ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD KERR BLACK HOLES

1. ORBITAL RESONANCE MODELS OF QPOsIN BRANEWORLD KERR BLACKHOLES Zdeněk Stuchlík and Andrea Kotrlová Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, CZECH REPUBLIC supported byCzech grantMSM 4781305903 Presentation download: www.physics.cz/research in sectionnews International conference on"High-energy sources at different time scales"September 29 – October 3, 2008, Kathmandu, Nepal

2. Outline 1. Braneworld, black holes & the 5th dimension 1.1. Rotating black hole with a tidal charge 2. Quasiperiodic oscillations (QPOs) 2.1. Black hole binaries and accretion disks 2.2. X-ray observations 2.3. QPOs 2.4. Non-linear orbital resonance models 2.5. Orbital motion in a strong gravity 2.6. Properties of the Keplerian and epicyclic frequencies 2.7. Digest of orbital resonance models 2.8. Resonance conditions 2.9. Strong resonant phenomena - "magic" spin 3. Application to microquasars 3.1. Microquasars data: 3:2 ratio 3.2. Results for GRO J1655-40 3.3. Results for GRS 1915+105 3.4. Conclusions 4. Application to the Sgr A* QPOs 5. References

3. Braneworld, black holes & the 5th dimension Braneworld model - Randall & Sundrum 1999: - our observable universe is a slice, a "3-brane" in 5-dimensional bulk spacetime • Aliev & Gümrükçüoglu 2005 (Phys. Rev. D 71, 104027): • exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld • The metric form on the 3-brane • assuming a Kerr-Schild ansatz for the metric on the branethe solution in the standard Boyer-Lindquist coordinates takes the form where

4. Braneworld, black holes & the 5th dimension Braneworld model - Randall & Sundrum 1999: - our observable universe is a slice, a "3-brane" in 5-dimensional bulk spacetime • Aliev & Gümrükçüoglu 2005 (Phys. Rev. D 71, 104027): • exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld • The metric form on the 3-brane • assuming a Kerr-Schild ansatz for the metric on the branethe solution in the standard Boyer-Lindquist coordinates takes the form where • looks exactly like the Kerr–Newman solution in general relativity, in which the square of the electricchargeQ2 is replaced by a tidal charge parameterβ

5. 1.1. Rotating black hole with a tidal charge • The tidal chargeβ • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge for for extreme horizon:

6. 1.1. Rotating black hole with a tidal charge • The tidal chargeβ • means an imprint of nonlocal gravitational effects from the bulk space, • may take on both positive and negative values! The event horizon: • the horizon structuredepends on the sign of the tidal charge This is not allowed in the framework of general relativity! for for extreme horizon: • The effects of the negative tidal chargeβ • tends to increase the horizon radius rh, the radii of the limitingphoton orbit (rph),the innermost bound (rmb) and the innermost stable circular orbits (rms)for both directand retrograde motions of the particles, • mechanism for spinning up the black hole so that its rotation parameter exceeds its mass. Such a mechanism is impossible in general relativity!

7. 1.1. Rotating black hole with a tidal charge Stable circular geodesics exist for rms – the radius of the marginally stable orbit, implicitly determined by the relation dimensionless radial coordinate: Extreme BH:

8. NaS BHs 1.1. Rotating black hole with a tidal charge

9. 1.1. Rotating black hole with a tidal charge • The effects of the negative tidal chargeβ: • tends to increase xh,xph,xmb,xms • mechanism for spinning up the black hole(a > 1)

10. NaS BHs 1.1. Rotating black hole with a tidal charge

11. NaS BHs 1.1. Rotating black hole with a tidal charge

12. 1.1. Rotating black hole with a tidal charge NaS BHs

13. 1.1. Rotating black hole with a tidal charge

14. 1.1. Rotating black hole with a tidal charge

15. 1.1. Rotating black hole with a tidal charge

16. 2. Quasiperiodic oscillations (QPOs) Blackhole high-frequency QPOsin X-ray Figs on this page:nasa.gov

17. radio “X-ray” and visible 2.1.Black hole binaries and accretion disks Figs on this page:nasa.gov

18. 2.2.X-ray observations Light curve: Intensity time Power density spectra (PDS): Power Frequency - a detailed view of the kHz QPOs in Sco X-1 Figs on this page:nasa.gov

19. 2.3. Quasiperiodic oscillations low-frequency QPOs high-frequency QPOs (McClintock & Remillard 2003)

20. 2.3.Quasiperiodic oscillations (McClintock & Remillard 2003)

21. 2.3.Quasiperiodic oscillations (McClintock & Remillard 2003)

22. 2.4. Non-linear orbital resonance models "Standard" orbital resonance models • were introduced by Abramowicz & Kluźniak (2000) who considered the resonance between radial and vertical epicyclic frequency as the possible explanation of NS and BH QPOs(this kind of resonances were, in different context, independently consideredbyAliev & Galtsov, 1981)

23. 2.5. Orbital motion in a strong gravity

24. 2.5. Orbital motion in a strong gravity • Parametric resonance frequencies are in ratio of small natural numbers(e.g, Landau & Lifshitz, 1976), which must hold also in the case of Epicyclic frequencies depend on generic mass as f ~ 1/M. • forced resonances

25. the Keplerian orbital frequency • and the related epicyclic frequencies (radial , vertical ): 2.6. Keplerian and epicyclic frequencies Rotating braneworld BH with massM, dimensionless spina, and the tidal chargeβ:the formulae for Stable circular geodesics exist for

26. 2.6. Properties of the Keplerian and epicyclic frequencies Local extrema of the Keplerian and epicyclic frequencies:

27. 2.6. Properties of the Keplerian and epicyclic frequencies Local extrema of the Keplerian and epicyclic frequencies: Keplerian frequency: can have a maximumat

28. 2.6. Properties of the Keplerian and epicyclic frequencies Local extrema of the Keplerian and epicyclic frequencies: Keplerian frequency: can have a maximumat • Could it be located above • the outher BH horizon xh • the marginally stable orbit xms?

29. 2.6.1. Local extrema of the Keplerian frequency

30. 2.6.1. Local extrema of the Keplerian frequency

31. 2.6.1. Local extrema of the Keplerian frequency

32. NaS BHs has a local maximum for all values of spin a - only for rapidly rotating BHs 2.6.2. Local extrema of the epicyclic frequencies the locations of the local extrema of the epicyclic frequenciesare implicitly given by

33. 2.6.2. Local extrema of the vertical epicyclic frequency NaS BHs • BHs: one local maximum at for • NaS: two or none local extrema

34. NaS BHs 2.6.2. Local extrema of the radial epicyclic frequency

35. 2.6.2. Local extrema of the epicyclic frequencies

36. 2.6.2. Local extrema of the epicyclic frequencies

37. 2.6.2. Local extrema of the epicyclic frequencies

38. 2.6.2. Local extrema of the epicyclic frequencies

39. 2.6.2. Local extrema of the epicyclic frequencies

40. 2.6.2. Local extrema of the epicyclic frequencies

41. 2.6.2. Local extrema of the epicyclic frequencies

42. 2.7. Digest of orbital resonance models

43. 2.8. Resonance conditions • determine implicitly the resonant radius • must be related to the radius of the innermost stable circular geodesic

44. 2.9. Strong resonant phenomena - "magic" spin

45. 2.9. Strong resonant phenomena - "magic" spin

46. 2.9. Strong resonant phenomena - "magic" spin

47. 2.9. Strong resonant phenomena - "magic" spin Resonances sharing the same radius for special values of BH spin a and brany parameter βstrong resonant phenomena(s, t, u – small natural numbers) - spin is given uniquely,- the resonances could be causally related and could cooperate efficiently(Landau & Lifshitz 1976)

48. 2.9. Strong resonant phenomena - "magic" spin

49. 3. Application to microquasars GRO J1655-40 GRS 1915+105

50. 3.1. Microquasars data: 3:2 ratio Török, Abramowicz, Kluzniak, Stuchlík 2005