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

We measure how young star clusters disperse over time in the two nearest star-forming complexes. Our sample comprises 44 SigMA-Gaia-defined clusters in Orion and 31 in Sco--Cen, with isochronal ages of 2--25,Myr. We extend each cluster's census with 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, 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 (proportional to t 0.6), cool with age, and approach equilibrium.