Institute of Physics, Silesian University in Opava

1. Determinationofcompact object parametersfrom observations of high frequency quasiperiodic oscillations Gabriel Török Institute ofPhysics, Silesian University in Opava GAČR 209/12/P740, CZ.1.07/2.3.00/20.0071Synergy, GAČR 14-37086G,SGS-11-2013, www.physics.cz CO-AUTHORS: Eva Šrámková, Martin Urbanec, Kateřina Goluchová, Andrea Kotrlová, Karel Adámek, Jiří Horák, Pavel Bakala, MarekAbramowicz, Zdeněk Stuchlík, WlodekKluzniak, Gabriela Urbancová, Tomáš Pecháček

2. Outline of our progress report • Introduction: neutron star rapid X-ray variability, quasiperiodicoscillations, twinpeaks • Measuring BH spin from HF QPOs • 2.1 BH spin fromgeodesicmodels • (summaryofsome olderresults byToroketal, 2011, A&A) • 2.2 Consideration of a>1 (Kotrlová etal 2014, A&A) • 2.3 Nongeodesic effects (Šrámková etal 2014, to be submitted) • Measuring NS spin from HF QPOs • 3.1 Mass-angular-momentum relations, EoSconsideration • (summaryofToroketal, ApJ, 2010, 2012, Urbanec etal 2010, A&A) • 3.2 DetailedconsiderationofEoSand spin(Toroketal 2014, in prep.) • 3.3 Torus model (Toroketal 2014, in prep.) • 3.4 Generalconstraints (Toroketal 2014, A&A)

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. 1.1 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)

8. 1.2. The ambition • 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 do not • Theambition /common to several oftheauthors/ is to relate HF QPOs to orbital motion in strong gravityand infer the compact object properties using the QPO measurements…

9. 1.3 Models and frequency relations considered here Several models imply observable frequencies that can be expressed in therms of combinations of frequencies of the Keplerian, radial and vertical epicyclic oscillations.In thesimple case ofgeodesicapproximationandKerrmetric these are

10. 1.3 Models and frequency relations considered here Here we only focus on the choice of fewhot-spot and disc-oscillation models: RP2

11. 1.3 Models and frequency relations considered here Here we only focus on the 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

12. 1.3 Models and frequency relations considered here Here we only focus on the 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

13. 1.3 Models and frequency relations considered here (or torus) Here we only focus on the 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

14. 1.3 Models and frequency relations considered here (or torus) Here we only focus on the 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

15. 2. 1 Models relating both of the 3:2 BH QPOs to a single radius (or torus) Here we only focus on the 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

16. 1.3 Models and frequency relations considered here Here we only focus on the choice of fewhot-spot and disc-oscillation models: MODEL : Characteristic Frequencies RP TD WD ER KR RP1 RP2

17. 2.1 BH spin from the geodesic QPO models • the (advantage of) BH HF QPOs: • (perhaps) constant frequencies, • exhibit the mysterious 3:2 ratio • The BH 3:2 QPO frequencies are rather 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);

18. 2.1 BH spin from the geodesic QPO models Different models associate QPOs to different radii… RP WD, TD ER

19. 2.1 BH spin from the geodesic QPO models One can easily calculate the frequency.mass functions for each of the models. Torok et al., (2011) A&A Spin a

20. 2.1 BH spin from the geodesic QPO models And compare the frequency.mass functions to the observation. For instance in the case of GRS 1915+105 (whichherewellrepresentsall 3:2 microquasars). Torok et al., (2011) A&A Spin a

21. 2.1 BH spin from the geodesic QPO models And compare the frequency.mass functions to the observation. For instance in the case of GRS 1915+105 (whichherewellrepresentsall 3:2 microquasars). Torok et al., (2011) A&A Clearly, differentmodelsimplyverydifferentspins… Spin a

22. 2.2Considerationofa>1 (naked sigularities or superspinars - NaS) When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thusmore complicated . Kotrlováet al., (2014) A&A

23. 2.2Considerationofa>1 (naked sigularities or superspinars - NaS) When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smoothfrequency.massfunctionsofthe spin. Kotrlováet al., (2014) A&A BH NaS

24. 2.2Considerationofa>1 (naked sigularities or superspinars - NaS) When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smoothfrequency.massfunctionsofthe spin. Someofthem not (discontinuitiesanddichotomiesappear…) Kotrlováet al., (2014) A&A

25. 2.2Considerationofa>1 (naked sigularities or superspinars - NaS) When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smoothfrequency.massfunctionsofthe spin. Someofthem not (discontinuitiesanddichotomiesappear…) Kotrlováet al., (2014) A&A The case ofepicyclic resonance model

26. 2.2Considerationofa>1 (naked sigularities or superspinars - NaS) When the dimensionless spin reads a>1, the behaviour of the epicyclic frequencies of orbital motion qualitatively changes. The situation is thus more complicated. Nevertheless, some models still imply smoothfrequency.massfunctionsofthe spin. Someofthem not (discontinuitiesanddichotomiesappear…) ZOOM Kotrlováet al., (2014) A&A The case ofepicyclic resonance model

27. 2.2.1 Summaryof BH spin estimates Kotrlováet al., (2014) A&A • Forblackholes, differentmodelsimplyverydifferent spin values. • Exceptone model, thereisalwaysanalternativecompatiblewith existence of asuperspinningcompactobject. Nevertheless, onlyepicyclic resonance model thenimplies spin close to unity, whileothersimplyvaluesthat are severaltimeshigher.

28. 2.3. Non-geodesiceffectsconsiderationwithin ER model Pressuresupported fluid tori Šrámková et al., (2014), in prep.

29. 2.3. Non-geodesiceffectsconsiderationwithin ER model Pressuresupported fluid tori – impactofpressure on theresonantfrequency Šrámková et al., (2014), in prep. Forlowspinstheresultsagreewithpseudonewtonian case investigated by Blaesetal 2008. Forhighspins, thesituationisdifferent. Resonantfrequencies are decreasinginsteadofincreasingas the torus thickness rises.

30. 2.3. Non-geodesiceffectsconsiderationwithin ER model Pressuresupported fluid tori – impactofpressure on theresonantfrequency Šrámková et al., (2014), in prep. Forlowspinstheresultsagreewithpseudonewtonian case investigated by Blaesetal 2008. Forhighspins, thesituationisdifferent. Resonantfrequencies are decreasinginsteadofincreasingas the torus thickness rises.

31. 3.1 NS mass-angularmomentum relations from HF QPO data

32. 3.1 NS mass-angularmomentum relations from HF QPO data • We have considered the relativistic precession (RP) twin peak QPOmodel to estimate the mass of central NS in CircinusX-1fromthe HF QPO data. We have shown that such an estimate results in a specificmass–angular-momentum (M–j) relation rather than a single preferred combination of M and j. • Later we confrontedour previous results with another binary, the atoll source 4U 1636–53 that displays the twin peak QPOs at very highfrequencies, and extend the consideration to various twin peak QPO models. In analogy to the RP model, we findthat these imply their own specific M–j relations. Toroket al., (2010) ApJ Toroket al., (2012) ApJ

33. 3.1 NS mass-angularmomentum relations from HF QPO data RP MODEL (4U 1636-53):Color-coded map of chi^2 [M,j,10^6 points] well agrees with rough estimate given by simple one-parameter fit. M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun Best chi^2 Torok et al., (2012) ApJ.

34. 3.1 NS mass-angularmomentum relations from HF QPO data Severalothermodelsandsources Torok et al., (2012) ApJ.

35. 3.1 NS mass-angularmomentum relations from HF QPO data

36. 3.2 DetailedconsiderationofEoSand NS spin • Onecancalculate M-j relations fromEoSand spin frequencyandcompare these to theresultsbased on QPOs. Torok et al., (2014) in prep.

37. 3.2 DetailedconsiderationofEoSand NS spin • Onecancalculate M-j relations fromEoSand spin frequencyandcompare these to theresultsbased on QPOs. Torok et al., (2014) in prep.

38. 3.2 DetailedconsiderationofEoSand NS spin • Anotherpossibilityis to INFER the spin fromthe QPO model. WhenEoS are considereddirectlyfor QPO modelling, the M-J degeneracyisbrokenand QPO modelsprovidechi-square minima. RP MODEL Torok et al., (2014) in prep.

39. 3.2 DetailedconsiderationofEoSand NS spin • Anotherpossibilityis to INFER the spin fromthe QPO model. WhenEoS are considereddirectlyfor QPO modelling, the M-J degeneracyisbrokenand QPO modelsprovidechi-square minima. VERY GOOD AGREEMENT ! X-rayburst RP MODEL Torok et al., (2014) in prep.

40. 3.3Torus Model • The models which we have considered so far always require tuning of parameters in order to provide good fits of the high-frequency sources data. • Here we attempt to fit the data with “modified RP model” which does not contain any new free parameters.

41. 3.3Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequency of the m=1 radial epicyclic oscillations of the torus.

42. 3.3Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequency of the m=1 radial epicyclic oscillations of the torus. Torok et al., (2014) in prep. BEST FITS TORUS MODEL RP MODEL

43. 3.3Torus Model We assume an inner pressure supported fluid torus with the CUSP CONFIGURATION. The torus moves in radial direction. The upper QPO frequency is given as the Keplerian frequency at the torus centre. The lower QPO frequency is the frequencyof the m=1 radial epicyclic oscillations of the torus. Torok et al., (2014) in prep. BEST FITS CUSP TORUS TORUS MODEL RP MODEL

44. 3.4 General constraints • The ISCO-NS distribution has thepeaksat the values of the spin which can be very different from the peak in the distribution of all NS. • High M -> peak at the original value of spin • Low M -> peak at the high value of spin • Inter. M -> twopeaks – POSSIBLE DISTRIBUTION OF QPO SOURCES ! Number of ISCO-NS [relative units] Torok et al., (2014)A&A Letters SPIN [Hz]

45. 3.5SummaryofNS spin estimates • When only Kerr or Hartle-Thorne spacetime is assumed, the HF QPO data and their individual models imply mass-spin relations instead of the preferred combinations of these quantities. • The degeneracy is broken when the EoS are considered. • The relativistic precession model implies NS spin frequency in a perfect agreement with the observation (4U 1636-53). • The modified RP model (torus model) provides good fits of the data without any tuning of parameters. • Strong constrainst are possible even from very general assumptions.

46. END Thankyouforyourattention…