Clusters, Clouds, and Correlations: Relating Young Clusters to Giant Molecular Clouds in M33 and M31

We use young clusters and giant molecular clouds (GMCs) in the galaxies M33 and M31 to constrain temporal and spatial scales in the star formation process. In M33, we compare the PHATTER catalogue of 1214 clusters with ages measured via colour-magnitude diagram (CMD) fitting to 444 GMCs identified from a new 35 pc resolution ALMA $^{12}$CO(2-1) survey. In M31, we compare the PHAT catalogue of 1249 clusters to 251 GMCs measured from a CARMA $^{12}$CO(1-0) survey with 20 pc resolution. Through two-point correlation analysis, we find that young clusters have a high probability of being near other young clusters, but correlation between GMCs is suppressed by the cloud identification algorithm. By comparing the positions, we find that younger clusters are closer to GMCs than older clusters. Through cross-correlation analysis of the M33 cluster data, we find that clusters are statistically associated when they are $\leq$10 Myr old. Utilizing the high precision ages of the clusters, we find that clusters older than $\approx 18$ Myr are uncorrelated with the molecular ISM. Using the spatial coincidence of the youngest clusters and GMCs in M33, we estimate that clusters spend $\approx$4-6 Myr inside their parent GMC. Through similar analysis, we find that the GMCs in M33 have a total lifetime of $\approx 11$-15 Myr. We also develop a drift model and show that the above correlations can be explained if the clusters in M33 have a 5-10 km s$^{-1}$ velocity dispersion relative to the molecular ISM.