Tidal Disruption Flares from Stars on Marginally Bound and Unbound Orbits

We study the mass fallback rate of tidally disrupted stars on marginally bound and unbound orbits around a supermassive black hole (SMBH) by performing three-dimensional smoothed particle hydrodynamic (SPH) simulations with three key parameters: the star is modeled by a polytrope with two different indexes ($n=1.5$ and $3$). The stellar orbital properties are characterized by five orbital eccentricities ranging from $e=0.98$ to $1.02$ and five different penetration factors ranging from $\beta=1$ to $3$, where $\beta$ represents the ratio of the tidal disruption to pericenter distance radii. We derive analytic formulae for the mass fallback rate as a function of the stellar density profile, orbital eccentricity, and penetration factor. Moreover, two critical eccentricities to classify tidal disruption events (TDEs) into five different types: eccentric ($e<e_{\rm crit,1}$), marginally eccentric ($e_{\rm crit,1}\lesssim{e}<1$), purely parabolic ($e=1$), marginally hyperbolic ($1<e<e_{\rm crit,2}$), and hyperbolic ($e\gtrsim{e_{\rm crit,2}}$) TDEs, are reevaluated as $e_{\rm crit,1}=1-2q^{-1/3}\beta^{k-1}$ and $e_{\rm crit,2}=1+2q^{-1/3}\beta^{k-1}$, where $q$ is the ratio of the SMBH to stellar masses and $0<k\lesssim2$. We find the asymptotic slope of the mass fallback rate varies with the TDE type. The asymptotic slope approaches $-5/3$ for parabolic TDEs, is steeper for marginally eccentric TDEs, and is flatter for marginally hyperbolic TDEs. The mass fallback rates of the marginally eccentric TDEs are much larger than the parabolic TDE case, while those of marginally hyperbolic TDEs are much smaller. Marginally unbound TDEs could be an origin of a very low density gas disk around a dormant SMBH.