1. STRONG RESONANT PHENOMENA IN BLACK HOLE SYSTEMS Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, CZECH REPUBLIC supported by Czech grant MSM 4781305903 Presentation download: www.physics.cz/research in section news Andrea Kotrlová, Zdeněk Stuchlík & Gabriel Török

2. 1. Motivation: Quasiperiodic oscillations (QPOs) in X-ray from the NS an BH systems - Black hole and neutron star binaries, accretion disks and QPOs 2. Non-linear orbital resonance models 2.1. "Standard orbital resonance models 3. "Exotic" multiple resonances at the common orbit 3.1. Triple frequencies and black hole spin a a)at different radii *) b)at the common radius - strong resonant phenomena **) 3.2. Necessary conditions 3.3. Classification - "magic" value of the black hole spin a = 0.983 4. A little "gamble" - The Galaxy centre source Sgr A* as a proper candidate system 5. Conclusions 6. References Outline *) Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole spin, in preparation **) Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena, submitted

3. 1. 1.Motivation

4. radio “X-ray” and visible 1.1. Black hole binaries and accretion disks Figs on this page: nasa.gov

5. t I Power Frequency 1.2. X-ray observations Light curve: Power density spectra (PDS): Figs on this page: nasa.gov

6. hi-frequency QPOs low-frequency QPOs (McClintock & Remillard 2003) 1.3. Quasiperiodic oscillations

7. (McClintock & Remillard 2003)

8. 2. Non-linear orbital resonance models

9. 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 considered by Aliev & Galtsov, 1981) 2.1. "Standard" orbital resonance models

11. rotating BHnon-rotating BH a=0 a~1

12. 2.1. "Standard" orbital resonance models

13. 3. "Exotic" multiple resonances at the common orbit Fig. on this page: nasa.gov

14. 3.1. Triple frequencies and black hole spin a From we can determine spin for various versions of the resonance model: a) a)Two resonances at different radii for special values of spin  common top, bottom, or mixed frequency  two frequency pairs reduce into a triple frequency ratio set - -possibility of highly precise determination of spin – given by the types of the two resonances and the ratios quite independently of the BH mass M (but not uniquely, as the same frequency set could correspond to more than one concrete spin a). b) b)Resonances sharing the same radius for special values of a  strong resonant phenomena – allow direct resonances at a given radius (s, t, u – small natural numbers) - for each triple frequency ratio set spin is given uniquely, - the resonances could be causally related and could cooperate efficiently (Landau & Lifshitz 1976). "top identity""bottom identity""middle identity"

15. Strong resonant phenomena - only for special values of spin a we consider BH when a ≤ 1  restriction on allowed values of s, t, u we have to search for the integer ratios s:t:u at x ≥ x ms and at the same radius  condition:  an explicit solution determining the relevant radius for any triple frequency ratio set s:t:u and the related BH spin: 3.2. Necessary conditions

16. Strong resonant phenomena - only for special values of spin a we consider BH when a ≤ 1  restriction on allowed values of s, t, u we have to search for the integer ratios s:t:u at x ≥ x ms and at the same radius  condition:  an explicit solution determining the relevant radius for any triple frequency ratio set s:t:u and the related BH spin: The solutions have been found for s ≤ 5 since the strength of the resonance and the resonant frequency width decrease rapidly with the order of the resonance (Landau & Lifshitz 1976) 3.2. Necessary conditions

17. Strong resonant phenomena - only for special values of spin a we consider BH when a ≤ 1  restriction on allowed values of s, t, u we have to search for the integer ratios s:t:u at x ≥ x ms and at the same radius  condition:  an explicit solution determining the relevant radius for any triple frequency ratio set s:t:u and the related BH spin: The solutions have been found for s ≤ 5 since the strength of the resonance and the resonant frequency width decrease rapidly with the order of the resonance (Landau & Lifshitz 1976) s:t:u = 3:2:1, 4:2:1, 4:3:1, 4:3:2, 5:2:1, 5:3:1, 5:3:2, 5:4:1, 5:4:2, 5:4:3. 3.2. Necessary conditions

18.  A      D C B E s:t:u = 3:2:1, 4:2:1, 4:3:1, 4:3:2, 5:2:1, 5:3:1, 5:3:2, 5:4:1, 5:4:2, 5:4:3. direct resonances realized only with combinational frequencies

19. A) arises for the so called "magic" spin a m = 0.983 the Keplerian and epicyclic frequencies are in the lowest possible ratio at the common radius any of the simple combinational frequencies coincides with one of the frequencies and are in the fixed small integer ratios the only case when the combinational frequencies (not exceeding ) are in the same ratios as the orbital frequencies we obtain the strongest possible resonances when the beat frequencies enter the resonances satisfying the conditions "Magic" spin a = 0.983 3.2. Classification

20. "Magic" spin a = 0.983 3.2. Classification A) the only case when the combinational frequencies (not exceeding ) are in the same ratios as the orbital frequencies we obtain the strongest possible resonances when the beat frequencies enter the resonances satisfying the conditions arises for the so called "magic" spin a m = 0.983 the Keplerian and epicyclic frequencies are in the lowest possible ratio at the common radius any of the simple combinational frequencies coincides with one of the frequencies and are in the fixed small integer ratios

21. 3.2. Classification the combinational frequencies give additional frequency ratios we can obtain the other three frequency ratio sets the four observable frequency ratio set is possible B) C) we can generate triple frequency sets involving the combinational frequencies two sets of four frequency ratios are possible we could obtain one set of five frequency ratio

22. 3.2. Classification this case leads to the triple frequency ratio sets and one four frequency ratio set we can obtain the triple frequency ratio sets the related four frequency ratio sets and one five frequency ratio set D) E)

23. 4. A little "gamble" Possible application to the Sgr A* QPOs Figs on this page: nasa.gov

24. The Galaxy centre source Sgr A* as a proper candidate system 4. A little "gamble" The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005): a) Considering the standard epicyclic resonance model:  - -it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):

25.  The Galaxy centre source Sgr A* as a proper candidate system 4. A little "gamble" The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005): a) Considering the standard epicyclic resonance model:  - -it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):

26.  The Galaxy centre source Sgr A* as a proper candidate system 4. A little "gamble" The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005): a) Considering the standard epicyclic resonance model:  - -it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):  - -it meets the allowed BH mass interval at its high mass end. b) Assuming the "magic" spin (Sgr A* should be fast rotating), with the frequency ratio at the sharing radius and identifying

27.  The Galaxy centre source Sgr A* as a proper candidate system 4. A little "gamble" The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005): a) Considering the standard epicyclic resonance model:  - -it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):  - -it meets the allowed BH mass interval at its high mass end. b) Assuming the "magic" spin (Sgr A* should be fast rotating), with the frequency ratio at the sharing radius and identifying

28.  The Galaxy centre source Sgr A* as a proper candidate system 4. A little "gamble" The three QPOs were reported for Sgr A* (Aschenbach 2004; Aschenbach et al. 2004; Török 2005): The model should be further tested, more precise frequency measurements are very important.  c) Using other versions of the multi-resonance model  best fit is for, with resonances at two different radii having common bottom frequency a) Considering the standard epicyclic resonance model:  - -it is in clear disagreement with the allowed range of the Sgr A* mass coming from the analysis of the orbits of stars moving within 1000 light hour of Sgr A* (Ghez et al. 2005):  - -it meets the allowed BH mass interval at its high mass end. b) Assuming the "magic" spin (Sgr A* should be fast rotating), with the frequency ratio at the sharing radius and identifying

29. Errors of frequency measurements The mass of the BH is related to the magnitude of the observed frequency set, not to its ratio. more precise measurement of the QPOs frequencies  more precise determination of the BH mass, method can work only accidentally, for the properly taken values of spin  precision of frequency measurement is crucial for determination of the BH mass. errors of frequency measurements  errors in the spin determination (depends on the concrete resonances occurring at a given radius) 4. A little "gamble" Ghez et al. (2005)

30. 5. Conclusions Conditions for strong resonant phenomena could be realized only for high values of spin (a ≥ 0.75)  idea probably could not be extended to the NS (where we expect a < 0.5). Allowing simple combinational frequencies (not exceeding )  observable QPOs with: the lowest triple frequency ratio set for the "magic" spin a = 0.983, but also for a = 0.866, 0.882, 0.962 (if the uppermost frequencies are not observed for some reasons), four frequency ratio set for a = 0.866, 0.882 and 0.962, five frequency ratio set for a = 0.882, 0.962. It is not necessary that all the resonances are realized simultaneously and that the full five (four) frequency set is observed at the same time. Generally, there exist few values of the spin a and the corresponding shared resonance radius allowed for a given frequency ratio set  detailed analysis of the resonance phenomena has to be considered and further confronted with the spin estimates coming from spectral analysis of the BH system (McClintock et al. 2006 and Middleton et al. 2006 for GRS 1915+105; Shafee et al. 2006 for GRO J1655-40), line profiles (Fabian & Miniutti 2005; Dovčiak et al. 2004; Zakharov 2003; Zakharov & Repin 2006), orbital periastron precession of some stars moving in the region of Sgr A* (Kraniotis 2005, 2007), very promising: studies of the energy dependencies of high-frequency QPOs determining the QPO spectra at the QPO radii (Życki et al. 2007).

31. 6. References Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Multi-resonance model of QPOs: possible high precision determination of black hole spin, in preparation Stuchlík, Z., Kotrlová, A., & Török, G. 2007: Black holes admitting strong resonant phenomena, submitted 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 Aschenbach, B. 2004, Astronomy and Astrophysics, 425, 1075 Aschenbach, B. 2006, Chinese Journal of Astronomy and Astrophysics, 6, 221 Fabian, A. C., & Miniutti, G. 2005, Kerr Spacetime: Rotating Black Holes in General Relativity (Cambridge Univ. Press) Ghez, A. M., Salim, S., Hornstein, S. D., Tanner, A., Lu, J. R., Morris, M., Becklin, E. E., & Duchene, G. 2005, Astrophys. J., 620, 744 Kraniotis, G. V. 2005, Classical Quantum Gravity, 22, 4391 Kraniotis, G. V. 2007, Classical Quantum Gravity, 24, 1775 Landau, L. D., & Lifshitz, E. M. 1976, Mechanics, 3rd edn. (Oxford: Pergamon Press) McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, Astrophys. J., 652, 518 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) Middleton, M., Done, C., Gierlinski, M., & Davis, S. W. 2006, Monthly Notices Roy. Astronom. Soc., 373, 1004 Török, G. 2005, Astronom. Nachr., 326, 856 Török, G., Abramowicz, M. A., Kluzniak,W. & Stuchlík, Z. 2005, Astronomy and Astrophysics, 436, 1 Török, G., & Stuchlík, Z. 2005a, 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), 315-338 Török, G., & Stuchlík, Z. 2005b, Astronomy and Astrophysics, 437, 775 Zakharov, A. F. 2003, Publications of the Astronomical Observatory of Belgrade, 76, 147 Zakharov, A. F., & Repin, S. V. 2006, New Astronomy, 11, 405 Życki, P. T., Niedzwiecki, A., & Sobolewska, M. A. 2007, Monthly Notices Roy. Astronom. Soc., in press THANK YOU FOR YOUR ATTENTION