Hydrodynamical evolutions of polytropic models of spherical
neutron stars can be used as test-bed computations for
multi-dimensional codes. Representative examples are the
simulations by [111], with pseudo-spectral methods, and by [244
] 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 [111] used a numerical code to detect, dynamically, the zero value of
the fundamental mode of a neutron star against radial
oscillations. Romero et al. [244] 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 [274] 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, crust and
core quakes, 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. As a first step
towards the study of pulsations of rapidly rotating relativistic
stars, Font, Stergioulas, and Kokkotas [97] developed an axisymmetric numerical code that integrates the
equations of general relativistic hydrodynamics in a fixed
background spacetime. The finite difference code is based on a
state-of-the-art approximate Riemann solver [70] and incorporates different second- and third-order TVD and ENO
numerical schemes. This code is capable of accurately evolving
rapidly rotating stars for many rotational periods, even for
stars at the mass-shedding limit. The test simulations reported
in [97
] showed that, for non-rotating stars, small amplitude
oscillations have frequencies that agree to better than 1% with
linear normal-mode frequencies (radial and non-radial) in the
so-called Cowling approximation (i.e., when the evolution of the
spacetime variables is neglected). Axisymmetric modes of
pulsating non-rotating stars are computed in [269
], both in Cowling and fully coupled evolutions. Contrary to the
2+1 approach followed by [97
], the code used in [269] 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 can be found in [88], where a comprehensive study of all low-order axisymmetric
modes of uniformly and rapidly rotating relativistic stars was
presented, using the code developed by [97]. The frequencies of quasi-radial and non-radial modes with
spherical harmonic indices
and 3 were computed through Fourier transforms of the time
evolution of selected fluid variables. This was done for a
sequence of appropriately perturbed stationary rotating stars,
from the non-rotating 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.
In [89], the first mode frequencies of uniformly rotating stars in full
general relativity and rapid rotation were obtained, using the
three-dimensional code
GR_ASTRO
described in Section
3.3.2
. Such frequencies were computed both in fixed spacetime
evolutions (Cowling approximation) and in coupled hydrodynamical
and spacetime evolutions. The simulations used a sequence of
(perturbed)
N
=1,
K
=100 () polytropes of central density
, in which the rotation rate
varies from zero to 97% of the maximum allowed rotational
frequency,
. The Cowling runs allowed a comparison with earlier results
reported by [88
], 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 [88].
Relativistic hydrodynamical simulations of nonlinear
r
-modes are presented in [279] (see also [155
] for Newtonian simulations). The gravitational radiation
reaction-driven instability of the
r
-modes might have important astrophysical implications, provided
that the instability does not saturate at low amplitudes by
nonlinear effects or by dissipative mechanisms. Using a version
of the
GR_ASTRO
code, Stergioulas and Font [279] found evidence that the maximum
r
-mode amplitude in isentropic stars is of order unity. The
dissipative mechanisms proposed by different authors to limit the
mode amplitude include shear and bulk viscosity, energy loss to a
magnetic field driven by differential rotation, shock waves, or
the slow leak of the
r
-mode energy into some short wavelength oscillation modes
(see [16
] and references therein). The latter mechanism would
dramatically limit the
r
-mode amplitude to a small value, much smaller than those found
in the simulations of [279,
155] (see [278] for a complete list of references on the subject). Energy leak
of the
r
-mode into other fluid modes has been recently considered
by [113] through Newtonian hydrodynamical simulations, finding a
catastrophic decay of the amplitude only once it has grown to a
value larger than that reported by [16].
The bar mode instability in differentially rotating stars in
general relativity has been analyzed by [261] by means of 3+1 hydrodynamical simulations. Using the code
discussed in Section
3.3.1, Shibata et al. [261] found that the critical ratio of rotational kinetic energy to
gravitational binding energy for compact stars with
is
, slightly below the Newtonian value
for incompressible Maclaurin spheroids. All unstable stars are
found to form bars on a dynamical timescale.
The accurate simulation of a binary neutron star coalescence
is, however, one of the most challenging tasks in numerical
relativity. These scenarios involve strong gravitational fields,
matter motion with (ultra-)relativistic speeds, and relativistic
shock waves. The numerical difficulties are exacerbated by the
intrinsic multi-dimensional 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 most of the (few)
available simulations have been attempted in the Newtonian (and
post-Newtonian) framework (see [235] 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 [246].
Concerning relativistic simulations, Wilson's formulation of
the hydrodynamic equations (see Section
2.1.2) was used in Refs. [303,
304
,
169
]. Such investigations assumed a conformally flat 3-metric, which
reduces the (hyperbolic) gravitational field equations to a
coupled set of elliptic (Poisson-like) equations for the lapse
function, the shift vector, and the conformal factor. 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 [169], after Flanagan [85] pointed out an error in the momentum constraint equation as
implemented by Wilson and coworkers [303,
304]. A summary of this controversy can be found in [235]. Subsequent numerical simulations with the full set of Einstein
equations (see below) did not find this effect.
Nakamura and coworkers have been pursuing a programme to
simulate neutron star binary coalescence in general relativity
since the late 1980's (see, e.g., [196]). The group developed a three-dimensional code that solves the
full set of Einstein equations and self-gravitating matter
fields [214]. The equations are finite-differenced in a uniform Cartesian
grid using van Leer's scheme [290] with TVD flux limiters. Shock waves 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 by
the study of the gravitational collapse of a rotating polytrope
to a black hole (comparing to the axisymmetric computation of
Stark and Piran [276]). Further work to achieve long term stability in simulations of
neutron star binary coalescence is under way [214]. We note that the hydrodynamics part of this code is at the
basis of Shibata's code (Section
3.3.1), which has successfully been applied to simulate the binary
coalescence problem (see below).
The head-on collision of two neutron stars (a limiting case of
a coalescence event) was considered by Miller et al. [183], who performed time-dependent relativistic simulations using
the code described in Section
3.3.2
. 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 [254] it was argued that in a head-on 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 [183], prompt collapse to a black hole was found in the head-on
collision of two
neutron stars modeled by a polytropic EOS with
and
. 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 movie in Figure
11
. 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
(t
=63.194 in code units). The violet dotted circles indicate the
trapped photons. The animation also shows a moderately
relativistic shock wave (Lorentz factor
) appearing at
(code units; yellow-white colors), which eventually is followed
by two opposite moving shocks (along the infalling
z
direction) that propagate along the atmosphere surrounding the
black hole.
The most advanced simulations of neutron star coalescence in
full general relativity are those performed by Shibata and
Uryu [258,
266
,
267
]. Their numerical code, briefly described in Section
3.3.1, allows the long-term simulation of the coalescences of both
irrotational and corotational binaries, from the innermost stable
circular orbit up to the formation and ringdown of the final
collapsed object (either a black hole or a stable neutron star).
Their code also includes an apparent horizon finder, and can
extract the gravitational waveforms emitted in the collisions.
Shibata and Uryu have performed simulations 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
quasi-equilibrium 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
conformally flat 3-metric and the existence of a helical Killing
vector (see [267
] 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
.
The comprehensive parameter space survey carried out by [258,
266,
267
] 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
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. The different outcome of the merger
is reflected in the gravitational waveforms [267
]. 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. Disk formation during binary
neutron star coalescence, a fundamental issue for cosmological
models of short duration GRBs, is enhanced for unequal mass
neutron stars, in which the less massive one is tidally disrupted
during the evolution (Shibata, private communication).
A representative example of one of the models simulated by
Shibata and Uryu is shown in Figure
12
. This figure is taken from [267]. The compactness of each star in isolation is
M
/
R
=0.14 and
. Additional properties of the initial model can be found in
Table 1 of [267
]. The figure shows nine snapshots of density isocontours and the
velocity field in the equatorial plane (z
=0) of the computational domain. At the end of the simulation a
black hole has formed, as indicated by the thick solid circle in
the final snapshot, representing the apparent horizon. The
formation timescale of the black hole is larger the smaller the
initial compactness of each star. The snapshots depicted in
Figure
12
show that once the stars have merged, the object starts
oscillating quasi-radially before the complete collapse takes
place, the lapse function approaching zero
non-monotonically [267
]. The collapse toward a black hole sets in after the angular
momentum of the merged object is dissipated through gravitational
radiation. Animations of various simulations (including this
example) can be found at Shibata's website [257].
To close this section we mention the work of Duez et
al. [72] where, through analytic modelling of the inspiral of
corotational and irrotational relativistic binary neutron stars
as a sequence of quasi-equilibrium configurations, the
gravitational wave-train from the last phase (a few hundred
cycles) of the inspiral is computed. These authors further show a
practical procedure to construct the entire wave-train through
coalescence by matching the late inspiral waveform to the one
obtained by fully relativistic hydrodynamical simulations as
those discussed in the above paragraphs [266]. Detailed theoretical waveforms of the inspiral and plunge
similar to those reported by [72] are crucial to enhance the chances of experimental detection in
conjunction with matched-filtering techniques.
Alternative: single figures.
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Numerical Hydrodynamics in General Relativity
José A. Font http://www.livingreviews.org/lrr-2003-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |