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1. Neutron star mass and spin implied by models of the oracular twin-peak quasiperiodic oscillations Gabriel Török, Pavel Bakala, Petr Celestian Čech, Zdeněk Stuchlík, Eva Šrámková &Martin Urbanec Institute ofPhysics, Faculty of Philosophy and Science, Silesian University in Opava,Bezručovo n.13, CZ-74601, Opava In collaboration: MAA; D. Barret (CESR); M. Bursa & J. Horák (CAS); W. Kluzniak (CAMK); J. Miller (SISSA). We also acknowledge the support of CZ grants MSM 4781305903, LC 06014, GAČR202/09/0772 and SGS. www.physics.cz

2. Outline The purpose of this presentationrely namely in the comparison between mass and spin predictions of several different orbital models of neutron star twin peak QPOs. The slides are organized as follows: • Introduction: neutron star rapid X-ray variability, quasiperiodic oscillations, twin peaks • QPO models under high mass approximation • - 2.1 Relativistic precession models,its implications (4U 1636-53) • - 2.2 Three other models, their implications (4U 1636-53) • - 2.3 Comparison to Circinus X-1 • 3. Summary for the four models • 4. Epicyclic resonance model and its implications

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 MAA & M. Bursa 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: • two correlated modes which • often 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); Lamb et al. (1985); Stella et al. (1999); Morsink & Stella (1999); Stella & Vietri (2002); Abramowicz & Kluzniak (2001); Kluzniak & Abramowicz (2001); Abramowicz et al. (2003a,b); Wagoner et al. (2001); Titarchuk & Kent (2002); Titarchuk (2002); Kato (1998, 2001, 2007, 2008, 2009a,b); Meheut & Tagger (2009); Miller at al. (1998a); Psaltis et al. (1999); Lamb & Coleman (2001, 2003); Kluzniak et al. (2004); Abramowicz et al. (2005a,b), Petri (2005a,b,c); Miller (2006); Stuchlík et al. (2007);Kluzniak (2008); Stuchlík et al. (2008);Mukhopadhyay (2009); Aschenbach 2004, Zhang (2005); Zhang et al. (2007a,b); Rezzolla et al. (2003); Rezzolla (2004); Schnittman & Rezzolla (2006); Blaes et al. (2007); Horak (2008); Horak et al. (2009); Cadez et al. (2008); Kostic et al. (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 most of the authors/ is to relate HF QPOs to strong gravity…. • Here we focus only on fewhot-spot or disc-oscillation models which we recall later…

8. 2. QPO models under high mass approximation using Kerrmetric 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.

9. 2.1 Relativistic precession model 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.

10. 2.1 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.

11. 2.1.1. 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

12. 2.1.1. 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 Torok et al., (2010) in prep.

13. 2.2 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

14. 2.3 Comparison to Circinus X-1 Upper vs. lower QPO frequencies in 4U1636-53 and Circinus X-1: Upper frequency [Hz] Lower frequency [Hz]

15. 2.3 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

16. 3. Summary for the four models

17. 3. Summary for the four models

18. 3. Summary for the four models

19. 3. Summary for the four models

20. 3. Summary for the four models

21. 3. Summary for the four models high chi^2 low chi^2 (except tidal model)

22. 3.1. 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

23. 3.1. 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.

24. 4. Epicyclic resonance model Within the group of non-linear models suggested by Abramowicz and Kluzniak there is one specific (often quted and discussed) model which relates QPOs to the axisymmetric vertical and radial accretion disc oscillations. These oscillations have frequencies equal to the vertical and radial frequency of the perturbed geodesic motion. Two distinct simplifications can be than assumed: a) Observed frequencies are roughly equal to resonant eigenfrequencies. This for NSs FAILS. b) Alternatively, there are large corrections to the resonant eigenfrequencies. Abramowicz et al., 2005 In the rest we focuse on this possibility. Fig: J. Horák

25. 4.1 NS mass and spin implied by the epicyclic resonance model For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.). j The solution related to the high mass (i.e. Kerr) approximation which we assumed till now thus cannot be for this model most likely belived…

26. 4.1 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, submitted 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).

27. 4.1 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, submitted Mass-spin relations calculated assuming several modern EOS (of both “Nuclear” and “Strange” type) and realistic scatter from 600/900 Hz eigenfrequencies.

28. 4.2 Paczynski modulation and implied restrictions (epicyclic resonance model) Urbanec et al., (2010) , A&A submitted 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)

29. 4.2 Paczynski modulation and implied restrictions (epicyclic resonance model) Urbanec et al., (2010) , A&A submitted 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…. (but what about nongeodesic…?)

30. 4.2 Paczynski modulation and implied restrictions (epicyclic resonance model) Urbanec et al., (2010) , A&A submitted 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 resonant mode eigenfrequencies being equal to geodesic radial and vertical epicyclic frequency…. (but what about nongeodesic…?) However, what about the recently discussed speculations on ELECTROWEAK STARS ??? (perhaps moderately rotating , 1.2-1.6 solar masses….) …the model is alive [suggestion based on Dai et al., Phys.Rev.L, subm. 2009arXiv0912.0520D]

31. E N D (of this presentation, the story continues…) Thank you for your attention. Special thanks belong to MAA for all the suggestions, help, and, above all, for the opportunity of enjoying the spirit of his approach to science and life…