A catalogue of open cluster radii determined from Gaia proper motions
  In this work we improve a previously published method to calculate in a
reliable way the radius of an open cluster. The method is based on the
behaviour of stars in the proper motion space as the sampling changes in the
position space. Here we describe the new version of the method and show its
performance and robustness. Additionally, we apply it to a large number of open
clusters using data from Gaia DR2 to generate a catalogue of 401 clusters with
reliable radius estimations. The range of obtained apparent radii goes from
Rc=1.4+-0.1 arcmin (for the cluster FSR 1651) to Rc=25.5+-3.5 arcmin (for
NGC~2437). Cluster linear sizes follow very closely a lognormal distribution
with a mean characteristic radius of Rc=3.7 pc, and its high radius tail can be
fitted by a power law as N \propto Rc^(-3.11+-0.35). Additionally, we find that
number of members, cluster radius and age follow the relationship Nc \propto
Rc^(1.2+-0.1) Tc^(-1.9+-0.4) where the younger and more extensive the cluster,
the more members it presents. The proposed method is not sensitive to low
density or irregular spatial distributions of stars and, therefore, is a good
alternative or complementary procedure to calculate open cluster radii not
having previous information on star memberships.
