4.3 Hydrodynamical evolution of neutron 4 Simulations of Astrophysical Phenomena4.1 Gravitational collapse

4.2 Accretion onto black holes 

The study of relativistic accretion and black hole astrophysics is currently a very active field of research, both theoretically and observationally (see, e.g., [30] and references therein). On the theoretical side, since the pioneering work by Shakura and Sunyaev [194Jump To The Next Citation Point In The Article] thin disk models, parametrized by the so-called tex2html_wrap_inline4039 viscosity, in which the gas rotates with Keplerian angular momentum which is transported radially by viscous stress, have been applied successfully to many astronomical objects. The thin disk model, however, is not valid for high luminosity systems, as it is unstable at high mass accretion rates. In this regime Abramowicz et al. [7] found a new solution, called the slim disk solution, which is stable against viscous and thermal instabilities. More recently, much theoretical work has been devoted to the problem of slow accretion, motivated by the discovery that many galactic nuclei are under-luminous (e.g. NGC 4258). Proposed accretion models involve the existence of advection-dominated accretion flows (ADAF solution; see, e.g., [155, 153]) or, more recently, advection-dominated inflow outflow solutions (ADIOS solution [31]). The self-similar global ADAF solutions provide an excellent description of systems such as the Galactic center SgrA tex2html_wrap_inline4041 and quiescent soft X-ray transients. In particular they can reproduce very accurately observed spectra from radio to X-ray frequencies. ADAF solutions have been very successfully applied to non-rotating black holes using a pseudo-Newtonian potential [167Jump To The Next Citation Point In The Article], and they have also been extended to the Kerr spacetime (e.g. [6]). Current investigations try to solve the uncertainty concerning the location of the transition radius in a two-component accretion flow, i.e. the radius between the inner ADAF solution and the outer geometrically thin disk.

On the other hand, advances in satellite instrumentation, e.g. the Rossi X-Ray Timing Explorer (RXTE), and the Advanced Satellite for Cosmology and Astrophysics (ASCA), are greatly stimulating, and are guiding theoretical research on accretion physics. The recent discovery of kHz quasi-periodic oscillations in low-mass X-ray binaries extends the frequency range over which these oscillations occur into timescales associated with the innermost regions of the accretion process (for a review see [218]). Moreover, in extragalactic sources spectroscopic evidence (broad iron emission lines) increasingly points to (rotating) black holes being the accreting central objects [215, 111, 42].

Additionally, most of the proposed theoretical models to explain tex2html_wrap_inline3549 -ray bursts involve a, possibly hyper-accreting, black hole at some point of the evolutionary paths [179]: neutron star-neutron star and black hole-neutron star binaries, collapsars, black hole-white dwarf binaries, or common envelope evolution of compact binary systems. Developing the capability of performing accurate numerical simulations of time-dependent accretion flows in regions of strong gravitational fields, possibly dynamic during the hyper-accreting phase, is indeed of enormous interest.

Accretion theory is primarily based on the study of (viscous) stationary flows and their stability properties through linearized perturbations thereof. A well-known example is the solution consisting of isentropic constant angular momentum tori, unstable to a variety of non-axisymmetric global modes, discovered by Papaloizou and Pringle [172] (see [15] for a review of instabilities in astrophysical accretion disks). Establishing the nature of flow instabilities requires highly resolved and accurate time-dependent non-linear numerical investigations in strong gravitational fields. Such simulations have only been attempted, in general relativity, in a few cases and only for inviscid flows.

For a wide range of accretion problems, the Newtonian theory of gravity is adequate for the description of the background gravitational forces (see, e.g., [77]). The extensive experience with Newtonian astrophysics has shown that explorations of the relativistic regime benefit from the use of model potentials. Among those, we mention the Paczynski-Wiita pseudo-Newtonian potential for a Schwarzschild black hole [167], which gives approximations of general relativistic effects with accuracy of tex2html_wrap_inline4045 in regions from the black hole larger than the marginally stable radius, which corresponds to three times the Schwarzschild radius. Nevertheless, for comprehensive numerical work, a full (i.e. three-dimensional) formalism is required, able to cover also the maximally rotating hole. In rotating spacetimes the gravitational forces cannot be captured fully with scalar potential formalisms. Additionally, geometric regions such as the ergo-sphere would be very hard to model without a metric description. Whereas the bulk of emission occurs in regions with almost Newtonian fields, only the observable features attributed to the inner region may crucially depend on the nature of the spacetime.

In the following we present a summary of illustrative time-dependent accretion simulations in relativistic hydrodynamics. We concentrate on multidimensional simulations. In the limit of spherical accretion, exact stationary solutions exist for both Newtonian gravity [38] and relativistic gravity [137]. Such solutions are commonly used as test-beds of time-dependent hydrodynamical codes, by analyzing whether stationarity is maintained during a numerical evolution [95Jump To The Next Citation Point In The Article, 125, 64, 184Jump To The Next Citation Point In The Article, 17].

4.2.1 Time-dependent disk accretion simulations 

Pioneering numerical efforts in the study of black hole accretion [226Jump To The Next Citation Point In The Article, 95Jump To The Next Citation Point In The Article, 92Jump To The Next Citation Point In The Article, 93Jump To The Next Citation Point In The Article] made use of the the so-called frozen star paradigm of a black hole. In this framework, the time ``slicing'' of the spacetime is synchronized with that of asymptotic observers far from the hole. Within this approach Wilson [226] first investigated numerically the time-dependent accretion of inviscid matter onto a rotating (Kerr) black hole. This was the first problem to which his formulation of the hydrodynamic equations, as presented in Section  2.1.2, was applied. Wilson used an axisymmetric hydrodynamical code in cylindrical coordinates studying the formation of shock waves and the X-ray emission in the strong-field regions close to the black hole horizon, being able to follow the formation of thick accretion disks during the simulations.

Wilson's formulation has been extensively used in time-dependent numerical simulations of disk accretion. In [95Jump To The Next Citation Point In The Article] (see also [92]) Hawley and collaborators studied, in the test-fluid approximation and axisymmetry, the evolution and development of non-linear instabilities in pressure-supported accretion disks formed as a consequence of the spiraling infall of fluid with some amount of angular momentum. Their initial models were computed following the analytic theory of relativistic disks presented by Abramowicz et al. [5]. The code used explicit second-order finite difference schemes with a variety of choices to integrate the transport terms of the equations (i.e. those involving changes in the state of the fluid at a specific volume in space). The code also used a staggered grid (with scalars located at the cell centers and vectors at the cell boundaries) for its suitability to difference the transport equations. Discontinuous solutions were stabilized with artificial viscosity terms.

With a three-dimensional extension of the axisymmetric code of Hawley, Smarr, and Wilson [94, 95Jump To The Next Citation Point In The Article], Hawley [93] studied the global hydrodynamic non-axisymmetric instabilities in thick, constant angular momentum accretion gas tori, orbiting around a Schwarzschild black hole. Such simulations showed that in radially wide, nearly constant angular momentum tori, the Papaloizu-Pringle instability saturates in a strong spiral pressure wave, not in turbulence. In addition, the simulations confirmed that accretion flows through the torus could reduce and even halt the growth of the global instability.

Igumenshchev and Beloborodov [102] have performed two-dimensional relativistic hydrodynamical simulations of inviscid transonic disk accretion onto a rotating (Kerr) black hole. The hydrodynamical equations follow Wilson's formulation but the code avoids the use of artificial viscosity. The advection terms are evaluated with an upwind algorithm which incorporates the PPM scheme [52] to compute the fluxes. Their numerical work confirms analytical expectations: (i) The structure of the innermost disc region strongly depends on the black hole spin, and (ii) the mass accretion rate goes as tex2html_wrap_inline4047, tex2html_wrap_inline4049 being the energy gap at the cusp of the torus (i.e. tex2html_wrap_inline4051, tex2html_wrap_inline4053 being the potential at the boundary of the torus) and tex2html_wrap_inline3731 the adiabatic index.

Yokosawa [237, 238], also using Wilson's formulation, studied the structure and dynamics of relativistic accretion disks and the transport of energy and angular momentum in magneto-hydrodynamical accretion onto a rotating black hole. In his code the hydrodynamic equations are solved using the Flux-Corrected-Transport (FCT) scheme [41] (a second-order flux-limiter method in smooth regions which avoids oscillations near discontinuities by reducing the magnitude of the numerical flux), and the magnetic induction equation is solved using the constrained transport method [66]. The code contains a parametrized viscosity based on the tex2html_wrap_inline3615 -model [194]. The simulations revealed different flow patterns, inside the marginally stable orbit, depending on the thickness h of the accretion disk. For thick disks with tex2html_wrap_inline4061, tex2html_wrap_inline4063 being the radius of the event horizon, the flow pattern becomes turbulent.

4.2.2 Wind accretion simulations 

The term ``wind'' or hydrodynamic accretion refers to the capture of matter by a moving object under the effect of the underlying gravitational field. The canonical astrophysical scenario in which matter is accreted in such a non-spherical way was suggested originally by Bondi, Hoyle and Lyttleton [97, 39], who studied, using Newtonian gravity, the accretion onto a gravitating point mass moving with constant velocity through a non-relativistic gas of uniform density. The matter flow inside the accretion radius, after being decelerated by a conical shock, is ultimately captured by the central object. Such a process applies to describe mass transfer and accretion in compact X-ray binaries, in particular in the case in which the donor (giant) star lies inside its Roche lobe and loses mass via a stellar wind. This wind impacts on the orbiting compact star forming a bow-shaped shock front around it. Such an accretion process is also thought to take place during common envelope evolution of a binary system.

Since those analytic studies numerical simulations by an increasing number of authors (see, e.g., [185, 25Jump To The Next Citation Point In The Article] and references therein) have extended the simplified analytic models and have helped to develop a thorough understanding of the hydrodynamic accretion scenario in its fully three-dimensional character. These investigations have revealed the formation of accretion disks and the appearance of non-trivial phenomena such as shock waves or flip-flop (tangential) instabilities.

Most of the existing numerical work has used Newtonian hydrodynamics to study the accretion onto non-relativistic stars. For compact accretors such as neutron stars or black holes the correct numerical modeling requires a general relativistic hydrodynamical description. Within the relativistic frozen star framework, wind accretion onto ``moving'' black holes was first studied in axisymmetry by Petrich et al. [175Jump To The Next Citation Point In The Article]. In this work Wilson's formulation of the hydrodynamic equations was adopted. The integration algorithm was taken from [212] with the transport terms finite-differenced following the prescription given in [95]. An artificial viscosity term of the form tex2html_wrap_inline4065, with a being a constant, was added to the pressure terms of the equations.

More recently, an extensive survey of the morphology and dynamics of relativistic wind accretion past a Schwarzschild black hole was performed in [71Jump To The Next Citation Point In The Article, 70Jump To The Next Citation Point In The Article]. This investigation differs from [175Jump To The Next Citation Point In The Article] in both the use of a conservative formulation for the hydrodynamic equations (the Valencia formulation; see Section  2.1.3) and the use of advanced HRSC schemes. Axisymmetric computations were compared to [175Jump To The Next Citation Point In The Article], finding major differences in the shock location, opening angle, and accretion rates of mass and momentum. The reasons for the discrepancies may be diverse and related to the use of different formulations, numerical schemes and, possibly, to the grid resolution. In [175] canonical grid sizes were extremely coarse, of tex2html_wrap_inline4069 zones in r and tex2html_wrap_inline4073 respectively. The simulations presented in [71, 70Jump To The Next Citation Point In The Article] used much finer grids in every direction.

Non-axisymmetric two-dimensional studies, restricted to the equatorial plane of the black hole, were first performed in [70], motivated by the non-stationary patterns found in Newtonian simulations (see, e.g., [25]). The relativistic computations revealed that initially asymptotic uniform flows always accrete onto the hole in a stationary way which closely resembles the previous axisymmetric patterns.


Click on thumbnail to view image

Figure 8: Exact (solid line) versus numerical (circles) solution for spherical accretion of a perfect fluid in Eddington-Finkelstein coordinates [169Jump To The Next Citation Point In The Article]. The solution extends inside the event horizon of the black hole at r=2 M, where all fields are smooth, only blowing up at the central singularity at r =0.

Papadopoulos and Font [169Jump To The Next Citation Point In The Article] have recently presented a procedure which considerably simplifies the numerical integration of the general relativistic hydrodynamic equations near black holes. Their procedure is based on identifying classes of coordinates in which the black hole metric is free of coordinate singularities at the horizon (unlike the commonly adopted Boyer-Lindquist coordinates), independent of time, and admits a spacelike decomposition. With those coordinates the innermost radial boundary can be placed inside the horizon, allowing for a clean treatment of the entire (exterior) physical domain. In [169] Michel's (spherical) solution was re-derived using a particular coordinate system adapted to the black hole horizon, the Eddington-Finkelstein system. In Fig.  8 a representative sample of hydrodynamical quantities is plotted for this stationary solution. The solid lines correspond to the exact solution and the symbols correspond to the numerical solution. The solution is regular well inside the horizon at r =2 M . The steepness of the hydrodynamic quantities dominates the solution only near the real singularity.

In [73Jump To The Next Citation Point In The Article, 74] this approach was applied to the study of relativistic wind accretion onto rapidly rotating (Kerr) black holes. The effects of the black hole spin on the flow morphology were found to be confined to the inner regions of the black hole potential well. Within this region, the black hole angular momentum drags the flow, wrapping the shock structure around. An illustrative example is depicted in Fig.  9 . The left panel of this figure corresponds to a simulation employing the Kerr-Schild form of the Kerr metric, regular at the horizon. The right panel shows how the accretion pattern would look like, were the computation performed using the more common Boyer-Lindquist coordinates. The transformation induces a noticeable wrapping of the shock around the central hole. The shock would wrap infinitely many times before reaching the horizon. As a result, the computation in these coordinates would be much more challenging than in Kerr-Schild coordinates.


Click on thumbnail to view image

Figure 9: Relativistic wind accretion onto a rapidly rotating Kerr black hole ( a=0.999 M, the hole spin is counter-clock wise) in Kerr-Schild coordinates (left panel). Isocontours of the logarithm of the density are plotted at the final stationary time t =500 M . Brighter colors (yellow-white) indicate high density regions while darker colors (blue) correspond to low density zones. The right panel shows how the flow solution looks like when transformed to Boyer-Lindquist coordinates. The shock appears here totally wrapped around the horizon of the black hole. The box is 12 M units long. The simulation employed a tex2html_wrap_inline3537 -grid of tex2html_wrap_inline3539 zones. Further details are given in [73].

4.3 Hydrodynamical evolution of neutron 4 Simulations of Astrophysical Phenomena4.1 Gravitational collapse

image Numerical Hydrodynamics in General Relativity
José A. Font
© Max-Planck-Gesellschaft. ISSN 1433-8351
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