The dependence of the hierarchical distribution of star clusters on galactic environment

We use the angular Two Point Correlation Function (TPCF) to investigate the hierarchical distribution of young star clusters in 12 local (3--18 Mpc) star-forming galaxies using star cluster catalogues obtained with the \textit{Hubble Space Telescope} (\textit{HST}) as part of the Treasury Program LEGUS (Legacy ExtraGalactic UV Survey). The sample spans a range of different morphological types, allowing us to infer how the physical properties of the galaxy affect the spatial distribution of the clusters. We also prepare a range of physically motivated toy models to compare with and interpret the observed features in the TPCFs. We find that, conforming to earlier studies, young clusters ($T \la 10\, \mathrm{Myr}$) have power-law TPCFs that are characteristic of fractal distributions with a fractal dimension $D_2$, and this scale-free nature extends out to a maximum scale $l_{\mathrm{corr}}$ beyond which the distribution becomes Poissonian. However, $l_{\mathrm{corr}}$, and $D_2$ vary significantly across the sample, and are correlated with a number of host galaxy physical properties, suggesting that there are physical differences in the underlying star cluster distributions. We also find that hierarchical structuring weakens with age, evidenced by flatter TPCFs for older clusters ($T \ga 10\, \mathrm{Myr}$), that eventually converges to the residual correlation expected from a completely random large-scale radial distribution of clusters in the galaxy in $\sim 100 \, \mathrm{Myr}$. Our study demonstrates that the hierarchical distribution of star clusters evolves with age, and is strongly dependent on the properties of the host galaxy environment.