The Maximum Gravity Model for partial Tidal Disruption Events: Mass Loss, Peak Fallback Rate and Dependence on Stellar Properties

A star entering the tidal sphere of a supermassive black hole (SMBH) can be partially stripped of mass, resulting in a partial tidal disruption event (TDE). Here we develop an analytical model for properties of these events, including the peak fallback rate, $\dot{M}_{\rm peak}$, the time at which the peak occurs, $t_{\rm peak}$, and the amount of mass removed from the star, $\Delta M$, for any star and any pericenter distance associated with the stellar orbit about the black hole. We compare the model predictions to 1276 hydrodynamical simulations of partial TDEs of main-sequence stars by a $10^6 M_\odot$ SMBH. The model yields $t_{\rm peak}$ predictions that are in good agreement (to within tens of percent) with the numerical simulations for any stellar mass and age. The agreement for $\dot{M}_{\rm peak}$ is weaker due to the influence of self-gravity on the debris stream dynamics, which remains dynamically important for partial TDEs; the agreement for $\dot{M}_{\rm peak}$ is, however, to within a factor of $\sim 2-3$ in the majority of cases considered, with larger differences for low-mass stars ($M_\star \lesssim 0.5 M_\odot$) on grazing orbits with small mass loss. We show that partial TDE lightcurves for disruptions caused by $\sim 10^6M_\odot$ SMBHs can span $\sim 20-100$ day peak timescales, whereas grazing encounters of high-mass stars with high-mass SMBHs can yield longer peak timescales ($t\gtrsim 1000$ days), associated with some observed transients. Our model provides a significant step toward an analytical prescription for TDE lightcurves and luminosity functions.