Włodek Kluźniak

Resonance in Forced Oscillations of an Accretion Disk and Kilohertz Quasi-periodic Oscillations

We have performed numerical simulations of a radially perturbed accretion torus around a black hole or neutron star and find that the torus performs radial and vertical motions at the appropriate epicyclic frequencies. We find clear evidence that vertical motions are excited in a nonlinear resonance when the applied perturbation is periodic in time. The strongest resonant response occurs when the frequency difference of the two oscillations is equal to one-half the forcing frequency, precisely as recently observed in the accreting pulsar SAX J1808.4-3658, where the observed kilohertz quasi-periodic oscillation peak separation is half the spin frequency of 401 Hz.

The Upper Kilohertz Quasi-periodic Oscillation: A Gravitationally Lensed Vertical Oscillation

We show that a luminous torus in the Schwarzschild metric oscillating along its own axis gives rise to a periodically varying flux of radiation, even though the source of radiation is steady and perfectly axisymmetric. This implies that the simplest oscillation mode in an accretion flow, axisymmetric up-and-down motion at the meridional epicyclic frequency, may be directly observable when it occurs in the inner parts of accretion flow around neutron stars and black holes. The high-frequency modulations of the X-ray flux observed in low-mass X-ray binaries at two frequencies (twin kHz QPOs) could then be a signature of strong gravity both because radial and meridional oscillations have different frequencies in non-Newtonian gravity, and because strong gravitational deflection of light rays causes the flux of radiation to be modulated at the higher frequency. Subject headings: X-rays: general 1. Highest frequency in accreting black holes and neutron stars The highest frequencies modulating the X-ray flux observed from accreting neutron stars and black holes continue to attract attention because their values are as high as those of orbital frequencies close to the neutron star surface or to the circular photon orbit around a black hole. The origin of the modulations, known as quasi-periodic oscillations (QPOs) because they are not quite coherent, still remains a major puzzle (see van der Klis 2000 for a review). Faculty of Mathematics and Physics, Charles University, CZ–18000 Prague, Czechia, bursa@sirrah.troja.mff.cuni.cz Theoretical Physics, Chalmers University S-412-96 Göteborg, Sweden, marek@fy.chalmers.se Astronomical Institute, Academy of Sciences, CZ–14131 Prague, Czechia, vladimir.karas@cuni.cz Institute of Astronomy, Zielona Góra University, Wieża Braniborska, Lubuska 2, PL-65-265 Zielona Góra, and Copernicus Astronomical Centre, Bartycka 18, PL-00-716 Warszawa, Poland, wlodek@camk.edu.pl Visiting scientists at the UKAFF supercomputer facility, University of Leicester, EnglandWe show that a luminous torus orbiting a Schwarzschild black hole gives rise to a periodically varying flux of radiation when oscillating along its own axis, even though the source of radiation is steady and perfectly axisymmetric. This implies that the simplest oscillation mode in an accretion flow, axisymmetric up and down motion at the meridional epicyclic frequency, may be directly observable when it occurs in the inner parts of accretion flow around neutron stars and black holes. The high-frequency modulations of the X-ray flux observed in low-mass X-ray binaries at two frequencies (twin kilohertz quasi-periodic oscillations) could then be a signature of strong gravity both because radial and meridional oscillations have different frequencies in non-Newtonian gravity and because strong gravitational deflection of light rays causes the flux of radiation to be modulated at the higher frequency.

The Importance of Discovering a 3:2 Twin-Peak Quasi-periodic Oscillation in an Ultraluminous X-Ray Source, or How to Solve the Puzzle of Intermediate-Mass Black Holes

Recently, twin-peak quasi-periodic oscillations (QPOs) have been observed in a 3 : 2 ratio for three Galactic black hole microquasars with frequencies that have been shown to scale as 1/M, as expected for general relativisitic motion near a black hole. It may be possible to extend this result to distinguish between the following two disparate models that have been proposed for the puzzling ultraluminous X-ray sources (ULXs): (1) an intermediate-mass black hole (M ~ 103 M☉) radiating very near the Eddington limit and (2) a conventional black hole (M ~10 M☉) accreting at a highly super-Eddington rate with its emission beamed along the rotation axis. We suggest that one could discriminate between these models by detecting the counterpart of a Galactic twin-peak QPO in a ULX: the expected frequency for the intermediate-mass black hole model is only about 1 Hz, whereas for the conventional black hole model the expected frequency would be the ~100 Hz value observed for the Galactic microquasars.

Epicyclic Orbital Oscillations in Newton's and Einstein's Dynamics

We apply Feynmans principle, “The same equations have the same solutions”, to Keplers problem and show that Newtons dynamics in a properly curved 3-D space is identical with that described by Einsteins theory in the 3-D optical geometry of Schwarzschilds spacetime. For this reason, rather unexpectedly, Newtons formulae for Keplers problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at r = rms.

Internal resonance in nonlinear disk oscillations and the amplitude evolution of neutron-star kilohertz QPOs

We study the properties of twin kilohertz quasiperiodic oscillations (QPOs) with a simple toy model consisting of two oscillation modes coupled with a general nonlinear force. We examine resonant effects by slowly varying the values of the tunable, and nearly commensurable, eigenfrequencies. The behavior of the true oscillation frequencies and amplitudes during a slow transition through the 3:2 resonance is examined in detail, and it is shown that both are affected significantly by the nonlinearities in the governing equations. In particular, the amplitudes of oscillations reflect a resonant exchange of energy between the modes, and as a result the initially weaker mode may become dominant after the transition. We note that a qualitatively similar behavior was reported for several neutron-star sources where the difference in the amplitudes of neutron-star twin-peak QPOs changes sign as the observed frequency ratio of the QPOs passes through the value 3:2.

Pulsing ULXs: tip of the iceberg?

We consider the three currently known pulsing ultraluminous X--ray sources (PULXs). We show that in one of them the observed spinup rate requires super--Eddington accretion rates at the magnetospheric radius, even if magnetar--strength fields are assumed. In the two other systems a normal--strength neutron star field implies super--Eddington accretion at the magnetosphere. Adopting super--Eddington mass transfer as the defining characteristic of ULX systems, we find the parameters required for self--consistent simultaneous fits of the luminosities and spinup rates of the three pulsed systems. These imply near--equality between their magnetospheric radii

Period doubling and non-linear resonance in the black hole candidate IGR J17091-3624?

R_M

Eddington capture sphere around luminous stars

and the spherization radii

Oscillations of the Eddington capture sphere

R_{rm sph}

Levitating atmospheres of Eddington-luminosity neutron stars

where radiation pressure becomes important and drives mass loss from the accretion disc. We interpret this near--equality as a necessary condition for the systems to appear as pulsed, since if it is violated the pulse fraction is small. We show that as a consequence all PULXs must have spinup rates