The situation is similar for rotating core collapse. Most
available studies use Newtonian hydrodynamics and account for
gravitational wave emission through the quadrupole formula. The
collapse of a non-rotating star is expected to bounce at nuclear
densities, but if the star is rotating the collapse can also
bounce at subnuclear densities because of the centrifugal force.
In each case, the emerging gravitational waves are dominated by a
burst associated with the bounce. But the waves that follow a
centrifugal bounce can also show large amplitude oscillations
that may be associated with pulsations in the collapsed core.
Such results have been obtained by Mönchmeyer et al. [151]. Some of their models show the presence of modes with different
angular dependence superimposed. Typically, these oscillations
have a period of a few ms and damp out in 20 ms. The
calculations also show that the energy in the higher multipoles
is roughly three orders of magnitude smaller than that of the
quadrupole. More recent simulations by Yamada and Sato [210] and Zwerger and Müller [215] also show post-bounce oscillations. The cited examples are
encouraging, and it seems reasonable that besides the fluid modes
the spacetime modes should also be excited in a generic case. To
show that this is the case one must incorporate general
relativity in the simulations of collapse and coalescence. As yet
there have been few attempts to do this, but an interesting
example is provided by the core collapse studies of Seidel et
al. [187,
186,
182,
183]. They considered axial (odd parity) or polar (even parity)
perturbations of a time-dependent background (that evolves
according to a specified collapse scenario). The extracted
gravitational waves are dominated by a sharp burst associated
with the bounce at nuclear densities. But there are also features
that may be related to the fluid and the
w
-modes. Especially for the axial case, since there are no axial
fluid modes for a non-rotating star, it is plausible that this
mode corresponds to one of the axial
w
-modes of the core. Furthermore, the power spectrum for one of
the simulations discussed in [187] (cf. their Fig. 2) shows some enhancement around 7 kHz.
Recently, Andersson and Kokkotas [18] have studied the excitation of axial modes by sending
gravitational wave pulses to hit the star (see figure
5). The results of this study are encouraging because they provide
the first indication that the
w
-modes can be excited in a dynamical scenario. Similar results
have recently been obtained by Borelli [48] for particles falling onto a neutron star. The excitation of
the axial QNMs by test particles scattered by a neutron star have
been recently studied in detail [203
,
84
,
27
] and some interesting conclusions can be drawn. For example,
that the degree of excitation depends on the details of the
particle's orbit, or that in the cases that we have strong
excitation of the quadrapole oscillations there is a comparable
excitation of higher multipole modes.
These results have prompted the study of an astrophysically
more relevant problem, the excitation of even parity (or polar)
stellar oscillations. In recent work Allen et al. [5] have studied the excitation of the polar modes using two sets
of initial data. First, as previously for the axial case, they
excited the modes by an incoming gravitational wave pulse. In the
second case, the initial data described an initial deformation of
both the fluid and the spacetime. In the first case the picture
was similar to that of the axial modes i.e. the star was excited
and emitted gravitational waves in both spacetime and the fluid
modes. Nearly all of the energy was radiated away in the
w
-modes.
This should be expected since the incoming pulse does not have enough time to couple to the fluid (which has a much lower speed of propagation of information). The infalling gravitational waves are simply affected by the spacetime curvature associated with the star, and the outgoing radiation contains an unmistakable w -mode signature. In the second case one has considerable freedom in choosing the degree of initial excitation of the fluid and the spacetime. In the study the choice was some ``plausible'' initial data inspired by the treatment of the problem in the Newtonian limit. Then the energy emitted as gravitational waves was more evenly shared between the fluid and the spacetime modes, and one could see the f, p, and w modes in the signal. The characteristic signal was (as expected) a short burst (w -modes) followed by a slowly damped sinusoidal wave (fluid modes).
The previous picture emerged as well in a recent study in the
close limit of head-on collision of two neutron stars [6]. There the excitation of both fluid and spacetime modes is
apparent but most of the energy is radiated via the fluid modes.
These new results show the importance of general relativity for
the calculation of the energy emission in violent processes. Up
to now the general way of calculating the energy emission has
been the quadrupole formula. But this formula only accounts for
the fluid deformations and motions, and as we have seen this
accounts only for a part of the total energy.
![]() |
Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |