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Vznik této prezentace byl podpořen projektem CZ.1.07/2.3.00/09.0138

Tato prezentace slouží jako vzdělávací materiál.

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.

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Mass and Spin Implications of High-Frequency QPOModels across Black Holes and Neutron Stars

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

G. Török, M. A. Abramowicz, P. Bakala, P. Čech,A. Kotrlová, Z. Stuchlík, E. Šrámková & M. Urbanec

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 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

Here we focus only to few of hot-spot or disc-oscillation models widely discussed for both classes of sources.(which we properly list and quote slightly later).

1. Data and their models: the choice of three sources

2. Near-extreme rotating black hole GRS 1915+105

a > 0.99(Remillard et al., 2009)

(McClintock & Remillard, 2003)

(Remillard et al., 2003)


a > 0.99

Relativistic precession [Stella et al. (1999)]

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

Relativistic precession [Stella et al. (1999)]

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

-1r, -2v disc-oscillation modes (frequency identification similar to the RP model)

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

Tidal disruption of large inhomogenities (mechanism similar to the RP model)Cadez et al. (2008); Kostic et al. (2009);

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

Oscillations of warped discs (implying for 3:2 frequencies the same characteristic radii as TD)Kato (1998,…, 2008)

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

3:2 non-linear disc oscillation resonancesAbramowicz & Kluzniak (2001), Török et. al (2005)

or

Abra

mow

icz e

t al.,

(201

0) in

pre

p.

Cour

tesy

of M

. Bur

sa


a > 0.99

Other non-linear disc oscillation resonancesAbramowicz & Kluzniak (2001), Török et al. (2005), Török & Stuchlík (2005)

Abra

mow

icz e

t al.,

(201

0) in

pre

p.


a > 0.99

Breathing modes

(here assuming constant angular momentum distribution)

Abra

mow

icz e

t al.,

(201

0) in

pre

p.

2. Near-extreme rotating black hole GRS 1915+105: summaryAb

ram

owic

z et a

l., (2

010)

in p

rep.

?

3. Neutron stars: high mass approximation through Kerr metric 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.

Toro

k et

al.,

(201

0) su

bmitt

ed

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.

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.

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:

M = 2.5….4 MSUN

3. Neutron stars: frequency relations implied by RP model

Toro

k et

al.,

(201

0) su

bmitt

ed

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

3. Neutron stars: RP model vs. the data of 4U 1636-53The 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.

Color-coded map of chi^2 [M,j,10^6 points] well agrees with rough estimate given by 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. Neutron stars: RP model vs. the data of 4U 1636-53

Toro

k et

al.,

(201

0) in

pre

p.

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

3. Neutron stars: other models vs. the data of 4U 1636-53For 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: models vs. the data of Circinus X-1For several models there are M-j relations having origin analogic to the case of RP model.

Modelatoll source 4U 1636-53 Z-source Circinus X-1

c2~ Mass RNS c2~ Mass RNS

rel.precessionnL= nK - nr,

nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms

tidal disruptionnL= nK + nr,

nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----

-1r, -2v reson.nL= nK - nr,

nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms

warp disc res.nL= 2(nK - nr,)nU= 2nK – nr

600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms

epic. reson.nL= nr,nU= nq

TBEL 1MSun[1+ ?? ] ~ rms x X ----

3. Neutron stars: nearly concluding table




nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----





nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----





nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----





nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----





nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----





nU= nK

300/20

1.8MSun[1+0.7(j+j2)] < rms15/10

2.2MSun[1+0.5(j+j2)] < rms


nU= nK

150/10

2.2MSun[1+0.7(j+j2)] < rms

bad1/M

X ----


nU= 2nK – nq

300/10 1.8MSun[1+(j+j2)] < rms

15/10

2.2MSun[1+0.7(j+j2)] < rms


600/20

2.5MSun[1+0.7(j+j2)] < rms15/10

1.3MSun[1+ ?? ] ~ rms


TBEL 1MSun[1+ ?? ] ~ rms x X ----


3. Neutron stars: M and j based on 3:2 epicyclic resonance model

Mass-spin inferred from epicyclic model assumingHartle-Thorne metric and 600:900Hz

Mass-spin after including several EOSand lower-eigenfrequency 580-680Hz

q/j2

jj

a)

b)

which FAILS

(Abramowicz et al., 2005)

Urbanec et al., (2010) in prep.

giving for j=0

Urb

anec

et a

l., (2

010)

in p

rep.

After

Abr

. et a

l., (2

007)

, Hor

ák (2

005)

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)

E N D

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