An analytical, fully relativistic framework for tidal disruption event streams in Schwarzschild geometry

We present an analytical and fully relativistic framework for studying the self-intersection of tidal disruption event (TDE) streams, restricting ourselves to the Schwarzschild spacetime. By taking advantage of the closed-form solution to the geodesic equations in the Schwarzschild metric, we calculate properties of the self-intersection without numerically evaluating the geodesic equations or making any post-Newtonian approximations. Our analytical treatment also facilitates geometric definitions of the orbital semimajor axis and eccentricity, as opposed to Newtonian formulas which lead to unphysical results for highly relativistic orbits. Combined with assumptions about energy dissipation during the self-intersection shock, our framework enables the calculation of quantities such as the fraction of material unbound during the self-intersection shock, and the characteristic semimajor axes and eccentricities of the material that remains in orbit after the collision. As an example, we calculate grids of post-intersection properties in stellar and supermassive black hole (SMBH) masses for disruptions of main-sequence stars, identifying regions where no material is ejected during self-intersection (e.g. SMBH mass $\lesssim 5\times 10^6\, {\rm M_\odot }$ for $1\, {\rm M_\odot }$ stars disrupted at the tidal radius), potentially explaining the TDEs observed by SGR/eROSITA that are visible in X-rays but not optical wavelengths. We also identify parameters for which the post-intersection accretion flow has low eccentricity (e ≲ 0.6), and find that the luminosity generated by self-intersection shocks only agrees with observed trends in the relationship between light curve decay time-scales and peak luminosities over a narrow range of SMBH masses.