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Paczyński Modulation: Diagnostics of the Neutron Star EOS?
Gabriel Török, Martin Urbanec, Karel Adámek, Pavel Bakala, Eva Šrámková, Zdeněk Stuchlík
Institute of Physics, Silesian University in Opava
CZ.1.07/2.3.00/20.0071 Synergy , GAČR 209/12/P740, 202/09/0772, SGS-01-2010, www.physics.cz
1. Outline
1. Introduction: QPOs
2. NS Compactness C (another introduction)
3. Epicyclic Resonance Model – Falsification using condition for
Paczynski Modulation, C < 1
4. General Implications of Paczynski Modulation Mechanism
(disc oscillation models): report on a work in progress
2. Introduction: QPOs
MOTIVATION
LMXBs
Compact object:
- black hole or neutron star (>10^10gcm^3)
LMXB Accretion disc
T ~ 10^6K
>90% of radiation
in X-ray
Companion:
• density comparable to the Sun
• mass in units of solar masses
• temperature ~ roughly as the T Sun
• more or less optical wavelengths
Observations: The X-ray radiation is absorbed by the 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. Introduction: QPOs
MOTIVATION
Sco X-1
power
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 time
evolution of a single oscillator.
• Low frequency QPOs (up to 100Hz)
frequency
• hecto-hertz QPOs (100-200Hz),...
• HF QPOs (~200-1500Hz):
Lower and upper QPO feature
forming twin peak QPOs
Fig: nasa.gov
The
HF QPO origin remains
questionable,
it is most often
expected that it is associated to
orbital motion in the inner part of the
accretion disc.
2. Introduction: QPOs
Quality factor Q indicates sharpness of
the peak, Q ~ h/w
Power
height h
width w at ½ h
Amplitude r indicates strength of peak
variability (its energy) in terms of “rms
amplitude” = percentual fraction (root
mean square fraction) of the peak
energy with the respect to the total
countrate
(r ~ area under peak)
Frequency
BH QPOs (Galactic microquasars):
frequencies up to 500Hz
low amplitude and Q : typically up to r~5% and Q~5
NS QPOs:
frequencies up to 1500Hz
often amplitudes up to r~20% and quality factors up to Q~200
Torok et al. (2010),ApJ
3. NS Compactness
The influence of NS oblateness on orbital frequenies has been extensively studied in last
decade, e.g.,
Morsink, Stella, 1999, ApJ;
Gondek-Rosińska, Stergioulas, Bulik, Kluźniak, Gourgoulhon, A&A (2001);
Amsterdamski, Bulik, Gondek-Rosińska, Kluźniak, A&A (2002),…
3. NS Compactness
C = RNS/Rms
3. NS Compactness
C = RNS/Rms
3. NS Compactness
1
C = RNS/Rms
3. NS Compactness
1
1
C = RNS/Rms
3. NS Compactness
1
1
1
1
high mass
MASS
C = RNS/Rms
3. NS Compactness
1
1
low mass
C = RNS/Rms
3. NS Compactness
3. Epicyclic Resonance Model for NS QPOs and NS Mass
Within the group of non-linear models suggested by Abramowicz and Kluzniak there is
one specific (often quted and discussed) model which relates QPOs to the axisymmetric
vertical and radial accretion disc oscillations (Abramowicz & Kluzniak 2001). These
oscillations have frequencies equal to the vertical and radial frequency of the perturbed
geodesic motion.
Two distinct simplifications can be than assumed (see Urbanec et al. 2010, for refs):
a) Observed frequencies are roughly equal to resonant eigenfrequencies.
This for NSs FAILS.
b) Alternatively, there are large corrections to the resonant eigenfrequencies.
Abramowicz et al., 2005
Fig: J. Horák
3. Epicyclic Resonance Model for NS QPOs and NS Mass
For a non-rotating approximation it gives NS mass about
(Bursa 2004, unp.).
j
The solution related to the high mass (i.e. Kerr) approximation thus cannot be trusted.
3. Epicyclic Resonance Model for NS QPOs and NS Mass
(Bursa 2004, unp.).
q/j2
j
Urbanec et al., (2010) , A&A
For a non-rotating approximation it gives NS mass about
Mass-spin relations inferred assuming Hartle-Thorne metric and various NS oblateness.
One can expect that the red/yellow region is allowed by NS equations of state (EOS).
3. Epicyclic Resonance Model for NS QPOs and NS Mass
j
(Bursa 2004, unp.).
Urbanec et al., (2010) , A&A
For a non-rotating approximation it gives NS mass about
Mass-spin relations calculated assuming several modern EOS (of both “Nuclear”
and “Strange” type) and realistic scatter from 600/900 Hz eigenfrequencies.
4. Paczynski Modulation and NS Compactness
Possible relation between
the X-ray QPO phenomenon
and general relativity
”….suggest that the unsteady flow
would make the boundary-layer
luminosity variable, possibly giving
rise to the X-ray quasi-periodic
oscillation (QPO) phenomenon.”
REQUIRED CONDITION:
C = RNS/Rms < 1
After Abr. et al., (2007), Horák (2005)
Bohdan Paczyński, 1987
1
high mass
MASS
C = RNS/Rms
4. Paczynski Modulation and NS Compactness
1
1
low mass
Urbanec et al., (2010) , A&A
4. Paczynski Modulation and Implied Restrictions
(Epicyclic Resonance Model)
The condition for modulation is fulfilled only for rapidly rotating strange stars, which most
likely falsifies the postulation of the 3:2 resonant mode eigenfrequencies being equal to
geodesic radial and vertical epicyclic frequency….
(Typical spin frequencies of discussed sources are about 200-700Hz; based on X-ray bursts)
5. Paczynski Modulation – General Implications
MASS [MSun]
Almost any disc-oscillation model requires C<1
Initial Distribution of NS
[C<>1] =>
Distribution of QPO Sources
SPIN [Hz]
5. Paczynski Modulation – General Implications
0
Almost any disc-oscillation model requires C<1
Initial Distribution of NS (one concrete EoS)
MASS [MSun]
1.5
Mass [Msun]
1
2
SPIN [Hz]
5. Paczynski Modulation – General Implications
0
Almost any disc-oscillation model requires C<1
Initial Distribution of NS (one concrete EoS)
MASS [MSun]
1.5
Mass [Msun]
1
2
SPIN [Hz]
5. Paczynski Modulation – General Implications
0
Resulting Distribution of QPO sources (the same EoS)
MASS [MSun]
1.5
Mass [Msun]
1
2
0 500 1000 1500
Spin [Hz]
SPIN [Hz]
5. Paczynski Modulation – General Implications
0
MASS [MSun]
1.5
Mass [Msun]
1
Resulting Distribution of QPO sources
(another example)
2
0 500 1000 1500
Spin [Hz]
SPIN [Hz]
5. Paczynski Modulation – General Implications
1.5
Mass [Msun]
1
Number of Sources
0
2
SPIN [Hz]
6. Conclusions
1.5
Mass [Msun]
1
Number of Sources
0
2
SPIN [Hz]
END
Thank you for your attention…