  In this paper we investigate scaling relations between star formation rates
and molecular gas masses for both local Galactic clouds and a sample of
external galaxies. We specifically consider relations between the star
formation rates and measurements of dense, as well as total, molecular gas
masses. We argue that there is a fundamental empirical scaling relation that
directly connects the local star formation process with that operating globally
within galaxies. Specifically, the total star formation rate in a molecular
cloud or galaxy is linearly proportional to the mass of dense gas within the
cloud or galaxy. This simple relation, first documented in previous studies,
holds over a span of mass covering nearly nine orders of magnitude and
indicates that the rate of star formation is directly controlled by the amount
of dense molecular gas that can be assembled within a star formation complex.
We further show that the star formation rates and total molecular masses,
characterizing both local clouds and galaxies, are correlated over similarly
large scales of mass and can be described by a family of linear star formation
scaling laws, parameterized by $f_{DG}$, the fraction of dense gas contained
within the clouds or galaxies. That is, the underlying star formation scaling
law is always linear for clouds and galaxies with the same dense gas fraction.
These considerations provide a single unified framework for understanding the
relation between the standard (non-linear) extragalactic Schmidt-Kennicutt
scaling law, that is typically derived from CO observations of the gas, and the
linear star formation scaling law derived from HCN observations of the dense
gas.
