Disk 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 nonaxisymmetric global modes,
discovered by Papaloizou and Pringle [313] (see [32] for a review of instabilities in astrophysical accretion
disks). Since the pioneering work by Shakura and Sunyaev [348
], thin disk models, parameterized by the
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. [2] found 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., [285, 284]) and advection-dominated
inflow-outflow solutions (ADIOS solution [48]). The importance of convection for low values of the
viscosity parameter
has also been discussed in the convection-dominated accretion flow
(CDAF solution; see [179] and references therein). The importance of magnetic fields and their
consequences for the stability properties of this solution are critically discussed in [31]. Magnetic
effects have also been long advocated in connection with outflows from black-hole–accretion-disk
systems, particularly when the mass accretion rate is much larger than the Eddington rate. Recent
Chandra observations of the stellar-mass black-hole binary GRO J1655-40 [260] provide strong
evidence supporting the magnetic nature of disk accretion onto black holes and its associated
magnetic-powered winds. Theoretically, MHD outflows (and relativistic jets) are believed to be
driven by energy flow through the accretion disk associated with the magnetic torque. The most
convincing argument for the existence of the torque is the presence of the MRI, which not
only drives the turbulent transport of angular momentum within the disk, self-regulating the
accretion process, but also amplifies the strength of a seed vertical field on a dynamic timescale
given by the inverse of the angular frequency of the disk. Extensive numerical work has been
carried out in the last few years for a better understanding of such complex dynamics (see
below).
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., [191]). Extensive experience with
Newtonian astrophysics has shown that explorations of the relativistic regime benefit from the
use of model potentials. In particular, the Paczyński–Wiita pseudo-Newtonian potential for
a Schwarzschild black hole [307] gives approximations of general-relativistic effects with an
accuracy of 10 – 20% outside the marginally stable radius (at ). Nevertheless, for
comprehensive numerical work, a three-dimensional formalism is required, that is also able to cover
the maximally-rotating black hole. In rotating spacetimes the gravitational forces cannot be
captured fully with scalar potential formalisms. Additionally, geometric regions such as the
ergosphere 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 (as the relativistic X-ray emission lines) may crucially depend on the nature of the
spacetime.
In the following we present a summary of representative time-dependent accretion simulations in relativistic
hydrodynamics and MHD. We concentrate on multidimensional simulations. In the limit of spherical
accretion, exact stationary solutions exist for both Newtonian gravity [57] and relativistic gravity [255].
Such solutions are commonly used to calibrate time-dependent hydrodynamic codes, by analyzing whether
stationarity is maintained during a numerical evolution [171, 237, 117, 339
, 36]. Correspondingly, similar
tests have been proposed for the GRMHD equations, such as the spherically-symmetric accretion
solution of a perfect fluid onto a Schwarzschild black hole in the presence of an (unphysical)
radial magnetic field (
) or the equatorial magnetized inflow solution in the
region between a Kerr black-hole horizon and the marginally stable orbit, derived by [393]
(see also [149
]). These analytic solutions have been used in the literature to assess numerical
codes [149
, 86
, 108
, 24
, 91].
Pioneering numerical efforts in the study of black-hole accretion [411, 171
, 167
, 168
] made use of the
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 black hole. Within this approach, Wilson [411] first
investigated numerically the time-dependent accretion of inviscid matter onto a rotating 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 hydrodynamic code in cylindrical coordinates to
study the formation of shock waves and X-ray emission in the strong-field regions close to the
black-hole horizon, and was 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
thick disk accretion. In a system formed by a black hole surrounded by a thick disk, the gas
flows in an effective (gravitational plus centrifugal) potential, whose structure is comparable
to that of a close binary. The Roche torus surrounding the black hole has a cusp-like inner
edge located at the Lagrange point , where mass transfer driven by the radial pressure
gradient is possible [3]. In [171
] (see also [167]) Hawley and collaborators studied, in the test-fluid
approximation and axisymmetry, the evolution and development of nonlinear instabilities in
pressure-supported accretion disks formed as a consequence of the spiraling infall of fluid with some amount
of angular momentum. 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 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 et al. [170, 171], Hawley [168]
studied the global hydrodynamic nonaxisymmetric instabilities in thick, constant–angular-momentum
accretion-gas tori orbiting around a Schwarzschild black hole. Such simulations showed that
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. Extensions to Kerr spacetimes are reported
in [84].
Igumenshchev and Beloborodov [180] performed two-dimensional relativistic hydrodynamic simulations
of inviscid transonic disk accretion onto a Kerr black hole. The hydrodynamic equations follow Wilson’s
formulation, but the code avoids the use of artificial viscosity. The advection terms are evaluated with an
upwind algorithm that incorporates the PPM scheme [80] to compute the fluxes. Their numerical work
confirms analytic expectations regarding the strong dependence of the structure of the innermost disk
region on the black-hole spin, and the functional form of the mass accretion rate with the potential barrier
between the inner edge of the disk and the cusp,
, with
being the adiabatic
index.
|
Thick accretion disks orbiting black holes, on the other hand, may be subject to the runaway instability,
as first proposed by Abramowicz et al. [1]. Starting with an initial disk filling its Roche lobe, so that mass
transfer is possible through the cusp located at the Lagrange point, two evolutions are feasible when
the mass of the black hole increases: either (i) the cusp moves inwards towards the black hole, which slows
down the mass transfer, resulting in a stable situation, or (ii) the cusp moves deeper inside the disk
material. In the latter case, the mass transfer speeds up, leading to runaway instability. This
instability, whose existence is still a matter of debate (see, e.g., [129
] and references therein),
is an important issue for most models of cosmic GRBs, where the central engine responsible
for the initial energy release is such a system consisting of a thick disk surrounding a black
hole.
In [129] Font and Daigne presented time-dependent simulations of the runaway instability of
constant-angular-momentum thick disks around black holes. The study was performed using a
fully-relativistic hydrodynamics code based on HRSC schemes and the conservative formulation discussed in
Section 2.1.3. The self-gravity of the disk was neglected and the evolution of the central black hole was
assumed to be that of a sequence of Schwarzschild black holes of varying mass. The black-hole mass increase
is determined by the mass accretion rate across the event horizon. In agreement with previous studies based
on stationary models, [129
] found that by allowing the mass of the black hole to grow, the disk becomes
unstable. For all disk–to–black-hole mass ratios considered (between 1 and 0.05), the runaway instability
appears very fast on a dynamic timescale of a few orbital periods (1
100), typically a few 10 ms and
never exceeding 1 s, for a
black hole and a large range of mass fluxes (
).
Extensions of this work to marginally stable (or even stable) constant angular momentum disks
were reported in Zanotti et al. [334] (animations can be visualized at the website listed in
[330]).
An example of the simulations performed by [129] appears in Figure 10. This figure shows eight
snapshots of the time-evolution of the rest-mass density, from
to
. The
contour levels are linearly spaced with
, where
is the maximum value of
the density at the center of the initial disk. In Figure 10
one can clearly follow the transition
from a quasi-stationary accretion regime (panels (1) to (5)) to the rapid development of the
runaway instability in about two orbital periods (panels (6) to (8)). At
, the
disk has almost entirely disappeared inside the black hole whose size has, in turn, noticeably
grown.
On the other hand, recent simulations with nonconstant angular momentum disks and rotating black
holes [128, 82
], show that the runaway instability is strongly suppressed in such cases. In the models built
and evolved by [128, 82
] the angular momentum of the disks is assumed to increase outwards with the
radial distance according to a power law,
, where
and
are constant, such that
corresponds to prograde disks (with respect to the black-hole rotation) and
to retrograde
disks. The results of [82] show that nonconstant angular momentum disks are dramatically
stabilized for very small values of the angular momentum slope
, much smaller than the
Keplerian value
. In light of these results further ingredients need to be incorporated
into the codes to finally discern the likelihood of the instability, namely magnetic fields and
self-gravity.
The impact of magnetic fields on the dynamics of accretion disks around black holes has
long been recognized, but only recently have the first systematic GRMHD simulations become
possible [86, 87
, 85, 149
, 200
, 173
, 24, 202
, 251
, 368
, 256
, 269
]. We note, however, that as early as 1977
Wilson [412] developed the first numerical GRMHD scheme to study the axisymmetric evolution of plasma
near a Kerr black hole. A key idea under intense numerical scrutiny currently is that the combined presence
of a magnetic field and of a differentially rotating Keplerian disk, can lead to the magneto-rotational
instability [32], generating effective viscosity and transporting angular momentum outwards through MHD
turbulence. The dynamo generated by the MRI amplifies the magnetic field on dynamic timescales and, in
turn, the associated magnetic torque is believed to be responsible for the generation of strong MHD
outflows (see below). The sustained level of activity in the numerical modelling of magnetized
accretion disks has also been accompanied by advances in the analytic front. In this respect,
Komissarov [202
] has derived an analytic solution for an axisymmetric, stationary, barotropic
torus with constant distribution of specific angular momentum and a toroidal magnetic field
configuration, which can be used to test multidimensional GRMHD codes in the strong gravity
regime.
Yokosawa [429, 430], using Wilson’s formulation, studied the structure and dynamics of relativistic
accretion disks and the transport of energy and angular momentum in MHD accretion on to a rotating
black hole. In his code, the hydrodynamic equations are solved using the FCT scheme [60] (a second-order
flux-limiter method that avoids oscillations near discontinuities by reducing the magnitude of the numerical
flux), and the magnetic induction equation is solved using the constrained transport method [119]. The
code contains a parameterized viscosity based on the -model [348]. The simulations revealed different
flow patterns inside the marginally-stable orbit, depending on the thickness,
, of the accretion disk. For
thick disks with
,
being the radius of the event horizon, the flow pattern becomes
turbulent.
|
The evolution of the MRI in both the Schwarzschild and Kerr metrics has been numerically investigated,
either in axisymmetry or in three dimensions, in a number of recent papers [86, 149
, 87
, 173
]. The
simulations have been performed with two of the codes listed in Table 2, namely HARM [149
]
and the code of De Villiers and Hawley [86
]. We note that both codes not only implement
different formulations of the GRMHD equations but they also use entirely different numerical
schemes (conservative and incomplete Riemann solver the former, nonconservative and artificial
viscosity the latter). Furthermore, the code of [149
] uses Kerr–Schild coordinates (modified to
increase the resolution where the disk lies), regular at the event horizon (see below), while the
code of [86] employs the more traditional Boyer–Lindquist coordinate system (singular at the
horizon). We note that the initial models in all these investigations are purely hydrodynamic,
constant–angular-momentum, thick disks in equilibrium, with an ad hoc poloidal magnetic field
distribution superposed. The simulations, carried out for a large sample of parameter values (such
as black-hole spin or initial magnetization of the plasma) reveal the following basic evolution
of the accretion process: first, there is an initial rapid growth of the toroidal magnetic field
driven by shear. This is followed by the development of the MRI until saturation. The accretion
process then continues through the quasi-stationary evolution of the disk by sustained MHD
turbulence, which redistributes the density and angular momentum of the disk. An example of
such an evolution is depicted in Figure 11
, taken from [149
]. This figure shows the first (left)
and last snapshot (after about 8 orbital periods) of the logarithm of the density field for a
magnetized, constant–angular-momentum torus around a Kerr black hole with
. The
exponential growth of the MRI happens within the first orbital period, after which the magnetic
field has been amplified to high enough levels to distort the torus and initiate the accretion
process.
De Villiers et al. [87] are able to identify five generic features in the accretion flow: a turbulent and
dense disk body with a roughly constant
(
and
being the pressure scale height and radius,
respectively), an inner disk consisting of an inner torus and a plunging inflow, a magnetized coronal
envelope, an evacuated axial funnel, and a jet-like outflow along the funnel wall. Overall, they
found a decrease in the accretion rate into the black hole with increasing spin parameter. The
accretion rate is highly variable in all models, which makes the characteristics of the inner
torus highly variable as well, which plays a key role in the accretion flow into the black hole.
The outflows are found to be launched from a region in the coronal envelope above the inner
torus, which is closer to the horizon as the black-hole spin increases, resulting in more powerful
jets.
Three-dimensional numerical work has also addressed recently the evolution of accretion disks that are
misaligned (tilted) with respect to the rotation axis of a Kerr black hole. Studies have been
carried out both for hydrodynamic flows [139] and for magnetized flows [140]. These studies are
motivated by the increasing evidence that misaligned black holes may actually exist in nature,
evidence collected both from observational data as from theoretical arguments (see [140
] and
references therein). The simulations have been done using the cosmos and cosmos++ codes listed
in Tables 1 and 2, respectively. For hydrodynamic flows it was found that Lense–Thirring
precession causes the disks to warp differentially, yet the precession timescale is not shorter than
other dynamic timescales. The late-time evolution of the disks showed solid-body precession at
rates consistent with some low frequency quasi-periodic oscillations (QPOs). Similarly, the MRI
turbulent tilted-disk simulations have not shown indication of a Bardeen–Petterson effect at
large.
Recent work on disk accretion has also focused on improving the microphysics of the tori, as in the
axisymmetric magnetohydrodynamic simulations of [368] for neutrino-cooled accretion tori around rotating
black holes. (See also [256] for comparisons between a simple ideal fluid EOS and the relativistic Synge
EOS in GRMHD accretion flows.) The EOS used takes into account neutronization, the nuclear statistical
equilibrium of a gas of free nucleons and -particles, black body radiation, and a relativistic Fermi gas
(neutrinos, electrons, and positrons), along with several cooling processes. It is found that the
neutrino luminosity is
for a torus of mass
and a black hole of
,
irrespective of its spin, a figure compatible with current estimates in short-hard GRBs. However,
the luminosity decreases with decreasing torus mass, and, in light of recent state-of-the-art
simulations of neutron-star binary mergers (see below), the formed black-hole–torus systems have
generally small torus masses (
). On the positive side, however, the simulations
predict a strong variability in the neutrino luminosity (on timescales of a few ms), due to the
MRI-turbulent accretion flow, which could help explain the observed variability of GRB light
curves.
The most promising processes for producing relativistic jets like those observed in AGNs, microquasars, and
GRBs involve the hydromagnetic centrifugal acceleration of material from the accretion disk [49], or the
extraction of rotational energy from the ergosphere of a Kerr black hole by magnetic processes [316
, 50
].
The Blandford–Payne model [49] is based upon the existence of a large-scale poloidal magnetic field
passing through the disk, whose matter dynamics is dominated by magnetic tension. Following
angular-momentum–conservation considerations, accelerated outflows are produced whenever the
magnetic field lines have a sufficient angle with respect to the disk. Correspondingly, in the
Blandford–Znajek mechanism [50
] the black hole is seen as a magnetized rotating conductor whose
rotational energy can be efficiently extracted by means of a magnetic torque, giving rise to a
Poynting flux jet. On the other hand, in the magnetic Penrose process [316] the magnetic field
lines across the ergosphere are twisted by frame dragging. The twist of the lines propagates
outwards as a torsional Alfvén wave train carrying electromagnetic energy and making the
total energy of the plasma near the black hole decrease to negative values. In addition, the
accretion of this plasma by the black hole reduces the black-hole rotational energy. As in the
Blandford–Znajek process, the type of outflows generated by this process is also a Poynting flux jet.
The viability of all these various mechanisms has been investigated numerically in the last few
years.
|
Koide et al. performed the first MHD simulations of jet formation in general relativity [196, 195, 197, 194
, 288
].
Their code uses the 3+1 formalism of general relativistic conservation laws of particle number, momentum,
and energy, and Maxwell equations with infinite electric conductivity. The MHD equations
are numerically solved in the test-fluid approximation (in the background geometry of Kerr
spacetime) using a finite difference symmetric scheme [83]. The Kerr metric is described in
Boyer–Lindquist coordinates, with a radial tortoise coordinate to enhance the resolution near the
horizon.
In [197, 194
], in particular, the GRMHD behavior of a plasma flowing into a rapidly-rotating black hole
(
) in a large-scale magnetic field was investigated numerically. The initial magnetic field is
uniform and strong,
,
being the mass density. The initial Alfvén speed is
. The simulation shows how the rotating black hole drags the inertial frames around in the
ergosphere. The azimuthal component of the magnetic field increases because of azimuthal twisting of
the magnetic field lines, as depicted in Figure 12
. This frame-dragging dynamo amplifies the
magnetic field to a value which, by the end of the simulation, is three times larger than the initial
one. The twist of the magnetic field lines propagates outwards as a torsional Alfvén wave.
The magnetic tension torques the plasma inside the ergosphere in a direction opposite to that
of the black-hole rotation. Therefore, the angular momentum of the plasma outside receives
a net increase. Even though the plasma falls into the black hole, electromagnetic energy is
ejected along the magnetic field lines from the ergosphere, due to the propagation of the Alfvén
wave. By total-energy–conservation arguments, Koide et al. [197
] conclude that the ultimate
result of the generation of an outward Alfvén wave is the magnetic extraction of rotational
energy from the Kerr black hole, a process the authors call the MHD Penrose process (but
see [201
]).
The first time-dependent GRMHD simulations of the magnetically-dominated monopole magnetospheres
of black holes were performed by [200], who found that the numerical solution evolves towards a stable
steady-state solution, which is very close to the corresponding force-free solution found by Blandford &
Znajek [50]. A similar result was reported by [252
] for weakly magnetized accretion disks around Kerr
black holes, who found numerical solutions consistent with the Blandford–Znajek solution in the low-density
funnel region around the black hole. The numerical simulations of [200] showed for the first time the
development of an ultrarelativistic particle wind, Poynting dominated, from a rotating black
hole.
Further numerical work on the Blandford–Znajek mechanism was pursued by Komissarov in [201] using
initial models similar to those employed by Koide et al. [197
, 194
]. The long-term solution shows properties
that are significantly different from those of the initial transient phase studied by [197
, 194
]. In particular,
no regions of negative hydrodynamic ‘energy at infinity’ are seen inside the ergosphere and the MHD
Penrose process does not operate. Similar conclusions were drawn by [252
], which highlights the fact that
the models of [197, 194] were evolved for too short a time to observe unbound mass outflows along the
funnel region (but see [288] for longer evolutions). The results of Komissarov [201
] indicate that the
rotational energy of the black hole continues to be extracted via the purely electromagnetic
Blandford–Znajek mechanism. This effect also operates when the black-hole spin is the maximum allowed,
(extreme Kerr), as has been recently shown by [206]. Perhaps the most important result of
these axisymmetric models has been not finding strong relativistic outflows from the black hole, contrary to
observational data indicating the existence of strongly relativistic jets in numerous scenarios (but see
below).
Ultrarelativistic outflows have not been found either in the comprehensive study of Hawley et
al. [87, 173, 88, 210, 169
]. Their third paper on the series [88], as well as [169], are devoted to studying
the formation of unbound outflows in the simulations. Such unbound outflows, which are produced
self-consistently from accretion disks that do not have large-scale magnetic fields as an initial condition, do
not require rotation in the black hole, yet their strength is greatly enhanced by the black-hole spin. If the
spin of the black hole is high, as in their almost extreme Kerr (Kepler disk) model KDJ with
, the energy ejected in the Poynting-flux–dominated outflows can be tens of percent of
the accreted rest mass. At low spin, kinetic energy and enthalpy of the matter dominate the
outflow energetics. These basic results are in qualitative agreement with axisymmetric simulations
performed by [252
] using the conservative, shock-capturing scheme HARM. The jets formed in the
three-dimensional simulations have two major components: a matter-dominated outflow, which
moves at moderate speed (
) along the centrifugal barrier surrounding an evacuated
axial funnel, and a highly-relativistic, Poynting-flux–dominated jet within the funnel. This jet
is accelerated and collimated by magnetic and gas pressure forces in the inner torus and the
surrounding corona. A critical discussion of the physical origin of the relativistic Poynting jet
formed in these numerical simulations has been conducted by Punsly [325], who points out
the importance of incorporating resistive MHD reconnection in the modelling to gain further
insight.
Axisymmetric models of jet formation allow one to extend the computational time of the simulations at
an affordable cost. This is the case for the simulations reported by [251], which extended the earlier work
of [252] by studying jets from a collapsar GRB model (see below) until and out to radial
distances of
. For this study a new version of the GRMHD code HARM was used. It was
found that, at such large spatial and temporal scales, the Poynting-flux–dominated jet is accelerated by
continuous mass-loading from the disk to reach bulk Lorentz factors of
and maximum terminal
Lorentz factors of
and it collimates to narrow half-opening angles of
. This remarkable result is
in concordance with the values observed in jets associated with GRBs, AGN, or X-ray binary
systems.
At stellar scales, relativistic thick disks (or tori) orbiting around stellar-mass black holes may form
after the gravitational collapse of the core of a rotating massive star () or after a
neutron-star–binary merger. Such an accreting torus–plus–black-hole system is the most likely candidate to
be the mechanism powering gamma-ray bursts. Part of the large amounts of energy released by accretion is
deposited in the low-density region along the rotation axis, where the heated gas expands in a jet-like
fireball. In connection with GRBs, van Putten and Levinson [405] have considered the theoretical
evolution of an accretion torus in suspended accretion. These authors claim that the formation
of baryon-poor outflows may be associated with a dissipative gap in a differentially-rotating
magnetic-flux tube supported by an equilibrium magnetic moment of the black hole. Numerical
simulations of nonideal MHD, incorporating radiative processes, are necessary to validate this
picture.
|
Numerical simulations of relativistic jets propagating through progenitor stellar models of collapsars
have been presented in [9, 435] (see also [266] for GRMHD simulations of jet formation in such context).
The collapsar scenario, proposed by [421], is currently the most favored model for explaining long duration
GRBs. The hydrodynamic simulations performed by [9
] employ the three-dimensional code genesis [7]
with a 2D spherical grid and equatorial-plane symmetry. The gravitational field of the black hole is
described by the Schwarzschild metric, and the relativistic hydrodynamic equations are solved in the
test-fluid approximation using a Godunov-type scheme. Aloy et al. [9] showed that the jet, initially formed
by an ad hoc energy deposition of a few
within a
cone around the rotation
axis, reaches the surface of the collapsar progenitor intact, with a maximum Lorentz factor of
.
Similarly, the genesis code has also been used to perform hydrodynamic simulations of relativistic jets
in the neutron-star–binary merger scenario. Figure 13 shows a snapshot at time
s of an
axisymmetric simulation of the launch and evolution of a relativistic outflow driven by energy deposition
through neutrino-antineutrino annihilation in the vicinity of a black-hole–plus–accretion-torus system
(model B1 of [8
]). The left panel displays the logarithm of the Lorentz factor in the entire computational
domain, while the right panel focuses on the innermost regions closer to the central black hole, depicting the
logarithm of the rest-mass density. Since in this simulation the system is the remnant of a
neutron-star–binary merger, it naturally provides a baryon-free channel along the rotation axis through
which the outflow can reach ultrarelativistic velocities. Indeed, terminal Lorentz factors of about
1000 are attained for initial energy deposition rates of some
erg/s per stereo-radian. As
shown in Figure 13
the outflow is strongly collimated due to the presence of the accretion
torus, which, in turn, makes the gamma-ray emission very weak outside of the polar cones.
The calculations reported by [8
] predict that about one out of a hundred neutron-star mergers
produces a detectable GRB, as its ultrarelativistic jet happens to be sent along the line of
sight 2.
It is worth noting that the simulations performed by [8] have also been used by [11] to illustrate a new
effect in relativistic hydrodynamics that can act as an efficient booster in jets and help explain the observed
ultrarelativistic speeds. This effect, concealed in the subtleties of the relativistic Riemann problem, is purely
hydrodynamic and occurs when large velocities tangential to a discontinuity are present in the
flow, yielding Lorentz factors
or larger in flows with moderate initial Lorentz
factors.
The term “wind” or hydrodynamic accretion refers to the capture of matter by a moving object under the effects of the underlying gravitational field. The canonical astrophysical scenario in which matter is accreted in such a nonspherical way was suggested originally by Bondi, Hoyle, and Lyttleton [175, 58], who studied, using Newtonian gravity, the accretion on to a gravitating point mass moving with constant velocity through a nonrelativistic 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. This process is also believed to occur during the common envelope phase in the evolution of a binary system.
Numerical simulations 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
(see, e.g., [341, 42
, 125
] and references therein). The numerical investigations have revealed the formation
of accretion disks and the appearance of nontrivial phenomena such as shock waves and tangential
(flip-flop) instabilities.
Most of the existing numerical work has used Newtonian hydrodynamics to study the accretion onto
nonrelativistic stars [341]. More recently, Newtonian MHD studies have also investigated the spin-down
of moving magnetars [399]. For compact accretors such as neutron stars or black holes, the
correct numerical modeling requires a general-relativistic (magneto-)hydrodynamic description.
Within the relativistic, frozen star framework, wind accretion onto “moving” black holes was
first studied in axisymmetry by Petrich et al. [317]. In this work, Wilson’s formulation of the
hydrodynamic equations was adopted. The integration algorithm was borrowed from [387] with
the transport terms finite-differenced following the prescription given in [171]. An artificial
viscosity term of the form
, with
a constant, was added to the pressure
terms of the equations in order to stabilize the numerical scheme in regions of sharp pressure
gradients.
|
An extensive survey of the morphology and dynamics of relativistic wind accretion past a Schwarzschild
black hole was performed by [133, 132
]. This investigation differs from [317
] in both the use of a
conservative formulation for the hydrodynamic equations and the use of Godunov-type HRSC schemes.
Axisymmetric computations were compared to [317], finding major differences in the shock location,
opening angle, and accretion rates of mass and momentum. The reasons for the discrepancies are related to
the use of different formulations, numerical schemes, and grid resolution, much higher in [133, 132
].
Nonaxisymmetric two-dimensional studies, restricted to the equatorial plane of the black hole, were
discussed in [132], motivated by the nonstationary patterns found in Newtonian simulations (see,
e.g., [42, 125]). The relativistic computations revealed that initially-asymptotic uniform flows always
accrete onto the black hole in a stationary way that closely resembles the previous axisymmetric
patterns.
In [308], Papadopoulos and Font presented a procedure that simplifies the numerical integration of the
general-relativistic hydrodynamics equations near black holes. This procedure is based on identifying classes
of coordinate systems 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 an unambiguous treatment of the entire (exterior) physical domain. In [135, 136] this
approach was applied to the study of relativistic wind accretion onto rapidly-rotating 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 Figure 14. 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 were the computation performed using the more common Boyer–Lindquist
coordinates. The transformation induces a noticeable wrapping of the shock around the central black
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. We note that such “horizon-penetrating” coordinates (or variations thereof) are
currently employed in most of the state-of-the-art codes developed to study accretion onto black
holes [149, 201, 140, 396].
Semi-analytical studies of finite-sized collections of dust, shaped in the form of stars or shells and falling
isotropically onto a black hole are available in the literature [282, 165, 350, 302, 318]. These early
investigations approximate gravitational collapse by a dust shell of mass falling into a
Schwarzschild black hole of mass
. These studies have shown that for a fixed amount of
infalling mass, the gravitational radiation efficiency is reduced compared to the point particle limit
(
), due to cancellations of the emission from distinct parts of the extended
object.
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In [310], such conclusions were corroborated with numerical simulations of the gravitational
radiation emitted during the accretion process of an extended object onto a black hole. The
first-order deviations from the exact black-hole geometry were approximated using curvature
perturbations induced by matter sources whose nonlinear evolution was integrated using a (nonlinear)
hydrodynamics code (adopting the conservative formulation of Section 2.1.3 and HRSC schemes).
All possible types of curvature perturbations are captured in the real and imaginary parts of
the Weyl tensor scalar (see, e.g., [77]). In the framework of the Newman–Penrose formalism,
the equations for the perturbed Weyl tensor components decouple, and when written in the
frequency domain, they even separate [397]. Papadopoulos and Font [310
] used the limiting
case for Schwarzschild black holes, i.e., the inhomogeneous Bardeen–Press equation [37]. The
simulations showed the gradual excitation of the black-hole quasi-normal–mode frequency by
sufficiently compact shells. Similar studies based on a metric formalism have recently been discussed
in [275, 276].
An example of the simulations of [310] appears in the movie of Figure 15
. This movie shows the time
evolution of the shell density (left panel) and the associated gravitational waveform during a complete
accretion/collapse event. The (quadrupolar) shell is parameterized according to
with
and
. Additionally,
denotes a logarithmic, radial (Schwarzschild)
coordinate. The animation shows the gradual collapse of the shell onto the black hole. This process triggers
the emission of gravitational radiation. In the movie, one can clearly see how the burst of the
emission coincides with the most dynamic accretion phase, when the shell crosses the peak of the
potential and is subsequently captured by the black hole. The gravitational wave signal coincides
with the quasinormal ringing frequency of the Schwarzschild black hole,
. The existence
of an initial burst, separated in time from the physical burst, is also noticeable in the movie.
It just reflects the gravitational radiation content of the initial data (see [310] for a detailed
explanation; this “junk” radiation is also a common inevitable feature in the recent black-hole–binary
simulations [323
]).
One-dimensional numerical simulations of a self-gravitating perfect fluid accreting onto a black hole are
presented in [312], where the effects of mass accretion during the gravitational wave emission from a black
hole of growing mass are explored. Using the conservative formulation outlined in Section 2.2.2 and HRSC
schemes, Papadopoulos and Font [312] performed the simulations adopting an ingoing null foliation of a
spherically-symmetric black-hole spacetime [311]. Such a foliation penetrates the black-hole horizon,
allowing for an unambiguous numerical treatment of the inner boundary. The essence of nonspherical
gravitational perturbations was captured by adding to the (characteristic) Einstein–perfect-fluid system, the
evolution equation for a minimally-coupled massless scalar field. The simulations showed the familiar
damped-oscillatory radiative decay, with both decay rate and frequencies being modulated by the
mass-accretion rate. Any appreciable increase in the horizon mass during the emission reflects on the
instantaneous signal frequency,
, which shows a prominent negative branch in the
evolution
diagram. The features of the frequency evolution pattern reveal key properties of the accretion
event, such as the total accreted mass and the accretion rate. To highlight the recent advances
accomplished in numerical relativity it is worth it linking this result with the 3D simulations of
black-hole formation performed by [30], which have provided information on the ring-down
gravitational signal through which the dynamic black hole relaxes to its stationary state (see
Section 5.1.2).
In a series of recent papers [434, 433
, 269
] it has also been shown that upon the introduction of
perturbations, stable, relativistic, accretion tori manifest a long-term oscillatory behavior lasting for tens of
orbital periods. The associated changes in the mass-quadrupole moment of such disks render these objects
promising sources of high-frequency, detectable gravitational radiation for ground-based interferometers and
advanced resonant bar detectors, particularly for Galactic systems and when the average disk density is
close to nuclear matter density. If the disks are instead composed of low-density material stripped from the
secondary star in Low-Mass X-Ray Binaries (LMXBs), their oscillations could help explain the
high-frequency QPOs observed in the spectra of X-ray binaries. Indeed, such HFQPOs can be explained
in terms of
-mode oscillations of a small-size torus orbiting around a stellar-mass black
hole [332].
The studies reported in the existing papers have considered both Schwarzschild and Kerr black holes, as well as constant and nonconstant (power-law) distributions of the specific angular momentum of the disks. The most recent investigation [269] has accounted for magnetic fields in the tori, which, in addition to satisfying a polytropic EOS and having a constant distribution of the specific angular momentum, were built with a nonzero toroidal magnetic field component, for which equilibrium configurations exist [202]. A representative sample of initial models was built and the dependence of their dynamics on the strength of the magnetic field was investigated in long-term GRMHD evolutions in timescales of 100 orbital periods. Overall, the dynamics of the magnetized tori considered was found to be strikingly similar to that found in the purely hydrodynamic case by [434, 433]. It should be noted that, since the axisymmetric initial models do not have a poloidal magnetic field component, the MRI, which could potentially alter the conclusions, can not develop.
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