1. Figs on this page: nasa.gov

2. Outline 1. Introduction: Quasi-periodic oscillations (QPOs) - Black-hole and neutron star binaries, accretion disks and QPOs 2. Non-linear resonance models 2.1. Orbital resonance model 2.2. Some properties of non-linear resonances 3. Black-hole observations 3.1. The 3:2 phenomenon 3.2. The 1/M scaling 4. Neutron-star observations 4.1. Frequency – frequency fits 4.2. The slope-shift anti-correlation 4.3. Evolution of RMS amplitudes across the resonance point 5. Summary and black hole – neutron star comparison 6. References Presentation download: www.physics.cz/research in sect. newswww.physics.cz/research

3. 1. Basic introduction: Quasiperiodic oscillations in X-ray Figs on this page: nasa.gov

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 and QPOs Light curve: Power density spectra (PDS): Figs on this page: nasa.gov

6. hi-frequency QPOs low-frequency QPOs frequency power 1.2. Quasiperiodic oscillations

7. The rms amplitude = percentual fraction (root mean square fraction) of the peak energy with the respect to the total countrate, the rms amplitude indicates the energy connected to the observed oscillations. Power Frequency 1.2. Quasiperiodic oscillations

8. 2. Non-linear resonance models Figs on this page: nasa.gov

9. in general relativity, all the three frequencies depend on generic mass as f ~ 1/M ! 2.1. Resonance of epicyclic frequencies (a particular non-linear resonance)

11. 2.1.1. Resonance of epicyclic frequencies (a particular non-linear resonance) frequencies are in ratio of small natural numbers [e.g, Landau & Lifshitz, 1976], which must hold also in the case of forced resonances ! animation by Bursa 2004

12. The 1/M scaling in epicyclic model follows from nature of Keplerian motion. On the other hand, frequencies in ratio of small natural numbers come from general properties of non-linear oscillators and have nothing to do with particular choice of resonant frequencies. One can check more general properties of non-linear resonances. 2.2. Some properties of non-linear oscillators

13. !

14. !

15. Of non-linear resonances: Eigenfrequencies in ratio of small natural numbers (Landau&Lifshitz, 1974) (Abramowicz etal., 2005/6, Török etal. 2006) Of particular epicyclic model: 1/M scaling Preferred 3:2 ratio (Horák 2004) Well defined predictions Lines in  plane: Their anticorrelation:

16. 3. Black-hole hi-frequency QPOs Figs on this page: nasa.gov, personal archiv

17. 3. Black-hole hi-frequency QPOs: the 3:2 ratio and 1/M scaling

18. Lachowicz, Czerny & Abramowicz (2006), astro-ph/0607594:

19. 3. Black-hole hi-frequency QPOs Of non-linear resonances: Eigenfrequencies in ratio of small natural numbers Anticorrelated Bursa-lines in frequency-frequency plane Frequencies seems to be constant, Bursa lines are not observed. Of particular epicyclic model: 1/M scaling Preferred 3:2 ratio Predictions vs. observations in the case

20. Figs on this page: nasa.gov, personal archiv 4. Neutron-star hi-frequency QPOs

21. kHz frequency – frequency relation for low mass X-ray binaries 4.1. Frequency – frequency fits NS BH

22. Psaltis et al. 1998 4.1. Frequency – frequency fits

23. Bursa 2002: NS are far away from  = 1.5 1, f-f relations differ for individual sources 4.1. Frequency – frequency fits

24. 4.2. Frequency – frequency fits: slope-shift anti-correlation The coefficients of linear fits for twelve neutron-star sources are anticorrelated [Abramowicz et al. 2006]

25. 4.2. Frequency – frequency fits and slope-shift anti-correlation The coefficients of linear fits for twelve neutron-star sources pointed to the eigenfrequency ratio equal to 3/2. Two weakly coupled oscillators with eigenfrequency ratio A^0: From the data for 12 sources,

26. 4.2. The rms amplitude evolution across the resonance point For few sources examined so far, the difference of lower and upper rms amplitudes changes its sign when the source pass the exact 3:2 ratio. The case of 4U 1636: When the ratio between upper and lower frequency is higher then 3:2, the upper oscillation is stronger. For the ratio of frequencies equal to 3:2, both oscillations have the same amplitudes. When the ratio between upper and lower frequency is lower then 3:2, the lower oscillation is stronger.

27. 4.3. The rms amplitude evolution across the resonance point When the ratio between upper and lower frequency is higher then 3:2, the upper oscillation is stronger. For the ratio of frequencies equal to 3:2, both oscillations have the same amplitudes. When the ratio between upper and lower frequency is lower then 3:2, the lower oscillation is stronger. (Török & Barret, Horák & Török, 2006 in prep. )

28. 4. Neutron-star hi-frequency QPOs Of non-linear resonances: Eigenfrequencies in ratio of small natural numbers Anticorrelated Bursa-lines in frequency-frequency plane + the effect of rms reverse Of particular epicyclic model: 1/M scaling Preferred 3:2 ratio Predictions vs. observations in the case + observation of the effect of the oscillation energy reverse

29. 5. Summary and black hole – neutron star comparison Both the black hole and neutron star QPOs shows the 3:2 ratio pointing to a resonance. The 1/M scaling indicating the orbital origin of hi-frequency QPOs holds /at least roughly/. The neutron star hi-frequency QPOs forms anticorrelated lines in frequency- frequency plane as predicted from the assumtion of non-linear weakly coupled oscillations. On the other hand, the black hole (microquasar) hi-frequencies seems to be rather fixed….

30. 5. Summary and black hole – neutron star comparison The range of frequencies for NS Bursa lines is in hundreds of Hz while the black hole frequencies are fixed. This could be due to scaling of range with the square of dimensionless amplitudes ? range ~ sq. of , NS range ~ 500 Hz but NS amplitudes are ~ 10 times higher thus BH range should be ~ 100 times lower, i.e. ~ 5Hz

31. 6. References Török, Abramowicz, Kluzniak, Stuchlík 2006, proc. of Albert Einstein Conf. Paris 2005, astro-ph: 0603847 (also contain complete list of references briefly mentioned in this presentation) Abramowicz, Barret, Bursa, Horák, Kluzniak, Olive, Rebusco, Török, 2006, proc. Of RAGtime 2005, download: ADS or www.physics.cz/research Abramowicz, Barret, Bursa, Horák, Kluzniak, Olive, Rebusco, Török 2006, submitted to MNRAS Lachowicz, Czerny & Abramowicz (2006), astro-ph/0607594 Presentation download: www.physics.cz/research in sect. newswww.physics.cz/research