  We develop a model for regulation of galactic star formation rates Sigma_SFR
in disk galaxies, in which ISM heating by stellar UV plays a key role. By
requiring simultaneous thermal and (vertical) dynamical equilibrium in the
diffuse gas, and star formation at a rate proportional to the mass of the
self-gravitating component, we obtain a prediction for Sigma_SFR as a function
of the total gaseous surface density Sigma and the density of stars + dark
matter, rho_sd. The physical basis of this relationship is that thermal
pressure in the diffuse ISM, which is proportional to the UV heating rate and
therefore to Sigma_SFR, must adjust to match the midplane pressure set by the
vertical gravitational field. Our model applies to regions where Sigma < 100
Msun/pc^2. In low-Sigma_SFR (outer-galaxy) regions where diffuse gas dominates,
the theory predicts Sigma_SFR \propto Sigma (rho_sd)^1/2. The decrease of
thermal equilibrium pressure when Sigma_SFR is low implies, consistent with
observations, that star formation can extend (with declining efficiency) to
large radii in galaxies, rather than having a sharp cutoff. The main parameters
entering our model are the ratio of thermal pressure to total pressure in the
diffuse ISM, the fraction of diffuse gas that is in the warm phase, and the
star formation timescale in self-gravitating clouds; all of these are (in
principle) direct observables. At low surface density, our model depends on the
ratio of the mean midplane FUV intensity (or thermal pressure in the diffuse
gas) to the star formation rate, which we set based on Solar neighborhood
values. We compare our results to recent observations, showing good agreement
overall for azimuthally-averaged data in a set of spiral galaxies. For the
large flocculent spiral galaxies NGC 7331 and NGC 5055, the correspondence
between theory and observation is remarkably close.
