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ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD BLACK HOLES

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ORBITAL RESONANCE MODELS OF QPOsIN BRANEWORLDBLACKHOLES

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

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RUSGRAV-1313 Russian Gravitational ConferenceInternational Conference on Gravitation, Cosmology and AstrophysicsJune 23-28, 2008, PFUR, Moscow, Russia

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. References

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

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

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:

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!

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:

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)

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

2.3.Quasiperiodic oscillations

(McClintock & Remillard 2003)

2.3.Quasiperiodic oscillations

(McClintock & Remillard 2003)

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)

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

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

2.6. Properties of the Keplerian and epicyclic frequencies

Local extrema of the Keplerian and epicyclic frequencies:

2.6. Properties of the Keplerian and epicyclic frequencies

Local extrema of the Keplerian and epicyclic frequencies:

Keplerian frequency:

can have a maximumat

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?

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

2.6.2. Local extrema of the vertical epicyclic frequency

NaS

BHs

- BHs: one local maximum at for
- NaS: two or none local extrema

- determine implicitly the resonant radius
- must be related to the radius of the innermost stable circular geodesic

2.9. Strong resonant phenomena - "magic" spin

Resonances sharing the same radius

for special values of BH spin a and brany parameter βstrong 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)

3.1. Microquasars data: 3:2 ratio

Török, Abramowicz, Kluzniak, Stuchlík 2005

3.1. Microquasars data: 3:2 ratio

From the observed twin peak frequencies and the known limits on the mass Mof the central BH, the dimensionless spin aand the tidal charge β can be related assuming a concrete version of the resonance model

The most recent angular momentumestimates from fits of spectral continua:

- Shafee et al. 2006

- McClintock et al. 2006

- Middleton et al. 2006

GRO J1655-40:a ~ (0.65 - 0.75)

GRS 1915+105:a > 0.98a~ 0.7

McClintock & Remillard 2004

Shafee et al. 2006

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 J1655-40.

The only model which matches the observational constraintsis the vertical-precession resonance (Bursa 2005)

McClintock & Remillard 2004

estimate 2

estimate 1

1 - Middleton et al. 2006

2 - McClintock et al. 2006

-1 < β< 0.51

-1 < β< 0.51

- there is not only one specific type of resonance model that could work for both sources simultaneously

- Stuchlík, Z. & Kotrlová, A. 2007, in: Proceedings of RAGtime 8/9: Workshops on black holes and neutron stars, Opava, Hradec nad Moravicí, 15–19/19–21 September 2006/2007, ed. S. Hledík & Z. Stuchlík (Opava: Silesian University in Opava), 323-361
- Stuchlík, Z., Kotrlová, A., & Török, G. 2007, in: Proceedings of RAGtime 8/9: Workshops on black holes and neutron stars, Opava, Hradec nad Moravicí, 15–19/19–21 September 2006/2007, S. Hledík and Z. Stuchlík (Opava: Silesian University in Opava), 363-416
- Stuchlík, Z., Kotrlová, A., & Török, G.: Black holes admitting strong resonant phenomena, 2007, subm.
- Stuchlík, Z. & Kotrlová, A.: Orbital resonances in discs around braneworld Kerr black holes, 2008, subm.
- Kotrlová, A., Stuchlík, Z., & Török, G.: QPOs in strong gravitational field around neutron stars testing braneworld models, 2008, subm.
- 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), 21-28
- 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, 104027
- 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), 39-45
- McClintock, J. E. & Remillard, R. A. 2004, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge Univ. 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), 253-263
- Török, G., Abramowicz, M. A., Kluzniak,W. & Stuchlík, Z. 2005, Astronomy and Astrophysics, 436, 1
- Török, G. 2005, Astronom. Nachr., 326, 856

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