An Analytic, Fully Relativistic Framework for Tidal Disruption Event Streams in Schwarzschild Geometry

Abstract

We present an analytic 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 analytic treatment also facilitates geometric definitions of the orbital semi-major 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 semi-major axes and eccentricities of the material which 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 \(\lesssim 5\times10^6\, {\rm M_\odot}\) for \(1\,{\rm M_\odot}\) stars disrupted at the tidal radius), potentially explaining the TDEs observed by SGR/eROSITA which are visible in X-rays but not optical wavelengths. We also identify parameters for which the post-intersection accretion flow has low eccentricity (\(e\lesssim0.6\)), and find that the luminosity generated by self-intersection shocks only agrees with observed trends in the relationship between light curve decay timescales and peak luminosities over a narrow range of SMBH masses.