ON MASS CONSTRAINTS IMPLIED BY THE RELATIVISTIC
PRECESSION MODEL OF TWIN-PEAK QPOs IN CIRCINUS X-1
Pavel Bakala, Gabriel Török, Eva Šrámková,
Petr Celestian Čech, Zdeněk Stuchlík & Martin Urbanec
Institute of Physics,
Faculty of Philosophy and Science,
Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava
In collaboration: D. Barret (CESR); M. Bursa & J. Horák (CAS);
M.A. Abramowicz & W. Kluzniak (CAMK).
We also acknowledge the support of CZ grants MSM 4781305903, LC 06014, GAČR202/09/0772 and SGS.
www.physics.cz
Introduction: accretion, quasiperiodic oscillations, twin peaks
LMXBs short-term X-ray variability:
peaked noise (Quasi-Periodic Oscillations)
Individual peaks can be related to a
set of oscillators as well as to a time
evolution of an oscillator.
power
Sco X-1
• Low frequency QPOs (up to 100Hz)
• hecto-hertz QPOs (100-200Hz)
• kHz QPOs (~200-1500Hz):
Lower and upper QPO mode
forming twin peak QPOs
Fig: nasa.gov
frequency
kHz QPO origin remains questionable,
it is often expected that they are
associated to the orbital motion in the
inner part of the disc.
Circinus X-1 and 4U 1636-53
We focuse on the two representative neutron star sources.
Upper vs. lower QPO frequencies in 1636-53 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.
(From : T. Belloni, M. Mendez, J.
Homan, 2007, MNRAS)
The geodesic frequencies are the functions of the parameters of
spacetime geometry (M, j, q) and the appropriate radial coordinate.
Circinus X-1 mass estimation based on RP
model and Schwarzschild geometry
Orbital QPO models under high mass approximation
through Kerr metric
Torok et al., (2010),ApJ
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.
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.
Ambiguity in M and j
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 = Ms[1+0.7(j+j^2)]
Torok et al., (2010), ApJ
M = 2.5….4 MSUN
Ms = 2.5 MSUN
M ~ Ms[1+0.75(j+j^2)]
For a mass Ms of the non-rotating neutron star there is always a set of similar curves
implying a certain mass-spin relation M (Ms, 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.
Relativistic precession model vs. data of Circinus X-1
Color-coded map of χ2 [M,j,10^6 points] well agrees with rough estimate given by
simple one-parameter fit.
M= Ms[1+0.55(j+j^2)], Ms = 2.2M_sun
Best χ2 numericaly
Best χ2 exact Kerr solution
Best χ2 linearized Kerr frequencies
Relativistic precession model vs. data of 4U 1636-53
Color-coded map of χ2 [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
2
chi^2 ~ 300/20dof
chi^2 ~ 400/20dof
Torok et al., (2010) in prep.
Best χ
3.1. Nongeodesic corrections
- 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.
The above naive correction improves the RP model fits for both classes of sources.
Nongeodesic corrections
- 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.
The above naive correction improves the RP model fits for both classes of sources.
Conclusions
Model
rel.precession
nL= nK - nr,
nU= nK
atoll source 4U 1636-53
c2~
300
/20
Mass
1.8MSun[1+0.7(j+j2)]
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
Mass
2.2MSun[1+0.5(j+j2)]
RNS
< rms

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…