In particular, the understanding of the magnetospheres of neutron stars has recently improved through
time-dependent numerical simulations, which allow one to investigate the stability of steady-state analytic
solutions of the pulsar equation. Existing attempts have used both ideal relativistic magnetohydrodynamics
and resistive force-free electrodynamics (see [203] and references therein). The investigation by [203]
permits one to rule out some stationary solutions, which are not naturally recovered by the
time-dependent simulation. Further work is needed to account for resistive relativistic MHD
computations.
Another remarkable investigation based on fully-relativistic MHD simulations by [205] has permitted us to understand the properties of the flow produced by a magnetized pulsar wind within a plerionic nebula. The results of the axisymmetric simulations have disclosed the complex dynamics of the post-shock flow, very different from the steady quasi-radial outflow assumed in earlier (spherically-symmetric) analytical models for plerions. In particular, the jet-torus pattern discovered by Chandra in the Crab nebula can be explained within the MHD approximation when the condition of spherical symmetry is no longer enforced, as shown by the simulated synchrotron X-ray images of the MHD numerical data.
General-relativistic MHD winds from rotating neutron stars have also been studied by [66].
Magnetically-dominated winds from stars are central to the angular momentum evolution of these objects,
and GRMHD studies, such as those carried out by [66], may help us understand the observed spin-down of
rotation-powered pulsars with strong magnetic fields.
The use of relativistic hydrodynamic codes to study the stability properties of neutron stars and to compute
mode frequencies of oscillations of such objects has increased in recent years (see, e.g., the Living Reviews
article by Stergioulas [388] and references therein). In the case of GRMHD codes, the first investigations are
only just starting. An obvious advantage of the hydrodynamic approach is that it includes, by construction,
nonlinear effects, which are important in situations where the linearized equations (commonly employed
in calculations of the mode frequencies of pulsating stars) break down. In addition, while for
nonrotating stars the frequencies of normal modes can be computed with perturbative methods and
a theory of gravitational wave asteroseismology has already been formulated, there exist no
accurate frequency determinations for rapidly rotating stars to date, nor has the theory of
gravitational-wave asteroseismology been extended to include the effects of rotation on the oscillation
frequencies. Due to the advanced status of current hydrodynamics codes, however, the computation
of mode frequencies in rapidly rotating relativistic stars might be easier to achieve through
nonlinear numerical evolutions than by using perturbative computations (see, e.g., the results
in [131
, 355, 364, 389
, 99
]).
Hydrodynamic evolutions of polytropic models of spherical neutron stars can be used as test-bed
computations for multidimensional codes. Representative examples are the simulations by [157], with
pseudo-spectral methods, and by [339
] with HRSC schemes. These investigations adopted radial-gauge
polar-slicing coordinates in which the general relativistic equations are expressed in a simple way that
resembles Newtonian hydrodynamics. Gourgoulhon [157] used a numerical code to detect, dynamically, the
zero value of the fundamental mode of a neutron star against radial oscillations. Romero et al. [339]
highlighted the accuracy of HRSC schemes by finding, numerically, a change in the stability behavior of two
slightly different initial neutron star models: for a given EOS, a model with mass
is stable
and a model of
is unstable. More recently, in [383] a method based on the nonlinear evolution
of deviations from a background stationary-equilibrium star was applied to study nonlinear radial
oscillations of a neutron star. The accuracy of the approach permitted a detailed investigation of
nonlinear features such as quadratic and higher-order mode coupling and nonlinear transfer of
energy.
Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core
collapse, crustquakes and corequakes, and binary mergers, and they could become detectable either via
gravitational waves or high-energy radiation. An observational detection of such pulsations would yield
valuable information about the EOS of relativistic stars. In recent years, the time evolution of the
nonlinear equations governing the dynamics of matter and spacetime has been introduced as a
promising new approach for computing mode frequencies [138, 130
, 131
, 390
, 389
, 99
]. For small
amplitudes, the obtained frequencies are in excellent agreement with those expected by linear
perturbation theory, while two-dimensional eigenfunctions can be obtained through a Fourier transform
technique.
As a first step towards the study of pulsations of rapidly-rotating relativistic stars, Font, Stergioulas,
and Kokkotas [138] developed an axisymmetric numerical code called ToniK (see Table 1) that integrates
the GRHD equations in a fixed-background spacetime. The finite-difference code is based on a
state-of-the-art approximate Riemann solver [102] and incorporates different second and third-order TVD
and ENO numerical schemes. This code can accurately evolve rapidly rotating stars for many rotational
periods, even for stars at the mass-shedding limit. The test simulations reported in [138
] show that, for
nonrotating stars, small amplitude oscillations have frequencies that agree to better than 1% with linear
normal-mode frequencies (radial and nonradial) in the Cowling approximation (i.e., when the evolution of
the spacetime variables is neglected). Axisymmetric modes of pulsating nonrotating stars are computed
in [378
], both in Cowling and in fully-coupled evolutions. Contrary to the 2+1 approach followed
by [138
], the code used in [378] evolves the relativistic stars on null spacetime foliations (see
Section 2.2.2).
Until very recently (see below), the quasi-radial modes of rotating relativistic stars had been
studied only under simplifying assumptions, such as in the slow-rotation approximation or in the
relativistic Cowling approximation. An example of the latter is presented in [130], where a
comprehensive study of all low-order
and
axisymmetric modes of uniformly and
rapidly-rotating relativistic stars was presented, using the code of [138]. This was done for a sequence of
appropriately-perturbed stationary rotating stars, from the nonrotating limit to the mass-shedding
limit. The frequencies of the axisymmetric modes are affected significantly by rotation only
when the rotation rate exceeds about 50% of the maximum allowed. As expected, at large
rotation rates, apparent mode crossings between different modes appear. A comparison of the
mode-frequencies computed with ToniK was carried out by [188] with his pizaa code (see Table 1),
which offers an improved treatment of quasistationary scenarios. The agreement found was
satisfactory.
In [131], the first mode frequencies of uniformly-rotating stars in full general relativity and rapid
rotation were obtained, using the three-dimensional code gr_astro. Such frequencies were computed both
in fixed spacetime evolutions and in coupled hydrodynamic and spacetime evolutions. The Cowling runs
allowed a comparison with earlier results reported by [130], obtaining agreement at the 0.5% level. The
fundamental mode frequencies and their first overtones obtained in fully coupled evolutions show a
dependence on the increased rotation, which is similar to the one observed for the corresponding frequencies
in the Cowling approximation [130].
Small-amplitude, nonlinear pulsations of both uniformly and differentially rotating neutron stars were
studied by [389] employing the ToniK code. The centrifugal forces and the degree of differential rotation
have significant effects on the mode eigenfunction and it was found that near the mass-shedding limit, the
pulsations are damped due to shocks forming at the surface of the star. This damping mechanism may
set a small saturation amplitude for modes that are unstable to the emission of gravitational
waves. It was also found that the fundamental quasi-radial mode is split, at least in the Cowling
approximation and mainly in differentially rotating stars, into two different sequences. This
study was revisited by [99] using the CoCoNuT code, which solves the GRHD equations for a
conformally-flat three-metric (see Section 4.4). Differential rotation significantly shifts mode
frequencies to smaller values, increasing the likelihood of detection by current gravitational wave
interferometric detectors. An extended avoided crossing between the
and
first overtones
was observed (previously known to exist from perturbative studies), which is important for
correctly identifying mode frequencies in case of detection. For uniformly rotating stars near
the mass-shedding limit, the existence of the mass-shedding-induced damping of pulsations
was confirmed, even though the effect is not as strong as was previously found in the Cowling
approximation [389].
Neutron stars following a core collapse supernova are rotating at birth and can be subject to various
nonaxisymmetric instabilities (see, e.g., [388] for a review). Among those, if the rotation rate is high enough
that the ratio of rotational kinetic energy
to gravitational potential energy
,
, exceeds
the critical value
inferred from studies with incompressible Maclaurin spheroids, the star is
subject to a dynamic bar-mode (
-mode) instability driven by hydrodynamics and gravity.
Its study is highly motivated these days as such an instability bears important implications in the
prospects of the detection of gravitational radiation from newly-born rapidly-rotating neutron
stars.
Newtonian simulations of the bar-mode instability from perturbed equilibrium models of rotating stars
have shown that and is quite independent of the stiffness of the EOS provided the star is not
strongly differentially rotating. The relativistic simulations of [358
] yield a value of
for
the onset of the instability, while the dynamics of the process closely resembles that found in Newtonian
theory, i.e., unstable models with large enough
develop spiral arms following the formation of bars,
ejecting mass and redistributing the angular momentum. As the degree of differential rotation becomes
higher and more extreme, Newtonian simulations have also shown that
can be as low as
.
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Further relativistic hydrodynamic simulations of the dynamic bar-mode instability have been performed
by [29] using the whisky code and extending the set of models considered by [358]. This study focused on
investigating the persistence of the bar deformation once the instability has reached its saturation and on
the precise determination of the parameter signalling the threshold for the onset of the instability.
Generic nonlinear mode-coupling effects were found to appear during the development of the
instability, which can severely limit the persistence of the bar deformation and even suppress
the instability altogether. An animation of one such simulation can be seen in the movie of
Figure 16
.
A word of caution is needed here, as it remains unclear whether the requirements inferred from
numerical simulations are at all met by core-collapse progenitors. As shown by [384] magnetic
torques can spin down the core of the progenitor, which leads to slowly rotating neutron stars
at birth ( 10 – 15 ms). The most recent, state-of-the-art computations of the evolution
of massive stars, which include angular momentum redistribution by magnetic torques and
spin estimates of neutron stars at birth [172], lead to core-collapse progenitors, which do not
seem to rotate fast enough to guarantee the unambiguous growth of the canonical bar-mode
instability.
Regarding the CFS instability, relativistic hydrodynamic simulations of nonlinear -modes are
presented in [390
], using a version of the gr_astro code (see also [224
] for Newtonian simulations). The
gravitational radiation reaction-driven instability of the
-modes might have important astrophysical
implications, provided the instability does not saturate at low amplitudes by nonlinear effects or by
dissipative mechanisms. Regarding the latter, existing studies have shown that the mode amplitude could
be limited by shear and bulk viscosity, by energy loss to a magnetic field driven by differential rotation, by
shock waves, or by the leak of the
-mode energy into some short wavelength oscillation
modes (see [26
] and references therein). The latter mechanism would dramatically limit the
-mode amplitude to a small value of
, much smaller than those found in the early
simulations of [390, 224] for isentropic stars (see [388] for a complete list of references on the
subject). Energy leak of the
-mode into other fluid modes was also considered by [160]
through Newtonian hydrodynamic simulations, finding a catastrophic decay of the amplitude
only once it has grown to a value larger than that reported by [26]. Detailed analysis showed
that this breakdown is due to nonlinear 3-mode coupling of the
-mode to two other inertial
modes. The saturation amplitude is of
. It is believed that saturation amplitudes of
this order are still appropriate to detect gravitational waves from the
-mode instability in
LMXBs.
As mentioned in Section 5.1.2 the axisymmetric collapse of differentially-rotating magnetized neutron
stars has been studied by [105, 360] in full general relativity. The effects of magnetic fields on the magnetic
braking of a magnetized differentially-rotating relativistic star have been studied by [114]. (Earlier works in
the literature dealt with idealized, incompressible, uniform-density stars.) In order to overcome the
long evolution (Alfvén) timescale involved, of the order of
for the models simulated,
[114
] adopted a perturbative metric approach and considered the limit of slow rotation and
weak magnetic fields (still astrophysically realistic). This approach, however, permitted [114
]
to use a sufficiently high spatial resolution to track the development of the MRI and see its
effects on the dynamics. The most important result found by [114] is that, independent of
resolution, magnetic braking indeed drives the star toward uniform rotation within the timescales
considered.
Many efforts in code development in numerical relativity and relativistic astrophysics aim to simulate the
coalescence of compact binaries, composed of either two black holes, two neutron stars, or a black hole and
a neutron star. The solution of the black-hole–binary problem has seen major breakthroughs in recent years,
to the point that long-term stable evolutions for many orbits are now possible, and for various combinations
of the parameter space of the problem, namely the spins and linear momentum of the black
holes and their mass ratio (see [323] for a review). In this section we focus on simulations of
neutron-star-binaries and neutron-star–black-hole configurations, for which progress has been equally
dramatic.
Neutron-star binaries are among the most promising sources of gravitational radiation to be detected by
the various ground-based interferometers worldwide. The computation of the gravitational waveform during
the most dynamic phase of the coalescence and plunge depends crucially on hydrodynamic, finite-size
effects. This phase begins once the stars, initially in quasi-equilibrium orbits of gradually smaller orbital
radius (due to the emission of gravitational waves) reach the innermost stable circular orbit (ISCO).
Approximately years after formation of the binary system, the gravitational wave frequency enters the
LIGO/VIRGO high frequency band. The final plunge of the two objects takes place on a dynamic timescale
of a few ms. During the last 15 minutes before the stars finally merge, the frequency is inside the
LIGO/VIRGO sensitivity range. About 16,000 cycles of waveform oscillation can be monitored, while
the frequency gradually shifts from
10 Hz to
1 kHz. A perturbative treatment of
gravitational radiation in the quadrupole approximation is valid as long as
and
simultaneously,
being the total mass of the binary,
the neutron star radius,
and
the separation of the two stars. As the stars approach each other and merge, both
inequalities are less valid and therefore, fully-relativistic hydrodynamic calculations become
necessary.
Duez et al. [104] carried out the analytic modelling of the inspiral of corotational and irrotational
relativistic neutron-star binaries as a sequence of quasi-equilibrium configurations, and computed the
gravitational wavetrain from the last phase (a few hundred cycles) of the inspiral. These authors further
showed a practical procedure to construct the entire wavetrain through coalescence by matching
the late inspiral waveform to the one obtained by fully-relativistic hydrodynamic simulations.
Detailed theoretical waveforms of the inspiral and plunge similar to those reported by [104] are
crucial to enhance the chances of experimental detection in conjunction with matched-filtering
techniques.
The accurate simulation of a neutron-star–binary coalescence is, however, one of the most challenging
tasks in numerical relativity. These scenarios involve strong gravitational fields, matter motion with (ultra)
relativistic speeds, relativistic shock waves, and strong magnetic fields. The numerical difficulties are
exacerbated by the intrinsic multidimensional character of the problem and by the inherent
complexities in Einstein’s theory of gravity, such as coordinate degrees of freedom and the possible
formation of curvature singularities (e.g., collapse of matter configurations to black holes). It is thus
not surprising that a large number of the available simulations have been attempted in the
Newtonian (and post-Newtonian) framework (see [329] for a review). Many of these studies
employ Lagrangian particle methods such as SPH, and only a few have considered (less viscous)
high-order finite-difference methods such as PPM [342]. Notwithstanding the difficulties, major
progress has been achieved recently in simulating neutron-star–binary mergers within a relativistic
framework.
Concerning relativistic simulations, Wilson’s formulation of the hydrodynamic equations (see
Section 2.1.2) was used in [416, 418
, 242
]. Such investigations assumed a conformally-flat 3-metric. These
early simulations revealed the unexpected appearance of a “binary-induced collapse instability” of the
neutron stars, prior to the eventual collapse of the final merged object. This effect was reduced, but not
eliminated fully, in revised simulations [242], after Flanagan [124] pointed out an error in the
momentum-constraint equation as implemented by Wilson et al. [416, 418]. A summary of this controversy
can be found in [329]. Subsequent numerical simulations with the full set of Einstein equations (see below)
did not find this effect.
The effect was neither found in the new set of CFC simulations performed by [300, 299]. In the first
investigation a new 3D SPH code was presented (see Section 4.4), tested, and applied to the coalescence of
neutron-star binaries modelled by relativistic polytropes with , and
. Not only the system
dynamics was studied but also gravitational-wave signals and luminosities were computed. The second
investigation improved the microphysical treatment of the neutron-star models by using several nonzero
temperature EOS, such as those of Shen et al. [351] and Lattimer–Swesty [214]. The results were
compared with earlier investigations employing the BSSN equations performed by [375
] (see
below). It was found that the dynamics and outcome of the models depends strongly on the EOS
and on the binary parameters (neutron-star masses and spins). Larger torus masses are found
for asymmetric systems (up to
for a neutron-star mass ratio of 0.55). Within tens
of dynamic timescales only the Lattimer–Swesty EOS leads to a collapse of the postmerger
remnant to a black hole. The gravitational wave emission from these simulations was analyzed
in [298], where the authors concluded that the peak frequency of the post-merger signal is a
sensitive indicator of the EOS, provided the total mass can be determined from the inspiral chirp
signal.
Nakamura et al. started a program in the late 1980’s to simulate neutron-star–binary coalescence in general relativity (see, e.g., [280]). The group developed a three-dimensional code that solves the full set of Einstein and hydrodynamics equations through finite differences in a uniform Cartesian grid, using van Leer’s scheme [403] with TVD flux limiters. Shockwaves are spread out using a tensor artificial-viscosity algorithm. The hydrodynamic equations follow Wilson’s Eulerian formulation and the ADM formalism is adopted for the Einstein equations. This code has been tested with the study of the gravitational collapse of a rotating polytrope to a black hole (comparing to the axisymmetric computation of Stark and Piran [386]). A few results of neutron-star–binary simulations including gravitational radiation are reported in [304].
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The headon collision of two neutron stars (a limiting case of a coalescence event) was considered by
Miller et al. [261], who performed time-dependent relativistic simulations using the gr_astro code. These
simulations analyzed whether the collapse of the final object occurs in prompt timescales (a few
milliseconds) or delayed (after neutrino cooling) timescales (a few seconds). In [349] it was argued that in a
headon collision event, sufficient thermal pressure is generated to support the remnant in quasi-static
equilibrium against (prompt) collapse, prior to slow cooling via neutrino emission (delayed collapse).
In [261], prompt collapse to a black hole was found in the headon collision of two
neutron stars modeled by a polytropic EOS with
. The stars, initially separated by a
proper distance of
, were boosted toward one another at a speed of
(the
Newtonian infall velocity). The simulation employed a Cartesian grid of
points. The time
evolution of this simulation can be followed in the Quicktime movie in Figure 17
. This animation
simultaneously shows the rest-mass density and the internal energy evolution during the on-axis collision.
The formation of the black hole in prompt timescales is signalled by the sudden appearance of
the apparent horizon at
(
in code units). The violet dotted circles
indicate the trapped photons. The animation also shows a moderately-relativistic shockwave
(Lorentz factor
) appearing at
(code units;
; yellow-white), which
eventually is followed by two opposite moving shocks (along the infalling
direction) that
propagate along the atmosphere surrounding the black hole. We note that, using the same code,
critical phenomena has recently been found by [187] in the headon collison of two neutron
stars.
Using the code of [109], Marronetti et al. [234] performed the first dynamic determination in relativistic
gravity of the ISCO of a neutron-star binary (modelled as polytropes). These authors were able to
bracket the location of the ISCO by distinguishing stable circular orbits from unstable plunges. It was found
that, for the simplified models considered, the ISCO frequency varies with compaction, but does not depend
strongly on the stellar spin.
The most comprehensive study of neutron star coalescence in full general relativity has been performed
by Shibata et al. [353, 373
, 374
, 357
, 372
, 370
]. The numerical code, briefly described in Section 4.4,
allows the long-term simulation of the coalescences of both irrotational and corotational binaries, from the
ISCO up to the formation and ringdown of the final collapsed object (either a black hole or a stable neutron
star). The code also includes an apparent horizon finder along with microphysical EOS, and can extract the
gravitational waveforms emitted in the collisions. The early simulations of Shibata and Uryu [374
]
accounted for a large sample of parameters of the binary system, such as the compactness of the
(equal mass) neutron stars (
), the adiabatic index of the
-law EOS
(
), and the maximum density, rest mass, gravitational mass, and total angular
momentum. The initial data correspond to quasiequilibrium states, either corotational or irrotational,
the latter being more realistic from considerations of viscous versus gravitational-radiation
timescales. These initial data are obtained by solving the Einstein constraint equations and
the equations for the gauge variables under the assumption of a conformallyflat 3-metric and
the existence of a helicoidal Killing vector (see [374
] for a detailed explanation). The binaries
are chosen at the innermost orbits for which the Lagrange points appear at the inner edge of
the neutron stars, and the plunge is induced by reducing the initial angular momentum by
2 – 3%.
|
The comprehensive parameter-space survey carried out by [353, 373, 374] shows that the final outcome
of the merger depends sensitively on the initial compactness of the neutron stars before plunge. Hence,
depending on the stiffness of the EOS, controlled through the value of
, if the total rest mass of the
system is
1.3 – 1.7 times larger than the maximum rest mass of a spherical star in isolation, the end
product is a black hole. Otherwise, a marginally-stable massive neutron star forms. In the latter case, the
star is supported against self-gravity by rapid differential rotation. The star could eventually
collapse to a black hole once sufficient angular momentum has dissipated via neutrino emission or
gravitational radiation. In turn, the different outcome of the merger is imprinted in the gravitational
waveforms [374]. Therefore, future detection of high-frequency gravitational waves could help to constrain
the maximum allowed mass of neutron stars. In addition, for prompt black-hole formation, a
disk orbiting around the black hole forms, with a mass less than 1% the total rest mass of the
system.
The input physics of those early simulations was later improved in [372] where hybrid EOS were
adopted to mimic realistic nuclear equations of state. In this approach the EOS is divided into two parts,
, where
is the cold part for which a fitting formula for realistic EOS of cold nuclear
matter was assigned. The simulations used, in particular, the SLy and FPS EOS for which the maximum
allowed ADM mass of cold and spherical neutron stars is
and
, respectively. Apart
from improving the neutron-star models, the simulations of [372
] also considered mergers of unequal-mass
neutron stars. The underlying motivation was to find out whether massive accretion disks form
depending on the initial mass ratio. Disk formation during neutron-star–binary coalescence, a
fundamental issue for cosmological models of short duration GRBs, was indeed found to be enhanced
for unequal-mass neutron stars, in which the less massive star is tidally disrupted during the
evolution. The simulations also showed, in broad agreement with the simplified models, that
depending on the threshold of the ADM mass of the system, which is also EOS dependent, the
outcome is a black hole or a hypermassive neutron star with large ellipticity. For an ADM mass
larger than the threshold, a black hole forms promptly in the merger irrespective of the mass
ratio.
An example of such findings is shown in Figure 18. Both panels in this figure correspond to equal-mass
neutron-star–binary mergers. The left panel is a
merger, while the right panel depicts
the
case. Likewise, both panels display the velocity field and rest-mass isodensity
contours in the equatorial plane at the final time of the evolution for the corresponding simulation. In
addition, the lowest area of each panel shows the topology of the lapse function at the final time of the
corresponding evolution. Despite the small difference of the initial neutron-star masses, the end product of
each merger is remarkably different. In the former case, a hypermassive neutron star is formed, supported
against collapse by centrifugal forces. In the latter case, the end product of this particular merger is
the delayed formation of a rotating black hole. Animations of such simulations are available
at [352].
In the formation of a hypermassive neutron star with an ellipsoidal figure, quasiperiodic gravitational
waves of large amplitude are emitted, with dimensionless strains of about at a distance
of 50 Mpc [357]. While these figures are within detectability for advanced laser-interferometric
gravitational-wave detectors, the frequency of those waves, between 3 and 4 kHz, may, however, be too
high. The eventual detection of such signals might help to further constrain the neutron star
EOS.
A large number of neutron-star–binary merger simulations with an emphasis on the the black-hole
formation case and on the resulting mass of the surrounding disk was presented by [370]. As in the
simulations of [372], hybrid EOS were adopted to mimic realistic stiff nuclear EOS. In order to
determine the mass of the disks that form, the simulations must be continued after black-hole
formation, which [370
] achieve with a simple excision technique. This technique, however, allows for
long enough computations to even extract the ring-down gravitational waves associated with
black-hole quasinormal modes. The resulting frequencies (too large) and amplitudes (too small at 50
Mpc) make unlikely the detection of this part of the signal by advanced laser interferometric
detectors. One of the main results of the simulations of [370
] was to find that the disk mass
steeply increases with decreasing mass ratio for given ADM mass and EOS, suggesting that such
small mass-ratio mergers are good candidates for producing the central engine of short-duration
GRBs. Regarding the end product of the simulations, we note that the results of [370] do not
agree with those of recent similar simulations of [297
], at least for high-mass binaries. The
reasons may be due to the different choice of nuclear EOS by both groups and also, perhaps more
importantly, to the suppression of radiation-reaction effects on the gravitational waves (which rapidly
dissipate angular momentum of the merged object) in the (approximate) CFC simulations
of [297].
It is also worth noting the new neutron-star–binary merger simulations performed by [13, 12],
particularly the latter, for the innovative aspect of dealing with magnetized stars for the first time (which
are, however, described by simple polytropes). The single simulation of a magnetized neutron star merger
reported shows the feasibility of the numerical approach, leading to a strongly differentially-rotating object
for the particular initial conditions used. Differences in the gravitational waveforms are found when
comparing with a purely hydrodynamic merger. Those are motivated by the extremely large
value of the magnetic field considered, which has a maximum magnitude of . In this
respect we point out that Newtonian SPH simulations of magnetized neutron-star mergers
carried out by [324] have revealed the amplification of the magnetic field beyond magnetar
field strength. Such ultrastrong magnetic fields are caused by Kelvin-Helmholtz instabilities
in the shear layer that forms between the neutron stars and lasts for a time scale of only 1
ms.
In recent years the first numerical simulations of the mixed binary system, i.e., that formed by a neutron
star and a black hole, have also been presented [228, 121
, 122
, 375
, 395
, 376
, 113
]. Future simulations of
such systems will benefit from the advances accomplished for the black-hole–binary case, and many are
already adopting the same numerical treatment to achieve long-term stability in black-hole spacetimes,
namely, the moving puncture approach (see [323
] and references therein).
The simulations of [228] considered only the headon collision case, but thanks to the use of six levels of fixed mesh refinement, the evolutions were carried over up to times far beyond the collision, which provided time for the extraction of the gravitational wave signal and for estimating the radiated energy.
Fully three-dimensional simulations of the mixed binary system were first performed by [122]. However,
two important simplifying assumptions were made in this study: first, the black hole was forced to remain
fixed in space by making its mass sufficiently large compared to that of the neutron star, and, secondly, the
treatment of the gravitational field equations was approximate due to the use of the conformal-flatness
approximation. The first assumption (extreme mass-ratio case) places the onset of tidal disruption of the
neutron star outside the ISCO. The initial data are quasiequilibrium models for synchronized
neutron-star polytropes generated as solutions of the conformal thin-sandwich decomposition of the
Einstein field equations (using the LORENE code [227]), from which relaxed SPH configurations
were built and evolved using an SPH code. The main result of this work was to show that
analytic models of mass transfer are not valid for neutron stars with low compactness, as they
are highly unstable. The evolutions led to the formation of accretion disks around the black
hole, which were discussed in [121] in the context of the mechanism that triggers short-hard
GRBs.
The first fully self-consistent simulations of mixed compact binaries have been performed by Shibata and
Uryu [375, 376
] and Shibata and Taniguchi [371
]. These works use quasicircular states as initial
conditions for the simulations, as the timescale of gravitational radiation reaction is a few times
longer than the orbital period, and the black hole is modeled by a nonspinning moving puncture
(see [323] and references therein for details). Furthermore, the first two works assume a corotating
velocity field for the neutron star while the latter also considers the irrotational case. A
-law
equation of state with
is used in all studies. High-order central schemes are used for the
hydrodynamics and the Einstein equations are formulated and solved in the BSSN approach. The single
simulation presented in [375
] showed that a black-hole–plus–massive-disk (
) system is not
formed from nonspinning black-hole–neutron-star binaries, later confirmed in [376
] for a slightly
larger sample of models. A systematic study of black-hole–neutron-star–binary mergers was
later reported in [371
], focusing on the case that the neutron star is tidally disrupted. For
all initial conditions, the neutron star was found to be tidally disrupted near the ISCO. The
larger the size of the neutron star the more massive the resulting torus. The largest value found
was
for the case of a black hole of mass
and a neutron star of mass
.
The most recent simulations of mixed compact binaries are those of [113], and are also based on moving
punctures. In this work, the assumptions adopted by the authors in their early work [122] are relaxed and
the simulations are fully relativistic and self-consistent and do not only cover the extreme mass ratio case.
They differ from those of [375
, 376
, 371
] mainly in the way the initial data are built. Here, instead of
quasicircular states, the conformal thin-sandwich formalism is used (see [395] for details). Other than this,
the same BSSN formulation is used for the evolution of the gravitational field together with a
conservative system for the hydrodynamics equations (that of [108]), which are solved using the HLL
approximate-Riemann solver. The coupling between both sets of equations is based on the Cactus
parallelization framework [244]. Despite different codes and initial data being used, the results of the
simulations of [113] are reassuringly similar to those of [375, 376, 371]; the disks that form have
masses, which may be too small to produce the required total energy output of a soft-hard
GRB.
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