Transcript M - Physics.cz

Relating high-frequency QPOs and neutron-star EOS
Gabriel Török*
*Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-74601 Opava, Czech Republic
The presentation refers to work in progress and draws
mainly from the collaboration with
M. Abramowicz*, P.Bakala*, M. Bursa, D. Barret, J. Horák, J. Miller, M. Urbanec*,
and Z. Stuchlík*
1. Low-mass X-ray binaries (LMXBs), accretion discs, variability
Artists view of LMXBs
“as seen from a hypothetical planet”
Compact object:
- black hole or neutron star (>10^10gcm^3)
LMXB Accretion T ~ 10^6K
Companion:
• density comparable to the Sun
• mass in units of solar masses
• temperature ~ roughly as the T Sun
• moreless optical wavelengths
disc
>90% of radiation
in X-ray
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).
Figs: space-art, nasa.gov
2. Short-term and rapid variability, kHz QPOs
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.
3. 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 corresponding to the frequencies of geodesic motion
(Kluzniak,
Abramowicz, 2001)
In next we focuse mostly on frequency identification given by relativistic
precession model,
(But we note that, in Schwarzschild spacetime, this identification correspond to
m= -1 radial and m= -2 vertical disc oscillation modes as well.)
4. Frequency relations given by the relativistic precession model
*
*For simplicity we consider Kerr spacetime on next few slides (while finaly we
apply a more realistic approach needed for rotating neutron stars).
One can solve above equations in order to obtain frequency relations nU(nL)
which can be compared to those observed.
4. Frequency relations given by the relativistic precession model
M=1.4M_sun, j=0.3
M=1.4M_sun, j=0
M=2M_sun, j=0
From the definition equations, there is always a unique curve for each different
combination of (M, j). On the other hand, it is the practical question whether one
can obtain curves which are rather similar for rather different combinations of (M,
j).
4. Frequency relations given by the relativistic precession model
M = Ms, j = 0
Ms=1M_sun
M = Ms(1+0.9*j)
j = 0.3
Ms=1.5M_sun
Ms=2M_sun
Ms=2.5M_sun
Ms=3M_sun
For a given mass Ms of the non-rotating neutron star there is a set of similar
curves given by the relation
M ~ Ms(1+0.9*j).
5. Fitting the data
It was previously noticed that the RP model fits the data qualitatively well but
always 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 Ms implies a rough mass-angular-momentum relation
M ~ Ms(1+0.9*j).
related to a “family of best fits” giving comparable chi^2.
We investigate this suggestion for the source 4U 1636-53.
5. Fitting the data
The best fit of 4U 1636-53 data for j = 0 is reached for Ms = 1.77 M_sun,
which implies
M= Ms(1+0.9*j), Ms = 1.77M_sun
Expected inaccuracy of this rough relation
(still analytic formulae)
5. Fitting the data
The best fit of 4U 1636-53 data for j = 0 is reached for Ms = 1.77 M_sun,
which implies
M= Ms(1+0.9*j), Ms = 1.77M_sun
Expected inaccuracy of this rough relation
(still analytic formulae)
Color-coded map of xi^2
for Hartle-Thorne metric
[M,j, q; 10^9 points; 40hours
on IBM Blade server]
well agrees with rough
estimate given by simple
one-parameter fit.
chi^2 ~ 300
chi^2 ~ 400
6. EOS
From X-ray burst lighcurves, the spin frequency was estimated (Sthromayer et.
al, 1998-2005) to be either 290 or 580Hz. We calculated relevant NS
configurations for several EOS.
7. Two more QPO models
We calculated also predictions of two other QPO model (frequency identifictions):
 Epicyclic model (one of models by Abramowicz & Kluzniak): nL = nr, nU = nq
 Warped disc oscillations model (S. Kato, 2008): nL = 2(nK - nr), nU = 2nK - nr
7. Conclusions
 Relativistic precession model:
M ~ 1.77 M_sun (1+0.9*j),
considering EOS M ~1.9-2 M_sun, j ~ 0.15 (280Hz) or M ~2-2.3 M_sun, j ~ 0.25
Neutron star radii below ISCO
 Warped disc oscillation model:
M ~ 2.3 M_sun (1+0.75*j)
considering EOS M ~2.4 M_sun, j ~ 0.15 (280Hz), no solution for 580Hz
Neutron star radii below ISCO
 Epicyclic model:
M ~ 0.95 M_sun (1+ j)
considering EOS M ~1.1 M_sun, j ~ 0.2 (280Hz), M ~1.2-1.3 M_sun, j ~ 0.4 (280Hz),
Neutron star radii highly above ISCO but slightly below the relevant resonant orbit
RNS ~ 8 – 10M vs. R3:2 ~ 10-11M
 The above results are preliminary and still require further investigation but the main
outlined quantitative differences between predictions of individual models are robust.