Modelling the structure of star clusters with fractional Brownian motion
  The degree of fractal substructure in molecular clouds can be quantified by
comparing them with Fractional Brownian Motion (FBM) surfaces or volumes. These
fields are self-similar over all length scales and characterised by a drift
exponent $H$, which describes the structural roughness. Given that the
structure of molecular clouds and the initial structure of star clusters are
almost certainly linked, it would be advantageous to also apply this analysis
to clusters. Currently, the structure of star clusters is often quantified by
applying $\mathcal{Q}$ analysis. $\mathcal{Q}$ values from observed targets are
interpreted by comparing them with those from artificial clusters. These are
typically generated using a Box-Fractal (BF) or Radial Density Profile (RDP)
model. We present a single cluster model, based on FBM, as an alternative to
these models. Here, the structure is parameterised by $H$, and the standard
deviation of the log-surface/volume density $\sigma$. The FBM model is able to
reproduce both centrally concentrated and substructured clusters, and is able
to provide a much better match to observations than the BF model. We show that
$\mathcal{Q}$ analysis is unable to estimate FBM parameters. Therefore, we
develop and train a machine learning algorithm which can estimate values of $H$
and $\sigma$, with uncertainties. This provides us with a powerful method for
quantifying the structure of star clusters in terms which relate to the
structure of molecular clouds. We use the algorithm to estimate the $H$ and
$\sigma$ for several young star clusters, some of which have no measurable BF
or RDP analogue.
