Structural parameters of 389 local Open Clusters
  The distribution of member stars in the surroundings of an Open Cluster (OC)
can shed light on the process of its formation, evolution and dissolution. The
analysis of structural parameters of OCs as a function of their age and
position in the Galaxy brings constraints on theoretical models of cluster
evolution. The Gaia catalogue is very appropriate to find members of OCs at
large distance from their centers. We aim at revisiting the membership lists of
OCs from the solar vicinity, in particular by extending these membership lists
to the peripheral areas thanks to Gaia EDR3. We used the clustering algorithm
HDBSCAN on Gaia parallaxes and proper motions to systematically look for
members up to 50 pc from the cluster centers. We fitted a King's function on
the radial density profile of these clusters and a Gaussian Mixture Model on
their two dimensional distribution of members. We also evaluated the degree of
mass segregation of the clusters. Our methodology performs well on 389 clusters
out of the 467 selected ones. We report the detection of vast coronae around
almost all the clusters and the detection of 71 OCs with tidal tails,
multiplying by more than four the number of such structures identified. We find
the size of the cores to be on average smaller for old clusters than for young
ones. Also, the overall size of the clusters seems to slightly increase with
age while the fraction of stars in the halo seems to decrease. As expected the
mass segregation is more pronounced in the oldest clusters but a clear trend
with age is not seen. OCs are more extended than previously expected,
regardless of their age. The decrease in the proportion of stars populating the
clusters halos highlights the different cluster evaporation processes and the
short timescales they need to affect the clusters. Reported parameters all
depend on cluster ages but can not be described as single functions of time.
