Non-axisymmetric relativistic Bondi—Hoyle accretion on to a Schwarzschild black hole

Jose A. Font 1 J. M. Iban ez 0 0 Departamento de Astronoma y Astrofsica, Universidad de Valencia , 46100 Burjassot (Valencia), Spain 1 Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut , Schlaatzweg 1, 14473 Potsdam, Germany A B S T R A C T We present the results of an exhaustive numerical study of fully relativistic non-axisymmetric Bondi-Hoyle accretion on to a moving Schwarzschild black hole. We have solved the equations of general relativistic hydrodynamics with a high-resolution shock-capturing numerical scheme based on a linearized Riemann solver. The numerical code was previously used to study axisymmetric flow configurations past a Schwarzschild black hole. We have analysed and discussed the flow morphology for a sample of asymptotically high Mach number models. The results of this work reveal that initially asymptotic uniform flows always accrete on to the hole in a stationary way, which closely resembles the previous axisymmetric patterns. This is in contrast with some Newtonian numerical studies where violent flip-flop instabilities were found. As discussed in the text, the reason can be found in the initial conditions used in the relativistic regime, as they cannot exactly duplicate the previous Newtonian setups where the instability appeared. The dependence of the final solution on the inner boundary condition as well as on the grid resolution has also been studied. Finally, we have computed the accretion rates of mass and linear and angular momentum. I N T R O D U C T I O N In a previous paper (Font & Iba nez 1998, hereafter Paper I) we studied the morphology and dynamics of relativistic BondiHoyle accretion in axisymmetric flows past a Schwarzschild black hole. The main conclusion of that work was to extend the validity of the BondiHoyle accretion picture (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944) to the relativistic regime, finding that the matter is always accreted on to the hole in a stationary way. Furthermore, if the flow was initially supersonic, the main feature of the accretion pattern was the presence of a shock cone in the solution. At the same time, we checked the validity of our numerical code, revisiting an existing previous calculation (Petrich et al. 1989) using more accurate numerical techniques specifically designed to capture discontinuities. In the present investigation we have extended those studies to account for non-axisymmetric configurations. We have only considered uniform flows at infinity focusing on studying whether or not the flow pattern ultimately reaches a stationary state. The main purpose of this work is, then, to extend previous nonaxisymmetric Newtonian computations to the realm of general relativity. As far as we know, this has not been studied so far. In particular, we plan to address if non-steady patterns, so prominent in some of the two-dimensional (2D) non-axisymmetric simulations in the classical regime, also arise here. During the past few years, a large number of wind accretion simulations past a gravitating body have been performed in Newtonian hydrodynamics (see for example the list of references in Paper I). One of the most interesting features, only revealed by numerical simulations, was the appearance of unstable accretion patterns, contrary to the theoretical (and simplified) BondiHoyle accretion picture (Bondi & Hoyle 1944). This highly non-steady behaviour on the wake of the accretor was only found in 2D nonaxisymmetric simulations, especially when the resolution employed was fine enough. These unstable patterns are characterized by the shock wave moving from side to side (the so-called flipflop instability) and also by the appearance of transient phases of disc formation where the angular momentum accreted by the central object significantly increases. This kind of behaviour has been found not only assuming local density and velocity gradients (Fryxell & Taam 1988; Taam & Fryxell 1989) but also when considering accretion of uniform flows at infinity (Matsuda et al. 1991). However, in detailed three-dimensional (3D) computations performed more recently (Ruffert & Arnett 1994; Ruffert 1994a,b, 1995, 1996) the accretion cone remains quite stable and no sign of flip-flop instability appears unless the flow is assumed to have explicit density gradients at infinity (Ruffert 1997). 2 E Q U AT I O N S , I N I T I A L S E T U P A N D N U M E R I C A L I S S U E S 2.1 Equations In order to study non-axisymmetric patterns in two dimensions we have restricted ourselves to an infinitesimally thin disc in the equatorial plane of the black hole. Therefore, we are using r; f coordinates, instead of the r; v used previously in Paper I. Although this configuration is somehow artificial, it suffices to try to understand the stability of the flow. This kind of setup has also been used in Newtonian simulations, assuming flow past an infinite cylinder (Fryxell & Taam 1988; Taam & Fryxell 1989; Benensohn et al. 1997). With the same definitions introduced in Paper I and using geometrized units (G c 1 with c the speed of light), the equations of (adiabatic) general relativistic hydrodynamics can be written, in the equatorial plane (v p=2) of the Schwarzschild metric, as Uw F rw F fw In this equation, the vector of primitive variables is defined as w r; vr; vf; ; where r and are, respectively, the rest-mass density and the specific internal energy, related to the pressure p via an equation of state, which we choose to be that of an ideal gas law p g 1r; In all previous studies it was shown that the key parameter that controls the appearance of the instability was the size of the accretor. In particular, this size should be a very small fraction of the accretion radius in order to find any evidence of flip-flop. For instance, in Sawada et al. (1989) and Matsuda et al. (1991), the flipflop instability appeared, in 2D, for a central size object of <0:0625ra, where ra is the accretion radius, the natural length scale of this problem, defined as Here, M is the mass of the accreting object, G is the gravitational constant and v is the asymptotic velocity of the fluid. They also found that the flow was stable if the size was larger than <0:125ra. More recently, Benensohn, Lamb & Taam (1997) have performed detailed 2D non-axisymmetric computations with a smaller accretor of size 0.0375ra, finding unstable behaviour as well. In addition, there are other parameters that, combined with the size of the accretor, can play a role in the development of the instability. In particular, one of these is the Mach number of the flow. The larger the Mach number at infinity, the more turbulent the wake becomes. Finally, another extremely important consideration is the position of the shock, namely if it is attached (tail shock) or detached (bow shock) with respect to the central object. This is basically controlled by the values of the asymptotic Mach number and the adiabatic index of the gas. For low Ma (...truncated)