Title: The longevity of the oldest open clusters: Structural parameters of NGC 188, NGC 2420, NGC 2425, NGC 2682, NGC 6791, NGC 6819

Abstract: Context: Open clusters' dynamical evolution is driven by stellar evolution, internal dynamics and external forces, which according to dynamical simulations, will evaporate them in a timescale of about 1 Ga. However, about 10\% of the known open clusters are older. They are special systems whose detailed properties are related to their dynamical evolution and the balance between mechanisms of cluster formation and dissolution. Aims:We investigate the spatial distribution and structural parameters of six open clusters older than 1 Ga in order to constrain their dynamical evolution, and longevity.} Methods: We identify members using Gaia EDR3 data up to a distance of 150 pc from each cluster's centre. We investigate the spatial distribution of stars inside each cluster to understand their degree of mass segregation. Finally, in order to interpret the obtained radial density profiles we reproduced them using the lowered isothermal model explorer with PYTHON Limepy and spherical potential escapers stitched SPES. Results: All the studied clusters seem more extended than previously reported in the literature. The spatial distributions of three of them show some structures aligned with their orbits. They may be related to the existence of extra tidal stars. In fact, we find that about 20 % of their members have enough energy to leave the systems or are already unbound. Together with their initial masses, their distances to the Galactic plane may play significant roles in their survival. We found clear evidences that the most dynamically evolved clusters do not fill their Roche volumes, appearing more concentrated than the others. Finally, we find a cusp-core dichotomy in the central regions of the studied clusters, which shows some similarities to the one observed among globular clusters.