MEASUREMENT OF BRANY BLACK HOLE PARAMETERS IN THE FRAMEWORK OF THE ORBITAL RESONANCE MODEL OF QPOs MEASUREMENT OF BRANY BLACK HOLE PARAMETERS IN THE FRAMEWORK OF THE ORBITAL RESONANCE MODEL OF QPOs Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ Opava, CZECH REPUBLIC supported by Czech grant MSM Presentation download: in section news Zdeněk Stuchlík and Andrea Kotrlová

Outline 1. Braneworld, black holes & the 5th dimension 1.1. Rotating braneworld black holes 2. Quasiperiodic oscillations (QPOs) 2.1. Black hole high-frequency QPOs in X-ray 2.2. Orbital motion in a strong gravity 2.3. Keplerian and epicyclic frequencies 2.4. Digest of orbital resonance models 2.5. Resonance conditions 2.6. Strong resonant phenomena - "magic" spin 3. Applications to microquasars 3.1. Microquasars data: 3:2 ratio 3.2. Results for GRO J Results for GRS Conclusions 4. References

1. 1.Braneworld, black holes & the 5 th dimension Braneworld model - Randall & Sundrum (1999): - our observable universe is a slice, a "3-brane" in 5-dimensional bulk spacetime

1.1. Rotating braneworld black holes The metric form on the 3-brane – –assuming a Kerr-Schild ansatz for the metric on the brane the solution in the standard Boyer-Lindquist coordinates takes the form Aliev & Gümrükçüoglu (2005): – –exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld where

1.1. Rotating braneworld black holes The metric form on the 3-brane – –assuming a Kerr-Schild ansatz for the metric on the brane the solution in the standard Boyer-Lindquist coordinates takes the form Aliev & Gümrükçüoglu (2005): – –exact stationary and axisymmetric solutions describing rotating BH localized on a 3-brane in the Randall-Sundrum braneworld where – –looks exactly like the Kerr-Newman solution in general relativity, in which the square of the electric charge Q 2 is replaced by a tidal charge parameter .

1.1. Rotating braneworld black holes The tidal charge  – –means an imprint of nonlocal gravitational effects from the bulk space, – –may take on both positive and negative values ! The event horizon: – –the horizon structure depends on the sign of the tidal charge condition: for for extreme horizon and

1.1. Rotating braneworld black holes The tidal charge  – –means an imprint of nonlocal gravitational effects from the bulk space, – –may take on both positive and negative values ! The event horizon: – –the horizon structure depends on the sign of the tidal charge condition: for for extreme horizon and

1.1. Rotating braneworld black holes The tidal charge  – –means an imprint of nonlocal gravitational effects from the bulk space, – –may take on both positive and negative values ! The event horizon: – –the horizon structure depends on the sign of the tidal charge condition: for This is not allowed in the framework ! of general relativity ! for extreme horizon and

1.1. Rotating braneworld black holes The tidal charge  – –means an imprint of nonlocal gravitational effects from the bulk space, – –may take on both positive and negative values ! The effects of negative tidal charge  – –tend to increase the horizon radius r h, the radii of the limiting photon orbit (r ph ), the innermost bound (r mb ) and the innermost stable circular orbits (r ms ) for both direct and 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 ! The event horizon: – –the horizon structure depends on the sign of the tidal charge condition: for This is not allowed in the framework ! of general relativity ! for extreme horizon and

2. Quasiperiodic oscillations (QPOs) Fig. on this page: nasa.gov Black hole hi-frequency QPOs in X-ray

hi-frequency QPOs low-frequency QPOs (McClintock & Remillard 2003) 2.1. Quasiperiodic oscillations

(McClintock & Remillard 2003)

2.2. Orbital motion in a strong gravity – –the Keplerian orbital frequency – –and the related epicyclic frequencies (radial, vertical ): ν ~ 1/M Rotating braneworld BH with mass M, dimensionless spin a, and the tidal charge : the formulae for has a local maximum for all values of spin a - only for rapidly rotating BHs x ms – radius of the marginally stable orbit Stable circular geodesics exist for

2.3. Keplerian and epicyclic frequencies

! - can have a maximum at x = x ex ! Notice, that reality condition must be satisfied 2.3. Keplerian and epicyclic frequencies

Can it be located above the outher BH horizon x h the marginally stable orbit x ms ? ! - can have a maximum at x = x ex ! Notice, that reality condition must be satisfied 2.3. Keplerian and epicyclic frequencies

Can it be located above the outher BH horizon x h the marginally stable orbit x ms ? ! - can have a maximum at x = x ex ! Extreme BHs: Notice, that reality condition must be satisfied 2.3. Keplerian and epicyclic frequencies

2.4. Digest of orbital resonance models

2.5. Resonance conditions – –determine implicitly the resonant radius – –must be related to the radius of the innermost stable circular geodesic

2.5. Resonance conditions

2.6. Strong resonant phenomena - "magic" spin

3. Applications to microquasars GRO J

3. Applications to microquasars GRS

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

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

3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge  can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge  can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge  can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge  can be related assuming a concrete version of the resonance model.

3.1. Microquasars data: 3:2 ratio Using known frequencies of the twin peak QPOs and the known mass M of the central BH, the dimensionless spin a and the tidal charge  can be related assuming a concrete version of the resonance model. The most recent angular momentum estimates from fits of spectral continua: GRO J : a ~ ( ) GRS : a > 0.98 a ~ Shafee et al McClintock et al Middleton et al. 2006

3.2. Results for GRO J

Shafee et al McClintock & Remillard 2004 Possible combinations of mass and spin predicted by individual resonance models for the high- frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J

3.2. Results for GRO J Possible combinations of mass and spin predicted by individual resonance models for the high- frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J Shafee et al McClintock & Remillard 2004

3.2. Results for GRO J The only model which matches the observational constraints is the vertical-precession resonance (Bursa 2005) Possible combinations of mass and spin predicted by individual resonance models for the high- frequency QPOs. Shaded regions indicate the likely ranges for the mass (inferred from optical measurements of radial curves) and the dimensionless spin (inferred from the X-ray spectral data fitting) of GRO J Shafee et al McClintock & Remillard 2004

3.2. Results for GRO J

3.3. Results for GRS

McClintock & Remillard 2004

3.3. Results for GRS estimate Middleton et al McClintock & Remillard 2004

3.3. Results for GRS estimate 1 estimate McClintock et al Middleton et al McClintock & Remillard 2004

3.3. Results for GRS

3.4. Conclusions

β = 0

3.4. Conclusions -1 < β < 0.51 (β max for a = 0.7)

3.4. Conclusions -1 < β < 0.51

3.4. Conclusions -1 < β < 0.51

3.4. Conclusions -1 < β < 0.51

3.4. Conclusions -1 < β < 0.51

3.4. Conclusions -1 < β < 0.51

3.4. Conclusions  there is no specific type of resonance model that could work for both sources simultaneously -1 < β < 0.51

THANK YOU FOR YOUR ATTENTION 4. References Abramowicz, M. A. & Kluzniak, W. 2004, in X-ray Timing 2003: Rossi and Beyond., ed. P. Karet, F. K. Lamb, & J. H. Swank, Vol. 714 (Melville: NY: American Institute of Physics), Abramowicz, M. A., Kluzniak, W., McClintock, J. E., & Remillard, R. A. 2004, Astrophys. J. Lett., 609, L63 Abramowicz, M. A., Kluzniak, W., Stuchlík, Z., & Török, G. 2004, in Proceedings of RAGtime 4/5: Workshops on black holes and neutron stars, Opava, 14-16/13-15 October 2002/2003, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 1-23 Aliev, A. N., & Gümrükçüoglu, A. E. 2005, Phys. Rev. D 71, Aliev, A. N., & Galtsov, D. V. 1981, General Relativity and Gravitation, 13, 899 Bursa, M. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September 2004/2005, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), McClintock, J. E. & Remillard, R. A. 2004, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge University Press) McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, Astrophys. J., 652, 518 Middleton, M., Done, C., Gierlinski, M., & Davis, S. W. 2006, Monthly Notices Roy. Astronom. Soc., 373, 1004 Randall, L., & Sundrum, R. 1999, Phys. Rev. Lett. 83, 4690 Shafee, R., McClintock, J. E., Narayan, R., et al. 2006, Astrophys. J., 636, L113 Stuchlík, Z. & Török, G. 2005, in Proceedings of RAGtime 6/7: Workshops on black holes and neutron stars, Opava, 16-18/18-20 September 2004/2005, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena, submitted Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole spin, in prep. Török, G., Abramowicz, M. A., Kluzniak,W. & Stuchlík, Z. 2005, Astronomy and Astrophysics, 436, 1 Török, G. 2005, Astronom. Nachr., 326, 856