A Model for Protostellar Cluster Luminosities and the Impact on the
  CO-H$_2$ Conversion Factor
  We construct a semi-analytic model to study the effect of far-ultraviolet
(FUV) radiation on gas chemistry from embedded protostars. We use the
Protostellar Luminosity Function (PLF) formalism of Offner & McKee (2011) to
calculate the total, FUV, and ionizing cluster luminosity for various
protostellar accretion histories and cluster sizes. We compare the model
predictions with surveys of Gould Belt star-forming regions and find the
Tapered Turbulent Core model matches best the mean luminosities and the spread
in the data. We combine the cluster model with the photo-dissociation region
astrochemistry code, {\sc 3d-pdr}, to compute the impact of the FUV luminosity
from embedded protostars on the CO to H$_2$ conversion factor, $X_{\rm CO}$, as
a function of cluster size, gas mass and star formation efficiency. We find
that $X_{\rm CO}$ has a weak dependence on the FUV radiation from embedded
sources for large clusters due to high cloud optical depths. In smaller and
more efficient clusters the embedded FUV increases X$_{\rm CO}$ to levels
consistent with the average Milky Way values. The internal physical and
chemical structure of the cloud are significantly altered, and $X_{\rm CO}$
depends strongly on the protostellar cluster mass for small efficient clouds.
