  We present new high resolution and dynamic range dust column density and
temperature maps of the California Molecular Cloud derived from a combination
of Planck and Herschel dust-emission maps, and 2MASS NIR dust-extinction maps.
We used these data to determine the ratio of the 2.2 micron extinction
coefficient to the 850 micron opacity and found the value to be close to that
found in similar studies of the Orion B and Perseus clouds but higher than that
characterizing the Orion A cloud, indicating that variations in the fundamental
optical properties of dust may exist between local clouds. We show that over a
wide range of extinction, the column density probability distribution function
(PDF$_N$) of the cloud can be well described by a simple power law with an
index that represents a steeper decline with column density than found in
similar studies of the Orion and Perseus clouds. Using only the protostellar
population of the cloud and our extinction maps we investigate the Schmidt
relation within the cloud. We show that the protostellar surface density,
$\Sigma_*$, is directly proportional to the ratio of the protostellar and cloud
pdfs. We use the cumulative distribution of protostars to infer the functional
forms for both $\Sigma_*$ and PDF$_*$. We find that $\Sigma_*$ is best
described by two power-law functions with steeper indicies than found in other
local GMCs. We find that the protostellar pdf is a declining function of
extinction also best described by two power-laws whose behavior mirrors that of
$\Sigma_*$. Our observations suggest that variations both in the slope of the
Schmidt relation and in the sizes of the protostellar populations between GMCs
are largely driven by variations in the slope of the cloud pdf. This confirms
earlier studies suggesting that cloud structure plays a major role in setting
the global star formation rates in GMCs.
