How many stars form in compact clusters in the local Milky Way?

Two main models coexist for the environment in which stars form. The clustered model stipulates that the bulk of star formation occurs within dense embedded clusters, but only a minority of them survive the residual gas expulsion phase caused by massive stellar feedback unbinding the clusters. On the other hand, the hierarchical model predicts that star formation happens at a range of scales and densities, where open clusters (OCs) only emerge from the densest regions. We aim to exploit a recent catalog of compact OCs, corrected for completeness, to obtain an updated estimation of the surface density star formation rate within OCs ($\sum_{\rm SFR,OC}$), which we compare with recent estimates of $\sum_{\rm SFR}$ to determine which model is more likely. We have applied two methods. The first one consisted of integrating over the power law that was fit for the mass function of the youngest OCs using a MC sampling. The second one consisted of counting the total compact mass within these youngest OCs within 1 kpc, so that the result could be directly compared with local values of $\sum_{\rm SFR}$. We estimated new $\sum_{\rm SFR,OC}$ values between $736^{+159}_{-176}$ and $875^{+34}_{-35}$ M$_{\odot}$ Myr$^{-1}$ kpc$^{-2}$, depending on the methodology. These results are significantly higher than previous $\sum_{\rm SFR,OC}$ estimates, which we attribute to the incompleteness of past catalogs, and are consistent with the majority ($\geq$ 50 \%) or even the vast majority ($\geq$ 80 \%) of the star formation occurring in initially compact clusters, through comparisons with $\sum_{\rm SFR}$ from the recent literature. Our new $\sum_{\rm SFR,OC}$ values are consistent with clustered formation being the most dominant mode of star formation.