Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ Opava, Czech Republic The presentation refers to work in progress. It also draws from a collaboration with D. Barret, J. Miller, J. Horák

1. Data and their models: LMXBs, accretion discs, variability density comparable to the Sun mass in units of solar masses temperature ~ roughly as the T Sun moreless optical wavelengths Artists view of LMXBs “as seen from a hypothetical planet” Companion: Compact object: - black hole or neutron star (>10^10gcm^3) >90% of radiation in X-ray LMXB Accretion disc Observations: The X-ray radiation is absorbed by 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). T ~ 10^6K Figs: space-art, nasa.gov

1.2 Data and their models: pairs of kHz QPOs Fig: nasa.gov LMXBs short-term X-ray variability: peaked noise (Quasi-Periodic Oscillations) Low frequency QPOs (up to 100Hz) hecto-hertz QPOs ( Hz) kHz QPOs (~ Hz): Lower and upper QPO mode forming twin peak QPOs frequency power Sco X-1 kHz QPO origin remains questionable, it is often expected that they are associated to the orbital motion in the inner part of the disc. Individual peaks can be related to a set of oscillators as well as to a time evolution of an oscillator.

1.3 Data and their models: frequency relations between kHz QPOs The two QPO frequencies seems to be well correlated, following a nearly linear relation specific for a given source.

1. Data and their models: orbital models of kHz QPOs  Several models have been proposed. Most of them relate QPOs to the orbital motion in inner parts of accretions disc. For instance,  Relativistic precession model, Stella, Vietri, 1999, relates the kHz QPOs to the frequencies of geodesic motion.  Some models relate the kHz QPOs to resonance between disc oscillation modes given by the frequencies of geodesic motion (Kluzniak, Abramowicz, 2001). On next few slides we focuse on frequency identification given by relativistic precession model, (Note that, in Schwarzschild spacetime, this identification correspond to m= -1 radial and m= -2 vertical disc oscillation modes as well.)

*For simplicity we consider Kerr spacetimes on few slides (while finaly we apply a more realistic approach needed for rotating neutron stars). Solving above equations one obtains frequency relations U ( L ) which can be compared to those observed. * 2. Relativistic precession model

2.1 Frequency relations given by the relativistic precession model M=1.4M_sun, j=0 M=2M_sun, j=0 M=1.4M_sun, j=0.3 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.

Uniqueness of the curves in frequency plot: Obviously, if there would be two different combinations of M and j implying from the RP model the same curve these combinations must imply also the same ISCO frequency. ISCO frequency is implicitly given by formulae determining the orbital frequency and the ISCO radius r ms, Solving these numerically one can find combinations M, j giving the same ISCO frequency and plot related curves. 2.1 Frequency relations given by the relativistic precession model

For a given mass M S of the non-rotating neutron star there is a set of similar curves given, within some approximation, by the relation M ~ M S [1+0.75(j+j^2)]. 2.1 Frequency relations given by the relativistic precession model 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 the curves are very similar. M = 2.5….4 M SUN

It was previously noticed that the RP model fits the data qualitatively well but often with non-negligible residuals (which arise especially on the top part of the correlation). It is often quoted that the model implies a high angular momentum (j>0.25) for which the residuals are somewhat lower (but still significant). Here we suggests that a fit for the non-rotating neutron star with only free parameter M s implies a rough mass-angular-momentum relation M ~ M S [1+0.75(j+j^2)]. related to a “family of best fits” giving comparable chi^2. We investigate this suggestion for the source 4U Fitting the data

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 2.2 Fitting the data

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 2.2 Fitting the data

- spin frequency of the source expected from x-ray bursts: either 290 or 580 Hz - Hartle-Thorne spacetimes, SKYRME EOS - RNS, LORENE… 2.3 Realistic configurations EOS 580Hz EOS 290Hz

3. Other models, other sources  We checked RP model for several other sources, the relation M = Ms[1+k(j+j^2)] With k = 0.75 well indicates the best chi M-j region in any high frequency source. For low-frequency sources the best fits are obtained for somewhat lower values of k=0.5 (Circinus X-1).  We also checked four other orbital models (listed later), for these there are also similar mass-angular momentum relations, in general it is M = Ms[1+k(j+j^2)], k=

3. Other models, other sources chi^2 maps [M,j, each 10^6 points]: 4U data

3. Other models, other sources chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data

6. Non-geodesic corrections ? - It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources

6. Non-geodesic corrections ? - It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources Circinus X-1 data 4U X-1 data