supported by the czech grants msm 4781305903, lc 06014 , ga Č r202/09/0772 and sgs -01-2010

Supported by the Czech grants MSM 4781305903, LC 06014, GAČR202/09/0772 and SGS-01-2010.

www.physics.cz

BH and NS spin estimates based on the models of oracular high-frequency quasiperiodic oscillations

Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo n.13, CZ-74601, Opava

Gabriel Török

Outline

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

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:

(The main task is to make the set of 44 slides in 15 minutes.)

• density comparable to the Sun• mass in units of solar masses• temperature ~ roughly as the T Sun• more or less optical wavelengths

Artists view of LMXBs“as seen from a hypothetical planet”

Companion:

Compact object:- black hole or neutron star (>10^10gcm^3)

>90% of radiation in X-ray

LMXB Accretion disc

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).

T ~ 10^6K

Figs: space-art, nasa.gov

1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs

Fig: nasa.gov

LMXBs short-term X-ray variability:peaked noise (Quasi-Periodic Oscillations)

• Low frequency QPOs (up to 100Hz)

• hecto-hertz QPOs (100-200Hz)

• HF QPOs (~200-1500Hz): Lower and upper QPO mode forming twin peak QPOs

frequency

pow

er

Sco X-1

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.

Individual peaks can be related to a set of oscillators, as well as to time evolution of the oscillator.

1. Introduction: LMXBs, quasiperiodic oscillations, HF QPOs

1.1 Black hole and neutron star HF QPOs

Lower frequency [Hz]

Upp

er fr

eque

ncy

[Hz]

Figure (“Bursa-plot”): after M. Bursa & MAA 2003, updated data

3:2

Figures -Left: after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003); Right: Torok (2009)

1.1 Black hole and neutron star HF QPOs

• BH HF QPOs: (perhaps) constant frequencies, exhibit the 3:2 ratio

• NS HF QPOs: 3:2 clustering, - two correlated modes which exchange the dominance when passing the 3:2 ratio

It is unclear whether the HF QPOs in BH and NS sources have the same origin.

Ampl

itude

diff

eren

ce

Frequency ratio

Upp

er fr

eque

ncy

[Hz]


3:2

3:2

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

1.2. The Desire

Figure:after Abramowicz&Kluzniak (2001), McClintock&Remillard (2003);

• the (advantage of) BH HF QPOs: (perhaps) constant frequencies, exhibit the mysterious 3:2 ratio

Upp

er fr

eque

ncy

[Hz]


• 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.

2. Spin from the models of the 3:2 BH QPOs

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:

RPTDWDERKRRP1RP2

MODEL : Characteristic Frequencies

Relativistic PrecessionStella et al. (1999); Morsink & Stella (1999); Stella & Vietri (2002)]

2. Models relating both of the 3:2 BH QPOs to a single radius


RPTDWDERKRRP1RP2


Cˇ adež et al. (2008), Kostic´ et al. (2009), and Germana et al.(2009)

Tidal DisruptionČadež et al. (2008), Kostič et al. (2009), Germana et al. (2009)

2. 1 Models relating both of the 3:2 BH QPOs to a single radius


RPTD

ERKRRP1RP2



Warped Disc Resonancea representative of models proposed by Kato (2000, 2001, 2004, 2005, 2008)


WD

(or torus)


RPTD

ERKRRP1RP2



Epicyclic Resonance, Keplerian Resonancetwo representatives of models proposed by Abramowicz, Kluzniak et al. (2000, 2001, 2004, 2005,…)


WD

(or torus)

RPTD

ERKRRP1RP2



Resonances between non-axisymmetric oscillation modes of a toroidal structure two representatives by Bursa (2005), Torok et al (2010) predicting frequencies close to RP model


WD

Here we focus just on a choice of few hot-spot and disc-oscillation models:(or torus)


RPTD

ERKRRP1RP2



WD


RP

ERWD, TD

Different models associate QPOs to different radii…


One can easily calculate frequency-mass functions for each of the models

Spin a


And compare the frequency-mass functions to the observation.For instance in the case of GRS 1915+105.

Spin a


Comparison with the independent spin measurements:

Considering the high spin measurement of 1915+105, some QPO models (e.g. relativistic precession) are clearly disfavoured.


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…

Toro

k et

al.

(201

1), A

&A

2.2 Discoseismic modesHere 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).

Wag

oner

et a

l. (2

001)

2.2 Discoseismic modes

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.

Toro

k et

al.

(201

1), 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.

Toro

k et

al.

(201

1), A

&A


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.

Taking into account the absolute values of QPO frequencies (and therefore the mass):

3. Spin from the models of the NS HF QPOsNS 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.

Toro

k et

al.,

(201

0),A

pJ

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. Spin from the models of the NS HF QPOs

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.

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:

Toro

k et

al.,

(201

0), A

pJ

M = 2.5….4 MSUN Ms = 2.5 MSUN M ~ Ms[1+0.75(j+j^2)]

3.1 Relativistic precession model

The best fit of 4U 1636-53 data (21 datasegments) for j = 0 is reached for Ms = 1.78

M_sun, which impliesM= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun

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.

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.

chi^2 ~ 300/20dof

chi^2 ~ 400/20dof

M= Ms[1+0.75(j+j^2)], Ms = 1.78M_sun

Best chi^2

3.1.1. Relativistic precession model vs. data of 4U 1636-53

chi^2 maps [M,j, each 10^6 points]: 4U 1636-53 data

Several models imply M-j relations having the origin analogic to the case of RP model.

3.2 Four models vs. data of 4U 1636-53

3.3 Comparison to Circinus X-1

chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data

Several models imply M-j relations having the origin analogic to the case of RP model.

- It is often believed that, e.g., RP model fits well the low-frequency sources but not the high-frequency sources.

RP m

odel

, figu

re fr

om To

rok

et a

l., (2

010)

, ApJ

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.

3.4. Quality of fits and nongeodesic corrections

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.

For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.).

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).

q/j2

j

Urb

anec

et a

l., (2

010)

, A&

A

3.5 NS mass and spin implied by the epicyclic resonance model

For a non-rotating approximation it gives NS mass about (Bursa 2004, unp.).

Mass-spin relations calculated assuming several modern EOS (of both “Nuclear” and “Strange” type) and realistic scatter from 600/900 Hz eigenfrequencies.

Urb

anec

et a

l., (2

010)

, A&

A

j

3.5 NS mass and spin implied by the epicyclic resonance model

Urb

anec

et a

l., (2

010)

, A&

A

After

Abr

. et a

l., (2

007)

, Hor

ák (2

005)

3.5 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 300-600Hz; based on X-ray bursts)

4. Summary (BHs)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.

Toro

k et

al.

(201

1), A

&A

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.

4. Summary (BHs)

Fundamental discoseismic modes vs. 3:2 frequency ratio (both QPOs):

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

We need a right model.

supported by the czech grants msm 4781305903, lc 06014 , ga Č r202/09/0772 and sgs -01-2010

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