An analytic Kerr-accretion model as a test solution for a new GR SPH code

This thesis discusses several aspects of the implementation and testing of a new general relativistic computer code based on the smoothed particle hydrodynamics (SPH) method. The code, which has been called SPHINCS (SPH IN Curved Spacetime), is still at the development stage and this is a progress report on some parts of the project. SPHINCS is intended for use in studying various astrophysical applications concerning accretion processes onto black holes (BHs) including: the tidal disruption of a star by a BH; study of the collapsing interiors of massive, rapidly rotating stars within the socalled collapsar scenario; and the interaction of a rotating BH with the inner parts of an accretion disc around it. In the rst part of this thesis we present the mathematical formulation on which SPHINCS is based. Using the language of the so-called 3+1 formalism, we show how the numerical variables and the corresponding evolution equations used in SPHINCS can be derived self-consistently from the Lagrangian of a perfect fluid. We then discuss the implementation of SPHINCS in Kerr spacetime and give explicit expressions for the acceleration terms due to the spacetime metric that enter into the evolution equations. In the second part of the thesis, we introduce an analytic toy model which is intended to be used as a test solution for benchmarking SPHINCS (and possibly other GR hydro codes). This model describes the steady and axisymmetric flow of a rotating gas cloud of non-interacting particles infalling towards a Kerr BH. We demonstrate the utility of the model as a test solution by showing some results of comparing it with several SPH simulations of an idealised collapsar-like setup that implement pseudo-Newtonian potentials for mimicking the e ects of Kerr and Schwarzschild spacetimes. Besides its use as a test solution for numerical codes, we also demonstrate that this model is a useful tool for highlighting purely general relativistic e ects in hydrodynamic flows. Finally, we discuss its potential use as a tool for exploring the parameter space in applications where the assumptions of the model are approximately valid.