Not all stars form in clusters - measuring the kinematics of OB
  associations with $Gaia$
  It is often stated that star clusters are the fundamental units of star
formation and that most (if not all) stars form in dense stellar clusters. In
this monolithic formation scenario, low density OB associations are formed from
the expansion of gravitationally bound clusters following gas expulsion due to
stellar feedback. $N$-body simulations of this process show that OB
associations formed this way retain signs of expansion and elevated radial
anisotropy over tens of Myr. However, recent theoretical and observational
studies suggest that star formation is a hierarchical process, following the
fractal nature of natal molecular clouds and allowing the formation of
large-scale associations in-situ. We distinguish between these two scenarios by
characterising the kinematics of OB associations using the Tycho-$Gaia$
Astrometric Solution catalogue. To this end, we quantify four key kinematic
diagnostics: the number ratio of stars with positive radial velocities to those
with negative radial velocities, the median radial velocity, the median radial
velocity normalised by the tangential velocity, and the radial anisotropy
parameter. Each quantity presents a useful diagnostic of whether the
association was more compact in the past. We compare these diagnostics to
models representing random motion and the expanding products of monolithic
cluster formation. None of these diagnostics show evidence of expansion, either
from a single cluster or multiple clusters, and the observed kinematics are
better represented by a random velocity distribution. This result favours the
hierarchical star formation model in which a minority of stars forms in bound
clusters and large-scale, hierarchically-structured associations are formed
in-situ.
