The statistics of triggered star formation: an overdensity of massive
  YSOs around Spitzer bubbles
  We present a detailed statistical study of massive star formation in the
environment of 322 Spitzer mid-infrared bubbles by using the RMS survey for
massive Young Stellar Objects (YSOs). Using a combination of simple surface
density plots and a more sophisticated angular cross-correlation function
analysis we show that there is a statistically significant overdensity of RMS
YSOs towards the bubbles. There is a clear peak in the surface density and
angular cross-correlation function of YSOs projected against the rim of the
bubbles. By investigating the autocorrelation function of the RMS YSOs we show
that this is not due to intrinsic clustering of the RMS YSO sample. RMS YSOs
and Spitzer bubbles are essentially uncorrelated with each other beyond a
normalised angular distance of two bubble radii. The bubbles associated with
RMS YSOs tend to be both smaller and thinner than those that are not associated
with YSOs. We interpret this tendency to be due to an age effect, with YSOs
being preferentially found around smaller and younger bubbles. We find no
evidence to suggest that the YSOs associated with the bubbles are any more
luminous than the rest of the RMS YSO population, which suggests that the
triggering process does not produce a top heavy luminosity function or initial
mass function. We suggest that it is likely that the YSOs were triggered by the
expansion of the bubbles and estimate that the fraction of massive stars in the
Milky Way formed by this process could be between 14 and 30%.
