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An Evolutionary Model for Collapsing Molecular Clouds and Their Star Formation Activity. II. Mass Dependence of the Star Formation Rate

We discuss the dependence of various properties of the star formation rate (SFR) and efficiency (SFE) in molecular clouds (MCs) on the maximum mass reached by the clouds, based on a previously-published model for MC and SFR evolution in which the clouds were assumed to be undergoing global collapse, and the SFR was controlled by ioniztion feedback. Because the model neglects various other processes, the results presented are upper limits. We find that clouds with $\Mmax \lesssim 10^4 \Msun$ end their lives with a mini-burst, at which the SFR reaches a peak of $\sim 10^4 ~\Msun \Myr^{-1}$, although its time average is only $\SFRavg \sim \hbox{ a few} \times 10^2 \Msun \Myr^{-1}$. The corresponding efficiencies are $\SFEmax \lesssim $60\%$ and $\SFEavg \lesssim $1\%$. For more massive clouds ($\Mmax \gtrsim 10^5 ~ \Msun$), the SFR first increases and then remains roughly constant for $\sim 10^7$ yr, because the clouds are influenced by the stellar feedback since earlier in their evolution. We find that $\SFRavg$ and $\SFEavg$ are well represented by the fits $\SFRavg \approx 100 (1+\Mmax/2 \times 10^5 ~ \Msun)^{2} ~ \Msun \Myr^{-1}$ and $\SFEavg \approx 0.024 (\Mmax/10^5 ~ \Msun)^{0.28}$, respectively. The massive model clouds follow the SFR-dense gas mass relation obtained by Gao \& Solomon for infrared galaxies, extrapolated down to MCs scales. Low-mass clouds fall above this relation, in agreement with recent observations. An integration of the model-predicted $\SFRavg$ over a Galactic GMC mass spectrum yields a realistic value for the Galactic SFR. Our results reinforce the suggestion that star-forming GMCs may be in global collapse, and still have low net SFRs and SFEs, due to the evaporation of most of the cloud material by the feedback from massive stars.