In most theoretical models, one or both kHz QPO frequencies are associated with the orbital motion of inhomogeneities or blobs in a thin accretion disk. In the actual calculations, the frequencies are computed in the approximation of an orbiting test particle, neglecting pressure terms. For most equations of state, stars that are massive enough possess an ISCO, and the orbital frequency at the ISCO has been proposed to be one of the two observed frequencies. To first order in the rotation rate, the orbital frequency at the prograde ISCO is given by (see Kluzniak, Michelson, and Wagoner [171])
where
. At larger rotation rates, higher order contributions of
j
as well as contributions from the quadrupole moment
Q
become important and an approximate expression has been derived
by Shibata and Sasaki [276], which, when written as above and truncated to the lowest order
contribution of
Q
and to
, becomes
where
.
Notice that, while rotation increases the orbital frequency at
the ISCO, the quadrupole moment has the opposite effect, which
can become important for rapidly rotating models. Numerical
evaluations of
for rapidly rotating stars have been used in [229] to arrive at constraints on the properties of the accreting
compact object.
In other models, orbits of particles that are eccentric and
slightly tilted with respect to the equatorial plane are
involved. For eccentric orbits, the periastron advances with a
frequency
that is the difference between the Keplerian frequency of
azimuthal motion
and the radial epicyclic frequency
. On the other hand, particles in slightly tilted orbits fail to
return to the initial displacement
from the equatorial plane, after a full revolution around the
star. This introduces a nodal precession frequency
, which is the difference between
and the frequency of the motion out of the orbital plane
(meridional frequency)
. Explicit expressions for the above frequencies, in the
gravitational field of a rapidly rotating neutron star, have been
derived recently by Markovic [222], while in [223] highly eccentric orbits are considered. Morsink and
Stella [231] compute the nodal precession frequency for a wide range of
neutron star masses and equations of state and (in a
post-Newtonian analysis) separate the precession caused by the
Lense-Thirring (frame-dragging) effect from the precession caused
by the quadrupole moment of the star. The nodal and periastron
precession of inclined orbits have also been studied using an
approximate analytic solution for the exterior gravitational
field of rapidly rotating stars [278]. These precession frequencies are relativistic effects and have
been used in several models to explain the kHz QPO
frequencies [291,
250,
2,
169,
5
].
It is worth mentioning that it has recently been found that an ISCO also exists in Newtonian gravity, for models of rapidly rotating low-mass strange stars. The instability in the circular orbits is produced by the large oblateness of the star [170, 339, 5].
In the spin-frequency interpretation, the increase in the oscillation frequency by a few Hz during the burst is explained as follows: The burning shell is supposed to first decouple from the neutron star and then gradually settle down onto the surface. By angular momentum conservation, the shell spins up, giving rise to the observed frequency increase. Cumming et al. [76] compute the expected spin-up in full general relativity and taking into account rapid rotation. Assuming that the angular momentum per unit mass is conserved, the change in angular velocity with radius is given by
where R is the equatorial radius of the star and all quantities are evaluated at the equator. The slow rotation limit of the above result was derived previously by Abramowicz et al. [3]. The fractional change in angular velocity during spin-up can then be estimated as
where
is the coordinate expansion of the burning shell, a quantity
that depends on the shell's composition. Cumming
et al.
find that the spin down expected if the atmosphere rotates
rigidly is a factor of two to three times smaller than observed
values. More detailed modeling is needed to fully explain the
origin and properties of burst oscillations.
Going further A very interesting topic is the modeling of the expected X-ray spectrum of an accretion disk in the gravitational field of a rapidly rotating neutron star as it could lead to observational constraints on the source of the gravitational field. See e.g. [303, 279, 280, 34, 33], where work initiated by Kluzniak and Wilson [172] in the slow rotation limit is extended to rapidly rotating relativistic stars.
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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 |