Panchromatic Hubble Andromeda Treasury XVI. Star Cluster Formation Efficiency and the Clustered Fraction of Young Stars

We use the Panchromatic Hubble Andromeda Treasury (PHAT) survey dataset to perform spatially resolved measurements of star cluster formation efficiency ($\Gamma$), the fraction of stellar mass formed in long-lived star clusters. We use robust star formation history and cluster parameter constraints, obtained through color-magnitude diagram analysis of resolved stellar populations, to study Andromeda's cluster and field populations over the last $\sim$300 Myr. We measure $\Gamma$ of 4-8% for young, 10-100 Myr old populations in M31. We find that cluster formation efficiency varies systematically across the M31 disk, consistent with variations in mid-plane pressure. These $\Gamma$ measurements expand the range of well-studied galactic environments, providing precise constraints in an HI-dominated, low intensity star formation environment. Spatially resolved results from M31 are broadly consistent with previous trends observed on galaxy-integrated scales, where $\Gamma$ increases with increasing star formation rate surface density ($\Sigma_{\mathrm{SFR}}$). However, we can explain observed scatter in the relation and attain better agreement between observations and theoretical models if we account for environmental variations in gas depletion time ($\tau_{\mathrm{dep}}$) when modeling $\Gamma$, accounting for the qualitative shift in star formation behavior when transitioning from a H$_2$-dominated to a HI-dominated interstellar medium. We also demonstrate that $\Gamma$ measurements in high $\Sigma_{\mathrm{SFR}}$ starburst systems are well-explained by $\tau_{\mathrm{dep}}$-dependent fiducial $\Gamma$ models.