Gaia DR3 reveals the complex dynamical evolution within star clusters

Context. Star clusters, composed of stars born from the same molecular cloud, serve as invaluable natural laboratories for understanding the fundamental processes governing stellar formation and evolution. Aims. This study aims to investigate correlations between the Mean Interdistance (D _i), Mean Closest Interdistance (D _c) and Median Weighted Central Interdistance (D _cc) with the age of star clusters, examining their evolutionary trends and assessing the robustness of these quantities as possible age indicators. Methods. We selected a sample of open clusters in the solar region and with a representative number of members (e.g. well populated and without outliers). The interdistances are derived from the spatial distribution of member stars within a cluster. Their evolution over time allows us to use them as age indicators for star clusters. Results. Our investigation reveals a high-significant correlation between the interdistances and cluster age. Considering the full sample of clusters between 7 and 9 kpc, the relationship is very broad. This is due to uncertainties in parallax, which increase with increasing distance. In particular, we must limit the sample to a maximum distance from the Sun of about 200 pc to avoid artificial effects on cluster shape and on the spatial distribution of their stars along the line of sight. Conclusions. By conservatively restraining the distance to a maximum of ~200 pc, we have established a relationship between the interdistances and the age of the clusters. In our sample, the relationship is mainly driven by the internal expansion of the clusters and is marginally affected by external perturbative effects. Such relation might enhance our comprehension of cluster dynamics and might be used to derive cluster dynamical ages.