Supported by the Czech grants MSM 4781305903, LC 06014 , GA Č R202/09/0772 and SGS -01-2010.

1. BH and NS spin estimates based on the models of oracular high-frequency quasiperiodicoscillations Gabriel Török Institute ofPhysics, Faculty of Philosophy and Science, Silesian University in Opava,Bezručovo n.13, CZ-74601, Opava Supported by the Czech grants MSM 4781305903, LC 06014, GAČR202/09/0772andSGS-01-2010. www.physics.cz

2. Outline The purpose of this presentationrely namely in the comparison between mass/spin predictions of several different orbital models of HF QPOs in LMXBs. The slides are organized as follows: • Introduction: neutron star rapid X-ray variability, quasiperiodicoscillations, twinpeaks • Measuring spin from HF BH QPOs: hot-spot models, disc oscillation models… • Measuring spin from HF NS QPOs: characteristic mass-spin relations • Summary (The main task is to make the set of 44 slides in 15 minutes.)

3. 1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs • Artists view of LMXBs • “as seen from a hypothetical planet” Compact object: - black hole or neutron star(>10^10gcm^3) • LMXB Accretion disc • T ~ 10^6K • >90% of radiation • in X-ray • Companion: • density comparable to the Sun • mass in units of solar masses • temperature ~ roughly as the T Sun • more or less optical wavelengths Observations: The X-ray radiation is absorbed by the Earth atmosphere and must be studied using detectors on orbiting satellites representing rather expensive research tool. On the other hand, it provides a unique chance to probe effects in the strong-gravity-field region (GM/r~c^2) and test extremal implications of General relativity (or other theories). Figs:space-art,nasa.gov

4. 1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs Individual peaks can be related to a set of oscillators, as well as to time evolution of the oscillator. LMXBsshort-term X-ray variability: peaked noise (Quasi-PeriodicOscillations) Sco X-1 power • LowfrequencyQPOs (up to 100Hz) • hecto-hertz QPOs (100-200Hz) • HF QPOs (~200-1500Hz): • Lower and upper QPO mode • forming twin peak QPOs frequency The HF QPO origin remains questionable, it is often expected that it is associated to orbital motion in the inner part of the accretion disc. Fig:nasa.gov

5. 1.1 Black hole and neutron star HF QPOs Upper frequency [Hz] 3:2 Lower frequency [Hz] Figure (“Bursa-plot”): after M. Bursa & MAA 2003, updated data

6. 1.1 Black hole and neutron star HF QPOs It is unclear whether the HF QPOs in BH and NS sources have the same origin. • BH HF QPOs: • (perhaps) constant frequencies, • exhibit the 3:2 ratio • NS HF QPOs:3:2 clustering, • - two correlated modes which • exchange the dominance • when passing the 3:2 ratio 3:2 3:2 Upper frequency [Hz] Amplitude difference Lower frequency [Hz] Frequency ratio Figures - Left: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); Right: Torok (2009)

7. 1.2. The Desire • There is a large variety of ideas proposed to explain the QPO phenomenon [For instance, Alpar & Shaham (1985); Lambetal. (1985); Stella etal. (1999); Morsink & Stella (1999); Stella & Vietri (2002); Abramowicz & Kluzniak (2001); Kluzniak & Abramowicz (2001); Abramowiczetal. (2003a,b); Wagoner et al. (2001); Titarchuk & Kent (2002); Titarchuk (2002); Kato (1998, 2001, 2007, 2008, 2009a,b); Meheut & Tagger (2009); Miller atal. (1998a); Psaltisetal. (1999); Lamb & Coleman (2001, 2003); Kluzniaketal. (2004); Abramowicz et al. (2005a,b), Petri (2005a,b,c); Miller (2006); Stuchlík etal. (2007);Kluzniak (2008); Stuchlík etal. (2008);Mukhopadhyay (2009); Aschenbach 2004, Zhang (2005); Zhangetal. (2007a,b); Rezzollaetal. (2003); Rezzolla (2004); Schnittman & Rezzolla (2006); Blaesetal. (2007); Horak (2008); Horaketal. (2009);Cadezetal. (2008); Kostic etal. (2009); Chakrabarti et al. (2009), Bachetti et al. (2010)…] • - in some cases the models are applied to both BHs and NSs, in some not • - some models accommodate resonances, some not • the desire /common to several oftheauthors/ is to relate HF QPOs to strong gravityand inferr the compact object properties using QPO measurements

8. 2. Spin from the models of the 3:2 BH QPOs • the (advantage of) BH HF QPOs: • (perhaps) constant frequencies, • exhibit the mysterious 3:2 ratio • The BH 3:2 QPO frequencies are stable which imply that they depend mainly on the geometry and not so much on the dirty physics of the accreted plasma. Upper frequency [Hz] Lower frequency [Hz] Figure: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003);

9. 2. Models relating both of the 3:2 BH QPOs to a single radius Here we use fewhot-spotand disc-oscillation models relating both QPOs to a single preferred radius in order to demonstrate the (potential) predictive power of QPOs.

10. 2. Models relating both of the 3:2 BH QPOs to a single radius Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD Relativistic Precession Stella etal. (1999); Morsink & Stella (1999); Stella & Vietri (2002)] WD ER KR RP1 RP2

11. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Tidal Disruption Čadež et al. (2008), Kostič et al. (2009), Germana et al.(2009) ER KR RP1 RP2

12. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius (or torus) Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD ER Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) WarpedDisc Resonance a representativeofmodelsproposed by Kato (2000, 2001, 2004, 2005, 2008) KR RP1 RP2

13. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius (or torus) Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD ER KR RP1 Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Epicyclic Resonance, Keplerian Resonance tworepresentativesofmodelsproposed by Abramowicz, Kluzniaketal. (2000, 2001, 2004, 2005,…) RP2

14. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius (or torus) Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD ER KR RP1 RP2 Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al. (2009) Resonancesbetween non-axisymmetricoscillationmodesof a toroidalstructuretworepresentatives by Bursa (2005), Toroketal (2010) predictingfrequenciesclose to RP model

15. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Here we focus just on a choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD ER KR RP1 RP2

16. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Different models associate QPOs to different radii… RP WD, TD ER

17. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius One can easily calculate frequency-mass functions for each of the models Spin a

18. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius And compare the frequency-mass functions to the observation. For instance in the case of GRS 1915+105. Spin a

19. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Comparison with the independent spin measurements: Considering the high spin measurement of 1915+105, some QPO models (e.g. relativistic precession) are clearly disfavoured.

20. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Because of rough observational 1/M scaling of the observed 3:2 frequencies, the spins inferred from QPOs are moreless common to each of the three microquasars, Consequently the QPO models favoured in 1915+105 does not fit continuum estimates for 1655-40 if these are very different from 1915+105 (0.7 vs. 0.98), which is rather a generic problem…

21. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius Because of rough observational 1/M scaling of the observed 3:2 frequencies, the spins inferred from QPOs are moreless common to each of the three microquasars, Torok et al. (2011), A&A Consequently the QPO models favoured in 1915+105 does not fit continuum estimates for 1655-40 if these are very different from 1915+105 (0.7 vs. 0.98), which is rather a generic problem…

22. 2.2 Discoseismic modes Here we attempt to identify both the 3:2 QPOs with (some) pair of fundamental disc oscillation modes . Wagoner et al. (2001) Note that the frequency ratio of two fundamental modes depends only on spin and (weakly) on the speed of sound (since each of the modes is located at its own radii).

23. 2.2 Discoseismic modes Ratio between fundamental g-mode and c-mode as it depends on the spin: Torok et al. (2011), A&A Assuming g- and c- modes, the 3:2 ratio can be reproduced only for spins either around 0.7-0.8 or 0.9-0.95.

24. 2.2 Discoseismic modes Ratio between g-mode and p-mode as it depends on the spin: Torok et al. (2011), A&A Assuming that the one QPO is g-mode and the other one p-mode, the 3:2 ratio can be reproduced for spins up to 0.8. Assuming p-modes (and one of the two others) the 3:2 ratio cannot be reproduced for high spins.

25. 2.2 Discoseismic modes Taking into account the absolute values of QPO frequencies (and therefore the mass): Except the combination identifying 3 and 2 as g-mode and c-mode the required masses do not overlap in a single case with those estimated from independent methods.

26. 3. Spin from the models of the NS HF QPOs NS spacetimes require three parametric description (M,j,Q), e.g., Hartle&Thorne (1968). However, high mass (i.e. compact) NS can be well approximated via simple and elegant terms associated to Kerr metric. This fact is well manifested on the ISCO frequencies: Torok et al., (2010),ApJ Several QPO models predict rather high NS masses when the non-rotating approximation is applied. For these models Kerr metric has a potential to provide rather precise spin-corrections which we utilize in next. A good example to start is the RELATIVISTIC PRECESSION MODEL.

27. 3. Spin from the models of the NS HF QPOs One can use the RP model definition equations to obtain the following relation between the expected lower and upper QPO frequency which can be compared to the observation in order to estimate mass M and “spin” j … The two frequencies scale with 1/M and they are also sensitive to j.In relation to matching of the data, there is an important question whether there are identical or similar curves for different combinations of M and j.

28. 3.1 Relativistic precession model One can find the combinations of M, j giving the same ISCO frequency and plot the related curves. The resulting curves differ proving thus the uniqueness of the frequency relations. On the other hand, they are very similar: M = 2.5….4 MSUN Ms = 2.5 MSUN M~ Ms[1+0.75(j+j^2)] Torok et al., (2010), ApJ For a mass M0of the non-rotating neutron star there is always a set of similar curves implying a certain mass-spin relation M (M0, j) (implicitly given by the above plot). The best fits of data of a given source should be therefore reached for combinations of M and j that can be predicted from just one parametric fit assuming j = 0.

29. 3.1.1. Relativistic precession model vs. data of 4U 1636-53 The best fits of data of a given source should be reached for the combinations of M and j that can be predicted from just one parametric fit assuming j = 0. The best fit of 4U 1636-53 data (21 datasegments) for j = 0 is reached for Ms = 1.78M_sun, which implies M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun

30. 3.1.1. Relativistic precession model vs. data of 4U 1636-53 Color-coded map of chi^2 [M,j,10^6 points] well agrees with the rough estimate given by a simple one-parameter fit. M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun Best chi^2 chi^2 ~ 300/20dof chi^2 ~ 400/20dof

31. 3.2 Four models vs. data of 4U 1636-53 Several models imply M-j relations having the origin analogic to the case of RP model. chi^2 maps [M,j, each 10^6 points]: 4U 1636-53data

32. 3.3 Comparison to Circinus X-1 Several models imply M-j relations having the origin analogic to the case of RP model. chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data

33. 3.4. Quality of fits and nongeodesic corrections - It is often believed that, e.g., RP model fits well the low-frequency sources but not the high-frequency sources. RP model, figure from Torok et al., (2010), ApJ • The difference however follows namely from • difference in coherence times (large and small errorbars) • position of source in the frequency diagram

34. 3.4. Quality of fits and nongeodesic corrections - It is often believed that, e.g., RP model fits well the low-frequency sources but not the high-frequency sources. The same non-geodesic corrections can be involved in both classes of sources. Circinus X-1 data 4U 1636-53 X-1 data The above naive correction improves the RP model fits for both classes of sources. Similar statement can be made for the other models.

35. 3.5 NS mass and spin implied by the epicyclic resonance model For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). q/j2 j Urbanec et al., (2010) , A&A Mass-spin relations inferred assuming Hartle-Thorne metric and various NS oblateness. One can expect that the red/yellow region is allowed by NS equations of state (EOS).

36. 3.5 NS mass and spin implied by the epicyclic resonance model For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). j Urbanec et al., (2010) , A&A Mass-spin relations calculated assuming several modern EOS (of both “Nuclear” and “Strange” type) and realistic scatter from 600/900 Hz eigenfrequencies.

37. 3.5 Paczynski modulation and implied restrictions (epicyclic resonance model) Urbanec et al., (2010) , A&A After Abr. et al., (2007), Horák (2005) The condition for modulation is fulfilled only for rapidly rotating strange stars, which most likely falsifies the postulation of the 3:2 resonant mode eigenfrequencies being equal to geodesic radial and vertical epicyclic frequency…. (Typical spin frequencies of discussed sources are about 300-600Hz; based on X-ray bursts)

38. 4. Summary (BHs) Spins of the three microquasars from the two hot-spot models and WD model are below a=0.5. The spins above a=0.9 are matched only by the resonance model (in our choice of models). Torok et al. (2011), A&A The QPO models favoured in 1915+105 does not fit for 1655-40. The reason is rather generic: 1/M scaling of the observed 3:2 frequencies.

39. 4. Summary (BHs) Fundamental discoseismic modes vs. 3:2 frequency ratio (both QPOs): • Assuming that the 3:2 QPOS in microquasars are related to the fundamentalp-modes and either to the g- modes or c-modes the spin cannot be high (a<0.8). • Assuming the combination of g-mode and c-mode, the spin must be about a=0.9-0.95. The problem with the 1/M scaling arises also for the discoseismic modes...

40. 4. Summary (NSs)

41. 4. Summary The list of questions which I have been asked to answer in this talk: what spin measurements have been made or attempted so far using the method what are the strengths and weaknesses of the method what future work is needed to improve the method

42. 4. Summary The list of questions which I have been asked to answer in this talk: what spin measurements have been made or attempted so far using the method what are the strengths and weaknesses of the method what future work is needed to improve the method • The BH 3:2 QPO frequencies are “stable” which imply that they depend mainly on the geometry and not so much on the dirty physics of the accreted plasma.

43. 4. Summary The list of questions which I have been asked to answer in this talk: what spin measurements have been made or attempted so far using the method what are the strengths and weaknesses of the method what future work is needed to improve the method • The BH 3:2 QPO frequencies are “stable” which imply that they depend mainly on the geometry and not so much on the dirty physics of the accreted plasma.

44. 4. Summary We need a right model.