Rotating black holes in the brany universe of the Randall-Sundrum type
with infinite additional dimension are described by the Kerr geometry
with a tidal charge b representing the interaction of the brany black
hole and the bulk spacetime. We investigate the role of the tidal charge
in the orbital resonance model of quasiperiodic oscillations (QPOs) in
black hole systems. The orbital Keplerian frequency ν_{K} and the
radial and vertical epicyclic frequencies ν_{r}, ν_{θ} of
the equatorial, quasicircular geodetical motion are discussed, and the
local maxima of their radial profiles related to Keplerian accretion
discs are given, assuming the inner edge of the disc located at loci of
the innermost stable circular geodesic. The resonant conditions are
given for possible direct (parametric) resonances of the oscillations
with the radial and vertical epicyclic frequencies and for some trapped
oscillations of the warped discs with resonant combinational frequencies
involving the Keplerian and radial epicyclic frequencies. It is shown,
how the tidal charge could influence matching of the observational data
indicating the 3:2 frequency ratio observed in GRS 1915+105 microquasar
with prediction of the orbital resonance model. The "magic"
dimensionless black hole spin enabling presence of strong resonant
phenomena at the radius, where ν_{K}:ν_{θ}:ν_{r} =
3:2:1, is determined in dependence on the tidal charge. Such strong
resonances could be relevant even in sources with highly scattered
resonant frequencies, as those expected in Sgr A*.
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Using known frequencies of the accretion disc twin peak quasiperiodic
oscillations (QPOs) and the known mass of the central black hole, the
black hole dimensionless spin a can be determined, assuming a concrete
version of the orbital resonance model. However, because of large range
of observationally limited values of the black hole mass, its spin can
be estimated with a low precision only. Higher precision of the black
hole spin measurement is possible in the framework of multi-resonance
model of QPOs inspired by complex high-frequency QPO patterns observed
in some black hole and neutron star systems. In the simple orbital
resonance models we determine the spin and mass dependence of the twin
peak frequencies for non-linear resonances of oscillations with the
epicyclic and Keplerian frequencies or their combinations in the case of
a general rational frequency ratio n : m, n > m. In the
multi-resonant model, the twin peak resonances are combined properly to
give the observed frequency set. The multi-resonant model is proposed in
three distinct versions. In the first one, related probably to the
neutron star binary systems, more instances of one resonance occur at
more specific radii. In the second case, more resonances are sharing one
specific radius, allowing for "cooperative" resonant phenomena in the
field of black holes with a specific value of spin. In the third
("ugly") case, more resonances occur at more specific radii; we restrict
our attention to the case of two such resonant radii. For special values
of the spin, only triple-set of frequencies is observed because of
coincidence of some frequencies, allowing determination of the spin from
the triple frequency ratio set. The spin is determined precisely, but
not uniquely as the same frequency set could be relevant for more than
one concrete spin and combination of resonant oscillations.
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We introduce the gravitational potential for the pseudo-Newtonian
description of the gravitational field around static and spherically
symmetric black holes in the universe with the repulsive cosmological
constant, described in terms of the general relativistic approach by the
Schwarzschild-de Sitter geometry. In order to demonstrate the accuracy
of the pseudo-Newtonian approach and its possible applications, we
construct the effective potential for a test particle motion and compare
its behaviour with its general relativistic counterpart. Our results
indicate that the pseudo-Newtonian potential could be useful in
applications of developed Newtonian theories of accretion disks in
astrophysically interesting situations in large galactic structures for
spacetimes with the cosmological parameter y = Λ M^2 / 3 ≤
10^{-6}.
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Marginally stable perfect fluid tori with uniform distribution of
specific angular momentum are determined in the
Reissner-Nordström-de Sitter black-hole and naked-singularity
spacetimes. Perfect fluid toroidal configurations are allowed only in
the spacetimes admitting existence of stable circular geodesics. The
configurations with equipotential surfaces crossing itself in a cusp
allow accretion (inner cusp) and/or excretion (outer cusp) of matter
from the toroidal configuration. The classification of the
Reissner-Nordström-de Sitter spacetimes according to the properties
of the marginally stable tori is given.
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Change of sign of the LNRF-velocity gradient has been found for
accretion discs orbiting rapidly rotating Kerr black holes with spin a
> 0.9953 for Keplerian discs and a > 0.99979 for marginally
stable thick discs. Such a "humpy" LNRF-velocity profiles occur just
above the marginally stable circular geodesic and could be related to
oscillations of accretion discs. The frequency of such "hump"-induced
oscillations can be identified with the maximal rate of change of the
orbital velocity within the "humpy" profile. Therefore, we introduce an
extended orbital resonance model (EXORM) of quasiperiodic oscillations
(QPOs) assuming non-linear resonant phenomena between oscillations with
the orbital epicyclic frequencies and the humpy frequency defined in a
fully general relativistic way. The EXORM is developed for both
Keplerian discs and perfect-fluid tori where the approximation of
oscillations with epicyclic frequencies is acceptable. Clearly, the
EXORM could be applied to the near-extreme Kerr black hole systems
exhibiting relatively complex QPO frequency patterns. Assuming a
Keplerian disc, it can be shown that in the framework of the EXORM, all
the QPOs observed in the microquasar GRS 1915+105 could be explained,
while it is not possible in the case of QPOs observed in the Galactic
Centre source Sgr A*.
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Relation between the lower and upper frequency mode of twin peak
quasiperiodic oscilations observed in neutron star X-ray binaries is
qualitatively well fitted by the frequency relation following from the
relativistic precession model. Assuming this model with no preferred
radius and the probability of a twin QPO excitation being uniform across
the inner edge of an accretion disc, we compare the expected and
observed twin peak QPO distribution in the case of atoll source 4U
1636-53. We find these two distributions highly incompatible. We also
find that the observed distribution roughly corresponds to the expected
one if an additional consideration of preferred resonant orbits is
included. We notice that our findings are relevant for some
disc-oscillation QPO models as well.
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The kHz quasiperiodic oscillations (QPOs) observed in low-mass X-ray
neutron star binaries are most likely connected to the orbital motion in
the accretion disc and show datapoint clustering of frequency ratio
between the upper and lower QPOs in small natural numbers. It is shown
for the atoll source 4U 1636-53 that using the Hartle-Thorne metric to
describe the neutron star spacetime, the data clustered around the
frequency ratios 3/2 and 5/4 could be fitted by three models
(Relativistic Precession, Vertical Precession and Total Precession)
involving the hot spot orbital motion with Keplerian, radial epicyclic
and vertical epicyclic frequencies. We demonstrate that with taking
into account the hotspots interaction with the neutron star magnetic
field the discussed three models can provide good fits implying
reasonable values of the neutron star mass and angular momentum.
Therefore the hypothesis of more instances of one orbital resonance has
the potential to explain the kHz QPO nature in the source 4U 1636-53.
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We further investigate the issue of clustering of kHz QPO frequency
ratios in neutron star low mass X-ray binaries. In this note we report
on the recent analysis of occurrences and properties of kHz QPOs in the
source 4U 1636-53. Assuming that kHz QPOs occur in pairs whose
frequencies are linearly correlated, we find a prominent frequency (or a
narrow frequency region) that separates upper and lower QPO
observations. The two QPO modes are then simultaneously detected mainly
in the vicinity of this transition points. We show that this can be
understood in terms of correlations of QPO properties with frequency,
such as quality factor and rms amplitude. We find that rms amplitudes
and quality factors of both QPOs nearly equal at the transition point.
In addition, the QPO frequencies are nearly commensurable there. We
investigate also five other atoll sources obtaining similar results.
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Several models have been outlined to explain the (upper and lower)
kilohertz quasi-periodic oscillations (QPOs) detected in many accreting
neutron star X-ray binaries. When facing the theory to observation,
rather limited attention has been payed to the mutual relations between
the (correlated) QPO amplitudes and quality factors till now. In this
paper we report on recent results on these relations. For six neutron
star atoll sources (namely 4U 1728-34, 4U 1608-52, 4U 1636-53, 4U
0614+09, 4U 1820-30 and 4U 1735-44) spanning wide range of frequencies
we investigate whether the relationship between the rms amplitudes and
quality factors of the observed kHz QPO modes ν _{L}, ν _{U}
display features that could have a significant meaning in terms of the
proposed QPO models. We find for all the six sources that after the twin
kHz QPOs pass a point (or the narrow interval) where their ratio R
equals 1.5 the lower/upper oscillation becomes stronger/weaker than
other one with increasing QPO frequency. Existence of a similar effect
close to R = 1.33 or R = 1.25 is also indicated. Moreover, for
increasing QPO frequency, shortly after passing 3/2 ratio, the
difference between QPO amplitudes as well as lower QPO quality factor
reaches its maxima on a narrow frequency interval where lower QPO is
much stronger than the upper one. This interval lies between
frequencies corresponding to 3/2 and 4/3 (or 5/4) frequency ratio. This
finding implies restrictions to the orbital QPO models (both hot spot-
and disc oscillations- like) and also to QPO modulation mechanism. In a
wider context, our results may indicate the existence of an energy
overflow between the upper and lower QPO mode when their ratio is close
to ratio of small integral numbers.
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Assuming a resonant origin of the twin peak quasiperiodic oscillations
observed in the X-ray neutron star binary systems, we apply a genetic
algorithm method for selection of neutron star models. It was suggested
that pairs of kilohertz peaks in the X-ray Fourier power density spectra
of some neutron stars reflect a non-linear resonance between two modes
of accretion disk oscillations. We investigate this concept for a
specific neutron star source. Each neutron star model is characterized
by the equation of state (EOS), rotation frequency Ω and central
energy density rho_{c}. These determine the spacetime structure
governing geodesic motion and position dependent radial and vertical
epicyclic oscillations related to the stable circular geodesics.
Particular kinds of resonances (KR) between the oscillations with
epicyclic frequencies, or the frequencies derived from them, can take
place at special positions assigned ambiguously to the spacetime
structure. The pairs of resonant eigenfrequencies relevant to those
positions are therefore fully given by KR, rho_{c}, Ω, EOS and can
be compared to the observationally determined pairs of eigenfrequencies
in order to eliminate the unsatisfactory sets (KR, rho_{c}, Ω,
EOS). For the elimination we use the advanced genetic algorithm. Genetic
algorithm comes out from the method of natural selection when subjects
with the best adaptation to assigned conditions have most chances to
survive. The chosen genetic algorithm with sexual reproduction contains
one chromosome with restricted lifetime, uniform crossing and genes of
type 3/3/5. For encryption of physical description (KR, rho_{c},
Ω, EOS) into the chromosome we use the Gray code. As a fitness
function we use correspondence between the observed and calculated pairs
of eigenfrequencies.
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