Vznik této prezentace byl podpořen projektem
CZ.1.07/2.3.00/09.0138
Tato prezentace slouží jako vzdělávací materiál.
Mass and Spin Implications of High-Frequency QPO
Models across Black Holes and Neutron Stars
G. Török, M. A. Abramowicz, P. Bakala, P. Čech,
A. Kotrlová, Z. Stuchlík, E. Šrámková & M. 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), W. Kluzniak (CAMK), J. Miller (SISSA).
Supported by OPVK CZ.1.07/2.3.00/09.0138, MSM 4781305903, LC 06014 and GAČR202/09/0772.
www.physics.cz
1. Data and their models: the choice of few models
High frequency quasiperiodic oscillations appears in X-ray fluxes of several
LMXB sources. Commonly to BH and NS they often behave in pairs. There is
a large variety of ideas proposed to explain this phenomenon (in some cases
applied to both BH and NS sources, in some not). The desire is to relate HF
QPOs to strong gravity….
[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); 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)…]
1. Data and their models: the choice of few models
High frequency quasiperiodic oscillations appears in X-ray fluxes of several
Here we focus only to few of hot-spot or disc-oscillation
LMXB sources. Commonly to BH and NS they often behave in pairs. There is
models widely discussed for both classes of sources.
a large variety of ideas proposed to explain this phenomenon (in some cases
(which we properly list and quote slightly later).
applied to both BH and NS sources, in some not). The desire is to relate HF
QPOs to strong gravity….
[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); 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)…]
1. Data and their models: the choice of three sources
2. Near-extreme rotating black hole GRS 1915+105
(Remillard et al., 2003)
(McClintock & Remillard, 2003)
a > 0.99
(Remillard et al., 2009)
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
Relativistic precession [Stella et al. (1999)]
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
Relativistic precession [Stella et al. (1999)]
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
-1r, -2v disc-oscillation modes (frequency identification similar to the RP model)
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
Tidal disruption of large inhomogenities (mechanism similar to the RP model)
Cadez et al. (2008); Kostic et al. (2009);
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
Oscillations of warped discs (implying for 3:2 frequencies the same characteristic radii as TD)
Kato (1998,…, 2008)
2. Near-extreme rotating black hole GRS 1915+105
3:2 non-linear disc oscillation resonances
Abramowicz & Kluzniak (2001), Török et. al (2005)
or
Courtesy of M. Bursa
Abramowicz et al., (2010) in prep.
a > 0.99
a > 0.99
Other non-linear disc oscillation resonances
Abramowicz & Kluzniak (2001), Török et al. (2005), Török & Stuchlík (2005)
Abramowicz et al., (2010) in prep.
2. Near-extreme rotating black hole GRS 1915+105
2. Near-extreme rotating black hole GRS 1915+105
Abramowicz et al., (2010) in prep.
a > 0.99
Breathing modes
(here assuming constant angular momentum distribution)
Abramowicz et al., (2010) in prep.
2. Near-extreme rotating black hole GRS 1915+105: summary
?
3. Neutron stars: high mass approximation through Kerr metric
Torok et al., (2010) submitted
NS 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 assumed on previous slides. This fact is well manifested on
ISCO frequencies:
Several 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. A good example to start is the
RELATIVISTIC PRECESSION MODEL.
3. Neutron stars: 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.
3. Neutron stars: frequency relations implied by RP model
One can find combinations M, j giving the same ISCO frequency and plot related curves.
Resulting curves differ proving thus the uniqueness of frequency relations. On the other
hand they are very similar:
Torok et al., (2010) submitted
M = 2.5….4 MSUN
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) here implicitly given by the above plot.
The best fits of data of given source should be therefore reached for combinations of M
and j which can be predicted just from a one parametric fit assuming j = 0.
3. Neutron stars: RP model vs. the data of 4U 1636-53
The best fits of data of given source should be therefore reached for combinations of M
and j which can be predicted just from a 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. Neutron stars: RP model vs. the data of 4U 1636-53
Color-coded map of chi^2 [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
chi^2 ~ 300/20dof
chi^2 ~ 400/20dof
Torok et al., (2010) in prep.
Best chi^2
3. Neutron stars: other models vs. the data of 4U 1636-53
For several models there are M-j relations having origin analogic to the case of RP model.
chi^2 maps [M,j, each 10^6 points]: 4U 1636-53 data
3. Neutron stars: models vs. the data of Circinus X-1
For several models there are M-j relations having origin analogic to the case of RP model.
chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data
3. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: nearly concluding table
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. Neutron stars: M and j based on 3:2 epicyclic resonance model
which FAILS
a)
b)
(Abramowicz et al., 2005)
giving
Mass-spin inferred from epicyclic model assuming
Hartle-Thorne metric and 600:900Hz
Mass-spin after including several EOS
and lower-eigenfrequency 580-680Hz
q/j2
j
for j=0
j
Urbanec et al., (2010) in prep.
Urbanec et al., (2010) in prep.
After Abr. et al., (2007), Horák (2005)
3. Neutron stars: epicyclic resonance model and Paczynski modulation
The condition for modulation is fullfilled only for rapidly rotating strange stars, which most
likely falsifies the postulation of 3:2 resonant resonant mode eigenfrequencies being equal
to the geodesic radial and vertical epicyclic frequency….
(this postulation on the other hand seems to work for GRS 1915 + 105)
END