Vznik této prezentace byl podpořen projektem CZ.1.07/2.3.00/ Tato prezentace slouží jako vzdělávací materiál.

Supported by grants OPVK CZ.1.07/2.3.00/ , MSM , LC 06014, GAČR202/09/0772, SGS/01/2010 and SGS/02/2010. ON MASS CONSTRAINTS IMPLIED BY THE RELATIVISTIC PRECESSION MODEL OF TWIN-PEAK QPOs IN CIRCINUS X-1 Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava Pavel Bakala, Gabriel Török, Eva Šrámková, Petr Celestian Čech, Zdeněk Stuchlík & Martin Urbanec

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. Introduction: accretion, quasiperiodic oscillations, twin peaks

Circinus X-1 and 4U We focuse on the two representative neutron star sources. Upper vs. lower QPO frequencies in and Circinus X-1: Clusters of detections : Circinus X1: 3:1 4U 1636 : 3:2, 5:4

Fitting the LMXBs kHz QPO data by relativistic precession model frequency relations The relativistic precesion model (in next RP model) introduced by Stella and Vietri, (1998, ApJ) indetifies the upper QPO frequency as orbital (keplerian) frequency and the lower QPO frequency as the periastron precesion frequency. The geodesic frequencies are the functions of the parameters of spacetime geometry (M, j, q) and the appropriate radial coordinate. (From : T. Belloni, M. Mendez, J. Homan, 2007, MNRAS)

Circinus X-1 mass estimation based on RP model and Schwarzschild geometry

Orbital QPO models under high mass approximation through Kerr metric 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 ISCO frequencies: Orbital 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. Torok et al., (2010),ApJ

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. Relativistic precession model

Torok et al., (2010), ApJ M = 2.5….4 M SUN M s = 2.5 M SUN M ~ M s [1+0.75(j+j^2)] For a mass M s of the non-rotating neutron star there is always a set of similar curves implying a certain mass-spin relation M (M s, j) (implicitly given by the above plot). The best fits of data of a given source should be therefore reached for the combinations of M and j which can be predicted just from one parametric fit assuming j = 0. For each pair of parameters M, j the relativistic precesion model gives a different curve in the frequency – frequency plane. On the other hand, one can find classes of very similar curves with parametres M,j bounded by the relation: M = M s [1+0.7(j+j^2)] Ambiguity in M and j

Color-coded map of χ 2 [M,j,10^6 points] well agrees with rough estimate given by simple one-parameter fit. M= M s [1+0.55(j+j^2)], M s = 2.2M_sun Best χ 2 numericaly Best χ 2 exact Kerr solution Best χ 2 linearized Kerr frequencies Relativistic precession model vs. data of Circinus X-1

Color-coded map of χ 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 χ 2 Torok et al., (2010) in prep. Relativistic precession model vs. data of 4U

- It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources. The same non-geodesic corrections can be involved in both classes of sources Nongeodesic corrections The above naive correction improves the RP model fits for both classes of sources.

- It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources. The same non-geodesic corrections can be involved in both classes of sources. Nongeodesic corrections The above naive correction improves the RP model fits for both classes of sources.

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 Conclusions  The estimate of mass calculated in the Schwarzschild geometry represents the lowest limit of mass estimate implied by the RP model.  The RP model is not able to provide independent mass and spin estimates based on the twin-peak kHz QPOs data..  Behavior of the twin-peak QPOs data fits by the RP model frequency relations indicates the existence of non-geodesic influence on the orbital frequencies.

Thank you for the attention…