The dispersal law of young star clusters: free expansion -- A Gaia/eROSITA census of Orion and Sco--Cen

Abstract: We measure how young star clusters disperse over time in the two nearest star-forming complexes. Our sample comprises 44 \texttt{SigMA}-\textit{Gaia}-defined clusters in Orion and 31 in Sco--Cen, with isochronal ages of 2--25\,Myr. We extend each cluster's census with \textit{eROSITA}-selected young stars that share its parallax and motion and lie within the distance they could have traveled ballistically since birth. By 10--30\,Myr, about one in three former cluster stars lies outside the cluster's cataloged boundary. Across both complexes and the full age range, we find that (1) the half-member radius increases linearly with age, $\rfifty \simeq \sigbirth t$, (2) the velocity dispersion shows no measurable evolution, and (3) all studied clusters are unbound. These results support free expansion. Young clusters in Orion and Sco-Cen emerge unbound from their birth clouds and coast at their birth velocity dispersion without significant relaxation or a characteristic dispersal timescale. We further show that memberships defined by density contrast preferentially omit the fastest stars, making the same clusters appear to expand more slowly ($\propto t<sup>{0.6}$), cool with age, and approach equilibrium.