Evolution of fractality in centrally concentrated young clusters

Abstract

We investigate the structural evolution of young star clusters forming within centrally concentrated molecular clouds. Our simulations use the Torch framework, which integrates the FLASH magnetohydrodynamics code with the AMUSE environment, enabling a self-consistent treatment of gas dynamics, star formation, stellar evolution, radiative transfer, and gravitational interactions. We quantify cluster structure using the $Q$ parameter for fractality and compute fractal dimensions via two methods: box-counting and correlation dimension. Our results show that clusters generally inherit fractal substructure from their parental clouds, which is typically erased within $\sim 2.5\,t_\mathrm{ff}$ through dynamical relaxation. Massive stars can induce the formation of secondary subclusters via feedback, with outcomes strongly dependent on stellar mass and formation timing. Interactions among subclusters, including mergers and dispersal, can extend fractal structure beyond $4\,t_\mathrm{ff}$. We also find systematic correlations between the fractality parameter $Q$ and the fractal dimension: fractality is positively correlated with both the correlation and box-counting dimensions, with the correlation dimension exhibiting a stronger correlation. These results demonstrate how stellar feedback and internal dynamics jointly shape the measurable fractal properties of embedded star clusters.

Figures (10)

• Figure 1: Left: Box-counting dimension $\ln N(L)$ vs. $-\ln L$ for the n6s1 model. The red solid line represents the best-fit linear regression, with its slope corresponding to the estimated fractal dimension $f_\mathrm{dim}$. The vertical dashed lines mark the fitting range of $[-\ln 2, \ln 2]$. The horizontal dashed line indicates the natural logarithm of the total number of member stars in the cluster $N_s$. Right: Correlation dimension $\ln C(r)$ vs. $\ln r$ for the same model. The red solid line represents the best-fit linear regression, with its slope corresponding to the estimated fractal dimension $f_\mathrm{dim}$. The vertical dashed lines mark the fitting range of $[-\ln 40, \ln 0.1]$. The horizontal dashed line indicates the natural logarithm of the normalized number of maximum neighbors, $\ln 1$.
• Figure 2: Gas density slices and projected stellar distributions for the n3s6, n3s7, and n4s1 simulations, all of which exhibit subcluster formation. For n3s6 and n3s7: (top row) onset of stellar feedback; (middle row) just before the formation of the second subcluster; (bottom row) after its formation. For n4s1: (top row) two subclusters are present; (middle row) they merge into a single system; (bottom row) the system separates again into two subclusters. Ionization fronts are marked by cyan lines. Black dots indicate individual stars with $M < 7\,M_\odot$, while massive stars ($M \geq 7\,M_\odot$) are shown in red, with symbol sizes proportional to stellar mass. Annotations in each panel show: (bottom left) scale bar; (bottom right) mass of the most massive star, total stellar mass within $5.5\,\mathrm{pc}$, and total number of stars; (top left) simulation label; and (top right) simulation time. The color scale indicates gas density in g cm$^{-3}$.
• Figure 3: Evolution of the $Q$ parameter over time for models that initially exhibit fractality but subsequently evolve to smooth distributions. Time is measured starting from the onset of star formation. Dashed purple lines show the $Q$ parameter calculated in 3D, while solid purple lines represent the mean $Q$ values computed from 2D projections ($xy$, $yz$, and $xz$), with shaded areas indicating the minimum and maximum range of each projection. Red circles represent the mass and time of formation of massive stars, with marker size and vertical position proportional to their mass; the secondary red$y$-axis on the right indicates the mass scale of the massive stars. em Vertical blue dashed lines denote multiples of the cloud's free-fall time $t_\mathrm{ff}$ and horizontal gray lines mark the fractality thresholds for the $Q$ parameter: 0.8 for 2D (dashed line) and 0.7 for 3D (dotted line) clusters.
• Figure 4: Evolution of the $Q$ parameter over time in models where subcluster formation maintained fractality after $4\,t_\mathrm{ff}$. All notation is the same as Figure \ref{['fig:6Qs']}.
• Figure 5: Mean interparticle distance $\bar{s}$ versus mean minimum spanning tree edge length $\bar{m}$ (see Section \ref{['sec:Q']}) for all simulations at different multiples of the free-fall time $t_\mathrm{ff}$, indicated in the upper right corner. Diamonds and dashed lines represent 3D results, while cross markers with solid lines represent mean values of 2D projections ($xy$, $yz$, and $xz$). Fractality is indicated by points lying below and to the right of the critical lines.