Transcript r - Physics.cz

Neutron star mass and spin implied by models of
the oracular twin-peak quasiperiodic oscillations
Gabriel Török, Pavel Bakala, Petr Celestian Čech,
Zdeněk Stuchlík, Eva Šrámková & Martin Urbanec
Institute of Physics,
Faculty of Philosophy and Science,
Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava
In collaboration: MAA; D. Barret (CESR); M. Bursa & J. Horák (CAS); W. Kluzniak (CAMK); J. Miller (SISSA).
We also acknowledge the support of CZ grants MSM 4781305903, LC 06014, GAČR202/09/0772 and SGS.
www.physics.cz
Outline
The purpose of this presentation rely namely in the comparison
between mass and spin predictions of several different orbital models of
neutron star twin peak QPOs. The slides are organized as follows:
1. Introduction: neutron star rapid X-ray variability, quasiperiodic
oscillations, twin peaks
2. QPO models under high mass approximation
- 2.1 Relativistic precession models, its implications (4U 1636-53)
- 2.2 Three other models, their implications (4U 1636-53)
- 2.3 Comparison to Circinus X-1
3. Summary for the four models
4. Epicyclic resonance model and its implications
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 MAA & M. Bursa 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:
two correlated modes which
often 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 most of the authors/ is to relate HF QPOs to strong
gravity….
Here we focus only on few hot-spot or disc-oscillation models which we
recall later…
2. QPO models under high mass approximation using 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 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.
2.1 Relativistic precession model
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.
2.1 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.
2.1.1. 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
2.1.1. 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
chi^2 ~ 300/20dof
chi^2 ~ 400/20dof
Torok et al., (2010) in prep.
Best chi^2
2.2 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
2.3 Comparison to Circinus X-1
Upper frequency [Hz]
Upper vs. lower QPO frequencies in 4U1636-53 and Circinus X-1:
Lower frequency [Hz]
2.3 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. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
2.5MSun[1+0.7(j+j2)]
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1MSun[1+ ?? ]
~ rms
x
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
X
RNS
< rms
----
< rms
~ rms
----
3. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
2.5MSun[1+0.7(j+j2)]
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1MSun[1+ ?? ]
~ rms
x
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
X
RNS
< rms
----
< rms
~ rms
----
3. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
2.5MSun[1+0.7(j+j2)]
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1MSun[1+ ?? ]
~ rms
x
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
X
RNS
< rms
----
< rms
~ rms
----
3. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
2.5MSun[1+0.7(j+j2)]
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1MSun[1+ ?? ]
~ rms
x
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
X
RNS
< rms
----
< rms
~ rms
----
3. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
2.5MSun[1+0.7(j+j2)]
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1MSun[1+ ?? ]
~ rms
x
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
X
RNS
< rms
----
< rms
~ rms
----
3. Summary for the four models
Model
atoll source 4U 1636-53
c2~
Mass
Z-source Circinus X-1
RNS
c2~
< rms
15
/10
< rms
bad
1/
M
rel.precession
nL= nK - nr,
nU= nK
300
/20
tidal disruption
nL= nK + nr,
nU= nK
150
/10
-1r, -2v reson.
nL= nK - nr,
nU= 2nK – nq
300/
1.8MSun[1+(j+j2)]
10
< rms
15
/10
warp disc res.
nL= 2(nK - nr,)
nU= 2nK – nr
600
/20
< rms
15
/10
epic. reson.
nL= nr,nU= nq
TBEL
1.8MSun[1+0.7(j+j2)]
2.2MSun[1+0.7(j+j2)]
2.5MSun[1+0.7(j+j2)]
1MSun[1+ ?? ]
high chi^2
~ rms
Mass
2.2MSun[1+0.5(j+j2)]
X
2.2MSun[1+0.7(j+j2)]
1.3MSun[1+ ?? ]
x
X
low chi^2 (except tidal model)
RNS
< rms
----
< rms
~ rms
----
3.1. 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
3.1. 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.
4. Epicyclic resonance model
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. These oscillations have frequencies equal
to the vertical and radial frequency of the perturbed geodesic motion.
Two distinct simplifications can be than assumed:
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
In the rest we focuse on this possibility.
Fig: J. Horák
4.1 NS mass and spin implied by the epicyclic resonance model
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 which we assumed till now
thus cannot be for this model most likely belived…
4.1 NS mass and spin implied by the epicyclic resonance model
(Bursa 2004, unp.).
q/j2
j
Urbanec et al., (2010) , A&A, submitted
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).
4.1 NS mass and spin implied by the epicyclic resonance model
j
(Bursa 2004, unp.).
Urbanec et al., (2010) , A&A, submitted
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 submitted
After Abr. et al., (2007), Horák (2005)
4.2 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…. (but what about nongeodesic…?)
Urbanec et al., (2010) , A&A submitted
After Abr. et al., (2007), Horák (2005)
4.2 Paczynski modulation and implied restrictions
(epicyclic resonance model)
Urbanec et al., (2010) , A&A submitted
After Abr. et al., (2007), Horák (2005)
4.2 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 resonant mode eigenfrequencies being
equal to geodesic radial and vertical epicyclic frequency…. (but what about nongeodesic…?)
However, what about the recently discussed speculations on ELECTROWEAK STARS ???
(perhaps moderately rotating , 1.2-1.6 solar masses….)
…the model is alive
[suggestion based on Dai et al., Phys.Rev.L, subm. 2009arXiv0912.0520D]
END
(of this presentation, the story continues…)
Thank you for your attention.
Special thanks belong to MAA for all the suggestions, help,
and, above all, for the opportunity of enjoying the spirit of
his approach to science and life…