ORBITAL RESONANCE MODELS OF QPOs
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ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD BLACK HOLES. Zdeněk Stuchlík and Andrea Kotrlov á. Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezru č ovo n á m. 13, CZ-74601 Opava, CZECH REPUBLIC. supported by Czech grant MSM 4781305903.

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ORBITAL RESONANCE MODELS OF QPOs IN BRANEWORLD BLACK HOLES

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Orbital resonance models of qpos in braneworld black holes

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

in sectionnews

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


Outline

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


Orbital resonance models of qpos in braneworld black holes

  • 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


Orbital resonance models of qpos in braneworld black holes

  • 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


Orbital resonance models of qpos in braneworld black holes

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:


Orbital resonance models of qpos in braneworld black holes

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!


Orbital resonance models of qpos in braneworld black holes

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:


Orbital resonance models of qpos in braneworld black holes

NaS

BHs

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

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)


Orbital resonance models of qpos in braneworld black holes

NaS

BHs

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

NaS

BHs

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

1.1. Rotating black hole with a tidal charge

NaS

BHs


Orbital resonance models of qpos in braneworld black holes

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

1.1. Rotating black hole with a tidal charge


Orbital resonance models of qpos in braneworld black holes

2. Quasiperiodic oscillations (QPOs)

Blackhole high-frequency QPOsin X-ray

Figs on this page:nasa.gov


Orbital resonance models of qpos in braneworld black holes

radio

“X-ray”

and visible

2.1.Black hole binaries and accretion disks

Figs on this page:nasa.gov


Orbital resonance models of qpos in braneworld black holes

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


Orbital resonance models of qpos in braneworld black holes

2.3. Quasiperiodic oscillations

low-frequency

QPOs

high-frequency

QPOs

(McClintock & Remillard 2003)


Orbital resonance models of qpos in braneworld black holes

2.3.Quasiperiodic oscillations

(McClintock & Remillard 2003)


Orbital resonance models of qpos in braneworld black holes

2.3.Quasiperiodic oscillations

(McClintock & Remillard 2003)


Orbital resonance models of qpos in braneworld black holes

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)


Orbital resonance models of qpos in braneworld black holes

2.5. Orbital motion in a strong gravity


Orbital resonance models of qpos in braneworld black holes

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


Orbital resonance models of qpos in braneworld black holes

  • 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


Orbital resonance models of qpos in braneworld black holes

2.6. Properties of the Keplerian and epicyclic frequencies

Local extrema of the Keplerian and epicyclic frequencies:


Orbital resonance models of qpos in braneworld black holes

2.6. Properties of the Keplerian and epicyclic frequencies

Local extrema of the Keplerian and epicyclic frequencies:

Keplerian frequency:

can have a maximumat


Orbital resonance models of qpos in braneworld black holes

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?


Orbital resonance models of qpos in braneworld black holes

2.6.1. Local extrema of the Keplerian frequency


Orbital resonance models of qpos in braneworld black holes

2.6.1. Local extrema of the Keplerian frequency


Orbital resonance models of qpos in braneworld black holes

2.6.1. Local extrema of the Keplerian frequency


Orbital resonance models of qpos in braneworld black holes

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


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the vertical epicyclic frequency

NaS

BHs

  • BHs: one local maximum at for

  • NaS: two or none local extrema


Orbital resonance models of qpos in braneworld black holes

NaS

BHs

2.6.2. Local extrema of the radial epicyclic frequency


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.6.2. Local extrema of the epicyclic frequencies


Orbital resonance models of qpos in braneworld black holes

2.7. Digest of orbital resonance models


Orbital resonance models of qpos in braneworld black holes

2.8. Resonance conditions

  • determine implicitly the resonant radius

  • must be related to the radius of the innermost stable circular geodesic


Orbital resonance models of qpos in braneworld black holes

2.9. Strong resonant phenomena - "magic" spin


Orbital resonance models of qpos in braneworld black holes

2.9. Strong resonant phenomena - "magic" spin


Orbital resonance models of qpos in braneworld black holes

2.9. Strong resonant phenomena - "magic" spin


Orbital resonance models of qpos in braneworld black holes

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)


Orbital resonance models of qpos in braneworld black holes

2.9. Strong resonant phenomena - "magic" spin


Orbital resonance models of qpos in braneworld black holes

3. Application to microquasars

GRO J1655-40

GRS 1915+105


Orbital resonance models of qpos in braneworld black holes

3.1. Microquasars data: 3:2 ratio

Török, Abramowicz, Kluzniak, Stuchlík 2005


Orbital resonance models of qpos in braneworld black holes

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


Orbital resonance models of qpos in braneworld black holes

3.2. Results for GRO J1655-40

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)


Orbital resonance models of qpos in braneworld black holes

3.2. Results for GRO J1655-40


Orbital resonance models of qpos in braneworld black holes

3.3. Results for GRS 1915+105

McClintock & Remillard 2004

estimate 2

estimate 1

1 - Middleton et al. 2006

2 - McClintock et al. 2006


Orbital resonance models of qpos in braneworld black holes

3.3. Results for GRS 1915+105


Orbital resonance models of qpos in braneworld black holes

3.4. Conclusions

-1 < β< 0.51


Orbital resonance models of qpos in braneworld black holes

3.4. Conclusions

-1 < β< 0.51

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


Orbital resonance models of qpos in braneworld black holes

4. References

  • 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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