Global rotational instabilities in fluids arise from non-axisymmetric modes , where
is
known as the “bar-mode” [239
, 6
]. It is convenient to parameterize a system’s susceptibility to
these modes by the stability parameter
. Here,
is the rotational kinetic
energy and
is the gravitational potential energy. Dynamical rotational instabilities, driven
by Newtonian hydrodynamics and gravity, develop on the order of the rotation period of the
object. For the uniform-density, incompressible, uniformly rotating MacLaurin spheroids, the
dynamical bar-mode instability sets in at
. For differentially rotating fluids with a
polytropic equation of state, numerical simulations have determined that the stability limit
is valid for initial angular momentum distributions that are similar to those of MacLaurin
spheroids [221
, 63, 162, 125, 192
, 118, 248]. If the object has an off-center density maximum,
could
be as low as
[247
, 260
, 192
, 49
]. General relativity may enhance the dynamical bar-mode instability
by slightly reducing
[223, 209]. Secular rotational instabilities are driven by dissipative processes such
as gravitational radiation reaction and viscosity. When this type of instability arises, it develops
on the timescale of the relevant dissipative mechanism, which can be much longer than the
rotation period (e.g., [225]. The secular bar-mode instability limit for MacLaurin spheroids is
.
In an attempt to reduce these high rotation requirements, there has been increasing work studying
bar-mode instabilities driven by dynamical sheer in differentially rotating neutron stars. Sheer instabilities
excite the co-rotating -mode. If viscous forces don’t damp this instability altogether, it is possible that
this instability can occur for
-values as low as
for stars with a large degree of differential
rotation.
In rotating stars, gravitational radiation reaction drives the -modes toward unstable growth [5, 77].
In hot, rapidly rotating neutron stars, this instability may not be suppressed by internal dissipative
mechanisms (such as viscosity and magnetic fields) [152]. If not limited, the dimensionless amplitude
of
the dominant (
)
-mode will grow to order unity within ten minutes of the formation of a
neutron star rotating with a millisecond period. The emitted GWs carry away angular momentum, and will
cause the newly formed neutron star to spin down over time. The spindown timescale and the strength of
the GWs themselves are directly dependent on the maximum value
to which the amplitude is
allowed to grow [153
, 154
]. Originally, it was thought that
. Later work indicated that
may be
[153
, 236, 213, 154
]. Some research suggests that magnetic fields, hyperon cooling, and
hyperon bulk viscosity may limit the growth of the
-mode instability, even in nascent neutron
stars [136, 135, 201, 202, 154, 151, 102, 6] (significant uncertainties remain regarding the efficacy of
these dissipative mechanisms). Furthermore, a study of a simple barotropic neutron star model by Arras et
al. [9] suggests that multimode couplings could limit
to values
. If
is indeed
(see also [97]),Update
GW emission from
-modes in collapsed remnants is
likely undetectable. For the sake of completeness, an analysis of GW emission from
-modes (which
assumes
) is presented in the remainder of this paper. However, because it is quite doubtful
that
is sizeable,
-mode sources are omitted from figures comparing source strengths
and detector sensitivities and from discussions of likely detectable sources in the concluding
section.
There is some numerical evidence that a collapsing star may fragment into two or more
orbiting clumps [96]. If this does indeed occur, the orbiting fragments would be a strong GW
source.
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