

Figure 13:
Time evolution of the rotational velocity profile for a
stationary, rapidly rotating relativistic star (in the Cowling
approximation), using the 3rd order PPM scheme and a
grid. The initial rotational profile is preserved to a high
degree of accuracy, even after 20 rotational periods.
(Figure 1 of Stergioulas and Font [294
].)
The long-term stable evolution of rotating relativistic stars
in 3D simulations has become possible through the use of
High-Resolution Shock-Capturing (HRSC) methods (see [103] for a review). Stergioulas and Font [294
] evolve rotating relativistic stars near the mass-shedding limit
for dozens of rotational periods (evolving only the equations of
hydrodynamics) (see Figure
13), while accurately preserving the rotational profile, using the
3rd order PPM method [65]. This method was shown to be superior to other, commonly used
methods, in 2D evolutions of rotating relativistic stars [106].
Movie 14:
Simulation of a stationary, rapidly rotating neutron star
model in full general relativity, for 3 rotational periods (shown
are iso-density contours, in dimensionless units). The stationary
shape is well preserved at a resolution of
. Simulation by Font, Goodale, Iyer, Miller, Rezzolla, Seidel,
Stergioulas, Suen, and Tobias. Visualization by W. Benger
and L. Rezzolla at the Albert Einstein Institute,
Golm [1].
Fully coupled hydrodynamical and spacetime evolutions in 3D
have been obtained by Shibata [270
] and by Font
et al.
[105
]. In [270], the evolution of approximate (conformally flat) initial data
is presented for about two rotational periods, and in [105
] the simulations extend to several full rotational periods (see
Movie
14), using numerically exact initial data and a monotonized central
difference (MC) slope limiter [315]. The MC slope limiter is somewhat less accurate in preserving
the rotational profile of equilibrium stars than the 3rd order
PPM method, but, on the other hand, it is easier to implement in
a numerical code.
New evolutions of uniformly and differentially rotating stars
in 3D, using different gauges and coordinate systems, are
presented in [93], while new 2D evolutions are presented in [273
].
Shibata, Baumgarte, and Shapiro [275
] study the stability of supramassive neutron stars rotating at
the mass-shedding limit, for a
polytropic EOS. Their 3D simulations in full general relativity
show that stars on the mass-shedding sequence, with central
energy density somewhat larger than that of the maximum mass
model, are dynamically unstable to collapse. Thus, the dynamical
instability of rotating neutron stars to axisymmetric
perturbations is close to the corresponding secular instability.
The initial data for these simulations are approximate,
conformally flat axisymmetric solutions, but their properties are
not very different from exact axisymmetric solutions even near
the mass-shedding limit [73]. It should be noted that the approximate minimal distortion
(AMD) shift condition does not prove useful in the numerical
evolution, once a horizon forms. Instead, modified shift
conditions are used in [275]. In the above simulations, no massive disk around the black
hole is formed, as the equatorial radius of the initial model is
inside the radius which becomes the ISCO of the final black hole.
This could change if a different EOS is chosen.
Shibata, Baumgarte, and Shapiro [274
] study the dynamical bar-mode instability in differentially
rotating neutron stars, in fully relativistic 3D simulations.
They find that stars become unstable when rotating faster than a
critical value of
. This is only somewhat smaller than the Newtonian value of
. Models with rotation only somewhat above critical become
differentially rotating ellipsoids, while models with
much larger than critical also form spiral arms, leading to mass
ejection (see Figure
15, and Movies
16
and
17). In any case, the differentially rotating ellipsoids formed
during the bar-mode instability have
, indicating that they will be secularly unstable to bar-mode
formation (driven by gravitational radiation or viscosity). The
decrease of the critical value of
for dynamical bar formation due to relativistic effects has been
confirmed by post-Newtonian simulations [259].
Figure 15:
Density contours and velocity flow for a neutron star model
that has developed spiral arms, due to the dynamical bar-mode
instability. The computation was done in full General Relativity.
(Figure 4 of Shibata, Baumgarte, and Shapiro [274]; used with permission).
Movie 16:
Simulation of the development of the dynamical bar-mode
instability in a rapidly rotating relativistic star. Spiral arms
form within a few rotational periods. The different colors
correspond to different values of the density, while the
computation was done in full general relativity. Movie produced
at the University of Illinois by T.W. Baumgarte,
S.L. Shapiro, and M. Shibata, with the assistance of
the Illinois Undergraduate Research Team [29
]; used with permission.
Movie 17:
Gravitational wave emission during the development of the
dynamical bar-mode instability in a rapidly rotating relativistic
star. The gravitational wave amplitude in a plane containing the
rotation axis is shown. At large distances, the waves assume a
quadrupole-like angular dependence. Movie produced at the
University of Illinois by T.W. Baumgarte, S.L. Shapiro,
and M. Shibata, with the assistance of the Illinois
Undergraduate Research Team [29]; used with permission.


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Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-2003-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to
livrev@aei.mpg.de
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