaia DR 4. As a consequence of the uncertainty in the proper motion (PM) of Gaia DR 3, PM measurements are strongly affected by unresolved binary stars (Pang et al. 2023). Therefore, Gaia PMs cannot adequately probe the internal kinematics of star clusters, unless the interpretation of the data is corrected for the presence of unresolved binaries. The larger kinematic dataset will allow further exploration of the intricate relationship between morphology and cluster dynamics.

Fig. 9 Fractal dimension for the spatial distribution of open clusters vs. distance. (a) In the solar neighborhood and (b) in the Galaxy. Note that the fractal dimension here is calculated based on the spatial distribution of individual clusters, rather than the internal distribution of member stars within each cluster, as discussed earlier. A cluster is represented as a single point in the 3D space, which is the median position of all stellar members. (a) Pink circles and orange triangles represent the fractal dimensions for the distribution of the Pang catalog clusters and Hunt catalog clusters, respectively. The blue bar indicates the extent of the Local Bubble. (b) Blue circles show the fractal dimension for the distribution of 3457 Hunt catalog open clusters across the Galaxy. The approximate locations and ranges of the Galactic arms are indicated in pink (Local Arm), brown (Perseus Arm), purple (Scutum-Centaurus Arm), red (Norma Arm), and orange (Outer Arm).

Data availability

Full Tables A.1 and A.2 are available in electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/695/A22.

Acknowledgements

We thank the anonymous referee for providing helpful comments and suggestions that helped to improve this paper. Xiaoying Pang acknowledges the financial support of the National Natural Science Foundation of China through grants 12173029 and 12233013. Antonella Vallenari acknowledges the support of INAF project CRA 1.05.23.05.19. This work made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).

Appendix A Tables

References

All Tables

All Figures

Fig. 1 Box-counting loge N vs. − loge L of Pleiades and NGC 3532. The slopes of the red solid lines fitted by robust regression represent the estimated fractal dimension fdim. The intervals defined by the vertical dashed lines correspond to the fitting range [− loge 2, loge 2]. The horizontal dashed line in each panel indicates the natural logarithm of the total number of member stars in the respective star clusters.

In the text

	Fig. 2 Dependence of the fractal dimension fdim,test on the number of member stars, ns, for three example clusters: NGC 3532 (a), NGC 6991 (b), and Pleiades (c). The blue triangles and the error bars indicate the corresponding mean fractal dimension and the standard deviation in the corresponding ns group of 100 iterations.
In the text

	Fig. 3 Fractal dimension vs. contamination rate for five example clusters, NGC 2516 (gray stars), NGC 3532 (red diamonds), NGC 6991 (orange triangles), Pleiades (pink circles), and Praesepe (brown squares). The blue crosses and the error bars are the mean fractal dimension and the standard deviation of five clusters at the same contamination rate.
In the text

Fig. 4 Relation between fractal dimension and the corrected distance of the Pang catalog clusters (left panels) and the Hunt catalog clusters (right panels) in the solar neighborhood for the all-member region (a and d), r ≤ rt (b and e), and r > rt (c and f). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value below 0.05 indicates a rejection of the null hypothesis.

In the text

Fig. 5 Histograms of the fractal dimension for open clusters. (a) Histogram of fractal dimension for the Pang catalog clusters. (b) Overlapping histograms of fractal dimension for the Pang catalog clusters at r ≤ rt (red) and r > rt (blue). (c) Histogram of fractal dimension for the Hunt catalog clusters. (d) Overlapping histograms of fractal dimension for the Hunt catalog clusters at r ≤ rt (red) and r > rt (blue).

In the text

Fig. 6 Histograms of the fractal dimension for the morphology- classified clusters. Left: histogram of fractal dimension for the allmember region in the 53 clusters from the Pang catalog, categorized by morphological type: (a) f1, (b) f2, (c) h, and (d) ${\overline{f}}_{\text{dim},\text{all}}$ represents the mean fractal dimension for each corresponding type. Right: overlapping histograms of fractal dimension for the 53 clusters from the Pang catalog at r ≤ rt (red) and at r > rt (blue), categorized by morphological type: (e) f1, (f) f2, (g) h, and (h) $\text{t}.{\overline{f}}_{\text{dim },\text{r}\le {\text{r}}_{\text{t}}}$ and ${\overline{f}}_{\text{dim, r}>{\text{r}}_{\text{t}}}$ represent the mean fractal dimension for each corresponding type of r ≤ rt (red) and r > rt (blue).

In the text

Fig. 7 Relation between the fractal dimension and the cluster age in Pang catalog clusters (left panels) and in Hunt catalog clusters (right panels) in the solar neighborhood for the all-member region (a and d), r > rt (b and e), and r ≤ rt (c and f). The colored open symbols and curves in panels a and d represent the evolution of the fractal dimension taken from the simulations by Ussipov et al. (2024) with SFEs of 0.15 (purple circles), 0.17 (red triangles), and 0.20 (orange crosses). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value less than 0.05 means rejection of the null hypothesis.

In the text

Fig. 8 Relation between fractal dimension and the observed total cluster mass in Pang catalog clusters (left panels) and Hunt catalog clusters (right panels) in the solar neighborhood for the all-member regions (a and d), for r ≤ rt (b and e), and for r > rt (c and f). The quantity s is Spearman’s rank correlation coefficient, and p is the probability of the null hypothesis in the correlation test. A p-value less than 0.05 means rejection of the null hypothesis.

In the text

Fig. 9 Fractal dimension for the spatial distribution of open clusters vs. distance. (a) In the solar neighborhood and (b) in the Galaxy. Note that the fractal dimension here is calculated based on the spatial distribution of individual clusters, rather than the internal distribution of member stars within each cluster, as discussed earlier. A cluster is represented as a single point in the 3D space, which is the median position of all stellar members. (a) Pink circles and orange triangles represent the fractal dimensions for the distribution of the Pang catalog clusters and Hunt catalog clusters, respectively. The blue bar indicates the extent of the Local Bubble. (b) Blue circles show the fractal dimension for the distribution of 3457 Hunt catalog open clusters across the Galaxy. The approximate locations and ranges of the Galactic arms are indicated in pink (Local Arm), brown (Perseus Arm), purple (Scutum-Centaurus Arm), red (Norma Arm), and orange (Outer Arm).

In the text