Dynamical evolution of fractal structures in star-forming regions
  The Q-parameter is used extensively to quantify the spatial distributions of
stars and gas in star-forming regions as well as older clusters and
associations. It quantifies the amount of structure using the ratio of the
average length of a minimum spanning tree, mbar, to the average length within
the complete graph, sbar. The interpretation of the Q-parameter often relies on
comparing observed values of Q, mbar and sbar to idealised synthetic
geometries, where there is little or no match between the observed star-forming
regions and the synthetic regions. We measure Q, mbar, and sbar over 10 Myr in
N-body simulations which are compared to IC 348, NGC 1333, and the ONC. For
each star-forming region we set up simulations that approximate their initial
conditions for a combination of different virial rations and fractal
dimensions. We find that dynamical evolution of idealised fractal geometries
can account for the observed Q, mbar, and sbar values in nearby star-forming
regions. In general, an initially fractal star-forming region will tend to
evolve to become more smooth and centrally concentrated. However, we show that
initial conditions, as well as where the edge of the region is defined, can
cause significant differences in the path that a star-forming region takes
across the mbar-sbar plot as it evolves. We caution that the observed
Q-parameter should not be directly compared to idealised geometries. Instead,
it should be used to determine the degree to which a star-forming region is
either spatially substructured or smooth and centrally concentrated.
