BH and NS spin estimates based on the models of
oracular high-frequency quasiperiodic oscillations
Gabriel Török
Institute of Physics,
Faculty of Philosophy and Science,
Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava
Supported by the CZ grants MSM 4781305903, LC 06014, GAČR202/09/0772, CZ.1.07/2.2.00/28.0271 & SGS-01-2010.
www.physics.cz
Outline
The purpose of this presentation rely namely in the comparison
between mass/spin predictions of several different orbital models of HF
QPOs in LMXBs. The slides are organized as follows:
1. Introduction: neutron star rapid X-ray variability, quasiperiodic
oscillations, twin peaks
2. Measuring spin from HF BH QPOs: hot-spot models, disc
oscillation models…
3. Measuring spin from HF NS QPOs: characteristic mass-spin
relations
4. Summary
Tento projekt je spolufinancován Evropským sociálním fondem a státním rozpočtem České republiky
1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs
Artists view of LMXBs
“as seen from a hypothetical planet”
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
1. Introduction: LMXBs, quasiperiodic oscillations, HF 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 time
evolution of the oscillator.
power
Sco X-1
• Low frequency QPOs (up to 100Hz)
• hecto-hertz QPOs (100-200Hz)
• HF QPOs (~200-1500Hz):
Lower and upper QPO mode
forming twin peak QPOs
Fig: nasa.gov
frequency
The HF QPO origin remains
questionable, it is often expected
that it is associated to orbital motion
in the inner part of the accretion disc.
Upper frequency [Hz]
1.1 Black hole and neutron star HF QPOs
3:2
Lower frequency [Hz]
Figure (“Bursa-plot”): after M. Bursa & MAA 2003, updated data
1.1 Black hole and neutron star HF QPOs
It is unclear whether the HF QPOs in BH and NS sources have the same origin.
3:2
Lower frequency [Hz]
• NS HF QPOs: 3:2 clustering,
- two correlated modes which
exchange the dominance
when passing the 3:2 ratio
Amplitude difference
Upper frequency [Hz]
• BH HF QPOs:
(perhaps) constant frequencies,
exhibit the 3:2 ratio
3:2
Frequency ratio
Figures Left: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); Right: Torok (2009)
1.2. The Desire
There is a large variety of ideas proposed to explain the QPO phenomenon
[For instance, Alpar & Shaham (1985); Lamb et al. (1985); Stella et al. (1999); Morsink &
Stella (1999); Stella & Vietri (2002); Abramowicz & Kluzniak (2001); Kluzniak &
Abramowicz (2001); Abramowicz et al. (2003a,b); Wagoner et al. (2001); Titarchuk & Kent
(2002); Titarchuk (2002); Kato (1998, 2001, 2007, 2008, 2009a,b); Meheut & Tagger
(2009); Miller at al. (1998a); Psaltis et al. (1999); Lamb & Coleman (2001, 2003); Kluzniak
et al. (2004); Abramowicz et al. (2005a,b), Petri (2005a,b,c); Miller (2006); Stuchlík et al.
(2007); Kluzniak (2008); Stuchlík et al. (2008); Mukhopadhyay (2009); Aschenbach 2004,
Zhang (2005); Zhang et al. (2007a,b); Rezzolla et al. (2003); Rezzolla (2004); Schnittman &
Rezzolla (2006); Blaes et al. (2007); Horak (2008); Horak et al. (2009); Cadez et al. (2008);
Kostic et al. (2009); Chakrabarti et al. (2009), Bachetti et al. (2010)…]
- in some cases the models are applied to both BHs and NSs, in some not
- some models accommodate resonances, some not
- the desire /common to several of the authors/ is to relate HF QPOs to
strong gravity and inferr the compact object properties using QPO
measurements
2. Spin from the models of the 3:2 BH QPOs
Upper frequency [Hz]
• the (advantage of) BH HF QPOs:
(perhaps) constant frequencies,
exhibit the mysterious 3:2 ratio
• The BH 3:2 QPO frequencies are stable
which imply that they depend mainly on
the geometry and not so much on the
dirty physics of the accreted plasma.
Lower frequency [Hz]
Figure:
after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003);
2. Models relating both of the 3:2 BH QPOs to a single radius
Here we use few hot-spot and disc-oscillation models relating both QPOs to
a single preferred radius in order to demonstrate the (potential) predictive
power of QPOs.
2. Models relating both of the 3:2 BH QPOs to a single radius
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
Characteristic Frequencies
RP
TD
Relativistic Precession
WD
Stella et al. (1999); Morsink & Stella (1999); Stella
ER
& Vietri (2002)]
KR
RP1
RP2
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
Characteristic Frequencies
RP
TD
WD
Tidal Disruption
ER
Čadež et al. (2008), Kostič et al. (2009), Germana
KR
adež et al. (2008), Kostic´ et al. (2009), and Germana et al.
etCˇal.
(2009)
(2009)
RP1
RP2
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
(or torus)
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
Characteristic Frequencies
RP
TD
WD
ER
Warped Disc Resonance
KR
a representative of models proposed by Kato
RP1
(2000,
2004,
2005,
2008)
Cˇ adež 2001,
et al. (2008),
Kostic´
et al.
(2009), and Germana et al.
(2009)
RP2
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
(or torus)
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
Characteristic Frequencies
RP
TD
WD
ER
KR
RP1
Epicyclic Resonance, Keplerian Resonance
RP2
two
representatives
ofetmodels
proposed
byet al.
Cˇ adež
et al. (2008), Kostic´
al. (2009),
and Germana
Abramowicz, Kluzniak(2009)
et al. (2000, 2001, 2004,
2005,…)
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
(or torus)
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
Characteristic Frequencies
RP
TD
WD
ER
KR
RP1
RP2
Resonances between non-axisymmetric
Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al.
oscillation
modes of
a toroidal structure
(2009)
two representatives by Bursa (2005), Torok et al
(2010) predicting frequencies close to RP model
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
Here we focus just on a choice of few hot-spot and disc-oscillation models:
MODEL :
RP
TD
WD
ER
KR
RP1
RP2
Characteristic Frequencies
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
Different models associate QPOs to different radii…
RP
WD, TD
ER
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
One can easily calculate frequency-mass functions for each of the models
Spin a
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
And compare the frequency-mass functions to the observation.
For instance in the case of GRS 1915+105.
Spin a
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
Comparison with the independent spin measurements:
Considering the high spin measurement of 1915+105, some QPO models
(e.g. relativistic precession) are clearly disfavoured.
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
Because of rough observational 1/M scaling of the observed 3:2
frequencies, the spins inferred from QPOs are moreless common to each of
the three microquasars,
Consequently the QPO models
favoured in 1915+105 does not fit
continuum estimates for 1655-40 if
these are very different from
1915+105 (0.7 vs. 0.98), which is
rather a generic problem…
Because of rough observational 1/M scaling of the observed 3:2
frequencies, the spins inferred from QPOs are moreless common to each of
the three microquasars,
Consequently the QPO models
favoured in 1915+105 does not fit
continuum estimates for 1655-40 if
these are very different from
1915+105 (0.7 vs. 0.98), which is
rather a generic problem…
Torok et al. (2011), A&A
2. 1 Models relating both of the 3:2 BH QPOs to a single radius
2.2 Discoseismic modes
Wagoner et al. (2001)
Here we attempt to identify both the 3:2 QPOs with (some) pair of fundamental
disc oscillation modes .
Note that the frequency ratio of two fundamental modes depends only on spin
and (weakly) on the speed of sound (since each of the modes is located at its own
radii).
2.2 Discoseismic modes
Torok et al. (2011), A&A
Ratio between fundamental g-mode and c-mode as it depends on the spin:
Assuming g- and c- modes, the 3:2 ratio can be reproduced only for spins either
around 0.7-0.8 or 0.9-0.95.
2.2 Discoseismic modes
Torok et al. (2011), A&A
Ratio between g-mode and p-mode as it depends on the spin:
Assuming that the one QPO is g-mode and the other one p-mode, the 3:2 ratio can
be reproduced for spins up to 0.8.
Assuming p-modes (and one of the two others) the 3:2 ratio cannot be reproduced
for high spins.
2.2 Discoseismic modes
Taking into account the absolute values of QPO frequencies (and therefore the mass):
Except the combination identifying 3 and 2 as g-mode and c-mode the required
masses do not overlap in a single case with those estimated from independent
methods.
3. Spin from the models of the NS HF QPOs
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 the ISCO frequencies:
Several QPO models predict 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. A good example to start is the
RELATIVISTIC PRECESSION MODEL.
3. Spin from the models of the NS HF QPOs
One can use the RP model definition equations
to obtain the following 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. In relation to
matching of the data, there is an important question whether there are identical or
similar curves for different combinations of M and j.
3.1 Relativistic precession model
One can find the combinations of M, j giving the same ISCO frequency and plot the
related curves. The resulting curves differ proving thus the uniqueness of the frequency
relations. On the other hand, they are very similar:
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 M0 of the non-rotating neutron star there is always a set of similar curves
implying a certain mass-spin relation M (M0, j) (implicitly given by the above plot).
The best fits of data of a given source should be therefore reached for combinations of M
and j that can be predicted from just one parametric fit assuming j = 0.
3.1.1. Relativistic precession model vs. data of 4U 1636-53
The best fits of data of a given source should be reached for the combinations of M and j
that can be predicted from just one parametric fit assuming j = 0.
The best fit of 4U 1636-53 data (21 datasegments) for j = 0 is reached for Ms = 1.78
M_sun, which implies
M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun
3.1.1. Relativistic precession model vs. data of 4U 1636-53
Color-coded map of chi^2 [M,j,10^6 points] well agrees with the rough estimate given
by a simple one-parameter fit.
M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun
Best chi^2
chi^2 ~ 300/20dof
chi^2 ~ 400/20dof
3.2 Four models vs. data of 4U 1636-53
Several models imply M-j relations having the origin analogic to the case of RP model.
chi^2 maps [M,j, each 10^6 points]: 4U 1636-53 data
3.3 Comparison to Circinus X-1
Several models imply M-j relations having the origin analogic to the case of RP model.
chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data
3.4. Quality of fits and nongeodesic corrections
RP model, figure from Torok et al., (2010), ApJ
- It is often believed that, e.g., RP model fits well the low-frequency sources but
not the high-frequency sources.
The difference however follows namely from
- difference in coherence times (large and small errorbars)
- position of source in the frequency diagram
3.4. Quality of fits and nongeodesic corrections
- It is often believed that, e.g., RP model fits well the low-frequency sources but
not the high-frequency sources. The same non-geodesic corrections can be
involved in both classes of sources.
Circinus X-1 data
4U 1636-53 X-1 data
The above naive correction improves the RP model fits for both classes of sources.
Similar statement can be made for the other models.
3.5 NS mass and spin implied by the epicyclic resonance model
(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).
3.5 NS mass and spin implied by the epicyclic resonance model
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.
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 300-600Hz; based on X-ray bursts)
Urbanec et al., (2010) , A&A
After Abr. et al., (2007), Horák (2005)
3.5 Paczynski modulation and implied restrictions
(epicyclic resonance model)
4. Summary (BHs)
Torok et al. (2011), A&A
Spins of the three microquasars from the two hot-spot models and WD
model are below a=0.5. The spins above a=0.9 are matched only by the
resonance model (in our choice of models).
The QPO models favoured in
1915+105 does not fit for 1655-40.
The reason is rather generic:
1/M scaling of the observed 3:2
frequencies.
4. Summary (BHs)
Fundamental discoseismic modes vs. 3:2 frequency ratio (both QPOs):
Assuming that the 3:2 QPOS in microquasars are related to the
fundamental p-modes and either to the g- modes or c-modes the spin
cannot be high (a<0.8).
Assuming the combination of g-mode and c-mode, the spin must be about
a=0.9-0.95.
The problem with the 1/M scaling arises also for the discoseismic modes...
4. Summary (NSs)
4. Summary
The list of questions which I have been asked to answer in this
talk:
(1) what spin measurements have been made or attempted so far using the
method
(2) what are the strengths and weaknesses of the method
(3) what future work is needed to improve the method
4. Summary
The list of questions which I have been asked to answer in this
talk:
(1) what spin measurements have been made or attempted so far using the
method
(2) what are the strengths and weaknesses of the method
(3) what future work is needed to improve the method
• The BH 3:2 QPO frequencies are
“stable” which imply that they depend
mainly on the geometry and not so much
on the dirty physics of the accreted
plasma.
4. Summary
The list of questions which I have been asked to answer in this
talk:
(1) what spin measurements have been made or attempted so far using the
method
(2) what are the strengths and weaknesses of the method
(3) what future work is needed to improve the method
• The BH 3:2 QPO frequencies are
“stable” which imply that they depend
mainly on the geometry and not so much
on the dirty physics of the accreted
plasma.
4. Summary
We need a right model.