In collaboration: D. Barret (CESR), M. Bursa & J. Horák (CAS), W. Kluzniak (CAMK), J. Miller (SISSA). We acknowledge the support of Czech grants MSM , LC and GAČR202/09/ Mass and Spin Implications of High-Frequency QPO Models across Black Holes and Neutron Stars Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava G. Török, M. A. Abramowicz, P. Bakala, P. Čech, A. Kotrlová, Z. Stuchlík, E. Šrámková & M. Urbanec

High frequency quasiperiodic oscillations appears in X-ray fluxes of several LMXB sources. Commonly to BH and NS they often behave in pairs. There is a large variety of ideas proposed to explain this phenomenon (in some cases applied to both BH and NS sources, in some not). The desire is to relate HF QPOs to strong gravity…. [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); 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)…] 1. Data and their models: the choice of few models

High frequency quasiperiodic oscillations appears in X-ray fluxes of several LMXB sources. Commonly to BH and NS they often behave in pairs. There is a large variety of ideas proposed to explain this phenomenon (in some cases applied to both BH and NS sources, in some not). The desire is to relate HF QPOs to strong gravity…. [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); 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)…] 1. Data and their models: the choice of few models Here we focus only to few of hot-spot or disc-oscillation models widely discussed for both classes of sources. (which we properly list and quote slightly later).

1. Data and their models: the choice of three sources

2. Near-extreme rotating black hole GRS a > 0.99 (Remillard et al., 2009) (McClintock & Remillard, 2003) (Remillard et al., 2003)

2. Near-extreme rotating black hole GRS a > 0.99 Relativistic precession [Stella et al. (1999)] Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > 0.99 Relativistic precession [Stella et al. (1999)] Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > r, -2v disc-oscillation modes (frequency identification similar to the RP model) Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > 0.99 Tidal disruption of large inhomogenities (mechanism similar to the RP model) Cadez et al. (2008); Kostic et al. (2009); Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > 0.99 Oscillations of warped discs (implying for 3:2 frequencies the same characteristic radii as TD) Kato (1998,…, 2008) Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > :2 non-linear disc oscillation resonances Abramowicz & Kluzniak (2001), Török et. al (2005) or Abramowicz et al., (2010) in prep. Courtesy of M. Bursa

2. Near-extreme rotating black hole GRS a > 0.99 Other non-linear disc oscillation resonances Abramowicz & Kluzniak (2001), Török et al. (2005), Török & Stuchlík (2005) Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS a > 0.99 Breathing modes (here assuming constant angular momentum distribution) Abramowicz et al., (2010) in prep.

2. Near-extreme rotating black hole GRS : summary Abramowicz et al., (2010) in prep. ?

3. Neutron stars: high mass approximation through Kerr metric NS 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 assumed on previous slides. This fact is well manifested on ISCO frequencies: Several QPO models predicts 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. Torok et al., (2010) submitted

3. Neutron stars: relativistic precession model One can solve the RP model definition equations Obtaining the 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. For matching of the data it is an important question whether there exist identical or similar curves for different combinations of M and j.

For a mass M 0 of the non-rotating neutron star there is always a set of similar curves implying a certain mass-spin relation M (M 0, j) here implicitly given by the above plot. The best fits of data of given source should be therefore reached for combinations of M and j which can be predicted just from a one parametric fit assuming j = 0. One can find combinations M, j giving the same ISCO frequency and plot related curves. Resulting curves differ proving thus the uniqueness of frequency relations. On the other hand they are very similar: M = 2.5….4 M SUN 3. Neutron stars: frequency relations implied by RP model Torok et al., (2010) submitted

The best fit of 4U data (21 datasegments) for j = 0 is reached for M s = 1.78 M_sun, which implies M= M s [1+0.75(j+j^2)], M s = 1.78M_sun 3. Neutron stars: RP model vs. the data of 4U The best fits of data of given source should be therefore reached for combinations of M and j which can be predicted just from a one parametric fit assuming j = 0.

Color-coded map of chi^2 [M,j,10^6 points] well agrees with rough estimate given by simple one-parameter fit. chi^2 ~ 300/20dof chi^2 ~ 400/20dof M= M s [1+0.75(j+j^2)], M s = 1.78M_sun Best chi^2 3. Neutron stars: RP model vs. the data of 4U Torok et al., (2010) in prep.

chi^2 maps [M,j, each 10^6 points]: 4U data 3. Neutron stars: other models vs. the data of 4U For several models there are M-j relations having origin analogic to the case of RP model.

chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data 3. Neutron stars: models vs. the data of Circinus X-1 For several models there are M-j relations having origin analogic to the case of RP model.

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

Model Model atoll source 4U Z-source Circinus X-1 2~2~2~2~ Mass Mass R NS 2~2~2~2~ Mass Mass R NS rel.precession L = K - r, L = K - r, U = K U = K300/20 1.8M Sun [1+0.7(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.5(j+j 2 )] < r ms tidal disruption L = K + r, L = K + r, U = K U = K150/10 2.2M Sun [1+0.7(j+j 2 )] < r ms bad 1/ M X r, -2v reson. L = K - r, L = K - r, U = 2 K –  U = 2 K –  300/ M Sun [1+(j+j 2 )] < r ms 15/10 2.2M Sun [1+0.7(j+j 2 )] < r ms warp disc res. L = 2( K - r, ) L = 2( K - r, ) U = 2 K – r U = 2 K – r600/20 2.5M Sun [1+0.7(j+j 2 )] < r ms 15/10 1.3M Sun [1+ ?? ] ~ r ms epic. reson. L = r, U =  L = r, U =  TBEL 1M Sun [1+ ?? ] 1M Sun [1+ ?? ] ~ r ms xX Neutron stars: nearly concluding table

3. Neutron stars: M and j based on 3:2 epicyclic resonance model Mass-spin inferred from epicyclic model assuming Hartle-Thorne metric and 600:900Hz Mass-spin after including several EOS and lower-eigenfrequency Hz q/j2q/j2 jj a) b) which FAILS (Abramowicz et al., 2005) Urbanec et al., (2010) in prep. giving for j=0

Urbanec et al., (2010) in prep. After Abr. et al., (2007), Horák (2005) 3. Neutron stars: epicyclic resonance model and Paczynski modulation The condition for modulation is fullfilled only for rapidly rotating strange stars, which most likely falsifies the postulation of 3:2 resonant resonant mode eigenfrequencies being equal to the geodesic radial and vertical epicyclic frequency…. (this postulation on the other hand seems to work for GRS )

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