Zdeněk Stuchlík, Gabriel Török, Petr Slaný

1. Multi-resonant models of quasi-periodic oscillations Zdeněk Stuchlík, Gabriel Török, Petr Slaný Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic

2. Introduction: Quasi-periodic oscillations (QPOs) in X-ray from the NS an BH systems and their resonant models Figs on this page:nasa.gov

3. 1.1 Quasiperiodic oscillations power hi-frequency (kHz) QPOs low-frequency QPOs frequency

4. 1.1 Quasiperiodic oscillations of the microquasar GRO 1655-40 from two different observations hi-frequency (kHz) QPOs

5. 1.2 kHz quasiperiodic oscillations: BH and NS

6. 2. Orbital and orbital resonance models of QPOs General belief dominating in the astrophysical community links the kHz QPOs to the orbital motion near the inner edge of an accretion disc. Class of models relates kHz QPOs to orbital resonances…. Figs on this page:nasa.gov

7. 2.1 Geodesic motion models: orbital motion in a strong gravity Imply the existence of the periastron and nodal (Lense-Thirring) precession Stella, L. \& Vietri, M. 1999, Phys. Rev. Lett., 82, 17 related the kHz QPOs to the Keplerian and periastron precession of the blobs close to the inner edge of an accretion disc. - Relativistic precession model

8. 2.2 Orbital resonance models Relativistic precession model (Stella, L. \& Vietri, M. 1999, Phys. Rev. Lett., 82, 17)‏ relates the kHz QPOs to the frequencies of geodesic motion (Keplerian and periastron precession of the blobs close to the inner edge of an accretion disc). Resonance model Kluzniak, W., Abramowicz, M. A., 2000, Phys. Rev. Lett. (submitted); Klu\'zniak, W., \& Abramowicz, M. A., 2001, Acta Physica Polonica B 32, 3605 [http://th-www.if.uj.edu.pl/acta/vol32/t11.htm] Orbital resonance model relates the kHz QPOs to disc oscillation modes corresponding to the frequencies of geodesic motion.

9. 2.2 Orbital resonance models

10. 2.2 Orbital resonance models

11. 2.2 Orbital resonance models

12. 2.2 Resonance frequency scatter

13. 2.2 Resonance in non-linear oscillations

14. 2.3 Orbital resonance models – possible gravitational excitation Investigated hypothesis: The gravitatational perturbations caused either by the surface features or by the binary companion may be relevant as an excitation mechanism which may also “feed“ the resonance. Talk - Hledík, Stuchlík, Konar, Miller

15. 2.3 Orbital resonance models – possible gravitational excitation Investigated hypothesis: The gravitatational perturbations caused either by the surface features or by the binary companion may be relevant as an excitation mechanism which may also “feed“ the resonance.

16. 2.3 Orbital resonance models – possible gravitational excitation Investigated hypothesis: The gravitatational perturbations caused either by the surface features or by the binary companion may be relevant as an excitation mechanism which may also “feed“ the resonance.

17. 2.3 Orbital resonance models – possible gravitational excitation Investigated hypothesis: The gravitatational perturbations caused either by the surface features or by the binary companion may be relevant as an excitation mechanism which may also “feed“ the resonance.

18. 2.3 Orbital resonance models – possible gravitational excitation The gravitatational perturbations caused either by the surface features or by the binary companion may be relevant as an excitation mechanism which may also “feed“ the resonance. The analysis presented for the binary companion can be applied also to black hole systems.

19. 3. Multi-resonance models Phenomenologically, there are two possibilities in the resonance models: one eigenfrequency pair hypothesis or, more eigenfrequency pairs hypothesis… Figs on this page:nasa.gov

20. 3. Multi-resonance models • orbital resonance models involving Keplerian and epicyclic oscillations: – a single resonance connected to one specific radius – more instances of one resonance occuring (excited) at (or close to) more specific radii – more resonances sharing one specific radius strong resonant phenomena – black holes with a specific spin – more resonances occuring (excited) at (or close to) more specific radii (the “ugly” case)‏ • extended resonance model with hump-induced oscillations

21. 3.1 One resonance occuring (excited) at (or close to) more radii Investigated hypothesis:the NS twin peak QPOs originate in a resonance between two modes having time-dependent eigenfrequencies determined by the frequencies of geodesic motion. Talks - Bakala,Torok, Stuchlík, Urbanec - Urbanec, Stuchlík, Torok, Bakala, Čermák

22. 3.1 One resonance occuring (excited) at (or close to) more radii Investigated hypothesis:the NS twin peak QPOs originate in a resonance between two modes having time-dependent eigenfrequencies determined by the frequencies of geodesic motion. For group of sources some of possible frequency relations considered in the Hartle-Thorne metric implies the neutron star mass M ~ 1.5-2M_sun and j ~ 0-0.2. 4U 1636-53 } ~0.3M

23. 3.1 One resonance occuring (excited) at (or close to) more radii For group of sources some of possible models (frequency relations) considered in the Hartle-Thorne metric implies the neutron star mass M ~ 1.5-2M_sun and j ~ 0-0.2. Concrete models give concrete restrictions to the neutron star structure. Genetic Algorithm

24. 3.2 More resonances sharing one specific radius Investigated hypothesis: For a special values of black hole spin more resonances can occur at the same radius. Strong resonance phenomena may arise when the Keplerian and epicyclic frequencies are in the lowest possible ratio. Talk – Kotrlová, Stuchlík, Torok

25. 3.2 More resonances sharing one specific radius Investigated hypothesis: For a special values of black hole spin more resonances can occur at the same radius. Strong resonance phenomena may arise when the Keplerian and epicyclic frequencies are in the lowest possible ratio.

26. 3.2 More resonances sharing one specific radius For a special values of black hole spin more resonances may occur at the same orbit. Strong resonance phenomena may occur in the situation arising for the so called "magic" spin a = 0.983 when the Keplerian and epicyclic frequencies are in the lowest possible ratio 3:2:1 at the common radius.

28. 3.3 Extended resonance model Investigated hypothesis:excitation of the epicyclic oscillations by the processes related to the LNRF orbital velocity hump arising in the case of near-extreme Kerr black holes. Talk – Slaný, Stuchlík, Torok

29. 3.3 Extended resonance model Investigated hypothesis: excitation of the epicyclic oscillations by the processes related to the LNRF orbital velocity hump arising in the case of near-extreme Kerr black holes:

30. 3.3 Extended resonance model In the relevant regions, for increasing rotational parameter the ratios of the epicyclic frequencies to the humpy frequency tend to the ratios of small integers. The regions are close to the 3:1 and 4:1 epicyclic resonant orbits.

31. 4. Conclusions Present: Measured eigenfrequencies of QPOs + properly chosen orbital resonance  black hole (neutron star) parameters Future: Mathematical theory of resonances + physical theory of accretion disks  connection of resonance parameters & physical properties of disks Observed details of QPOs (scatter of resonant frequencies, details of Török´s energy switch effect, etc.) imply restrictions on physical properties of accretion disk (or internal structure of neutron stars)‏