Black Hole-Neutron Star Mergers and Short Gamma-ray Bursts: A Relativistic Toy Model to Estimate the Mass of the Torus

The merger of a binary system composed of a black hole and a neutron star may leave behind a torus of hot, dense matter orbiting around the black hole. While numerical-relativity simulations are necessary to simulate this process accurately, they are also computationally expensive and unable at present to cover the large space of possible parameters, which include the relative mass ratio, the stellar compactness and the black hole spin. To mitigate this and provide a first reasonable coverage of the space of parameters, we have developed a method for estimating the mass of the remnant torus from black hole-neutron star mergers. The toy model makes use of the relativistic affine model to describe the tidal deformations of an extended tri-axial ellipsoid orbiting around a Kerr black hole and measures the mass of the remnant torus by considering which of the fluid particles composing the star are on bound orbits at the time of the tidal disruption. We tune the toy model by using the results of fully general-relativistic simulations obtaining relative precisions of few a percent and use it to investigate the space of parameters extensively. In this way we find that the torus mass is largest for systems with highly spinning black holes, small stellar compactnesses and large mass ratios. As an example, tori as massive as $M_{b,\text{tor}} \simeq 1.33\,M_{\odot}$ can be produced for a very extended star with compactness $C\simeq 0.1$ inspiralling around a black hole with dimensionless spin $a=0.85$ and mass ratio $q\simeq 0.3$. However, for a more astrophysically reasonable mass ratio $q \simeq 0.14$ and a canonical value of the stellar compactness $C\simeq 0.145$, the toy model sets a considerably smaller upper limit of $M_{b,\text{tor}} \lesssim 0.34\,M_{\odot}$