gabriel török, p.bakala, e. Šrámková, z. stuchlík, m. urbanec mass and spin of ns implied by...
TRANSCRIPT
![Page 1: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/1.jpg)
Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec
Mass and spin of NS implied by models of kHz QPOs
*Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic
The presentation refers to work in progress. It also drawsfrom a collaboration with
D. Barret, J. Miller, J. Horák
![Page 2: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/2.jpg)
1. Data and their models: LMXBs, accretion discs, variability
• density comparable to the Sun• mass in units of solar masses• temperature ~ roughly as the T Sun• moreless 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 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
![Page 3: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/3.jpg)
1.2 Data and their models: pairs of kHz 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)
• kHz QPOs (~200-1500Hz): Lower and upper QPO mode forming twin peak QPOs
frequency
pow
er
Sco X-1
kHz QPO origin remains questionable, it is often expected that they are associated to the orbital motion in the inner part of the disc.
Individual peaks can be related to a set of oscillators as well as to a time evolution of an oscillator.
![Page 4: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/4.jpg)
1.3 Data and their models: frequency relations between kHz QPOs
The two QPO frequencies seems to be well correlated, following a nearly linear relation specific for a given source.
![Page 5: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/5.jpg)
1. Data and their models: orbital models of kHz QPOs
Several models have been proposed. Most of them relate QPOs to the orbital motion in inner parts of accretions disc. For instance,
Relativistic precession model, Stella, Vietri, 1999, relates the kHz QPOs to the frequencies of geodesic motion.
Some models relate the kHz QPOs to resonance between disc oscillation modes given by the frequencies of geodesic motion (Kluzniak, Abramowicz, 2001).
On next few slides we focuse on frequency identification given by relativistic precession model,
(Note that, in Schwarzschild spacetime, this identification correspond to m= -1 radial and m= -2 vertical disc oscillation modes as well.)
![Page 6: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/6.jpg)
*For simplicity we consider Kerr spacetimes on few slides (while finaly we apply a more realistic approach needed for rotating neutron stars).
Solving above equations one obtains frequency relations U(L) which can be compared to those observed.
*
2. Relativistic precession model
![Page 7: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/7.jpg)
2.1 Frequency relations given by the relativistic precession model
M=1.4M_sun, j=0
M=2M_sun, j=0
M=1.4M_sun, j=0.3
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.
![Page 8: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/8.jpg)
Uniqueness of the curves in frequency plot: Obviously, if there would be two different combinations of M and j implying from the RP model the same curve these combinations must imply also the same ISCO frequency.
ISCO frequency is implicitly given by formulae determining the orbital frequency and the ISCO radius rms ,
Solving these numerically one can find combinations M, j giving the same ISCO frequency and plot related curves.
2.1 Frequency relations given by the relativistic precession model
![Page 9: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/9.jpg)
For a given mass MS of the non-rotating neutron star there is a set of similar curves given, within some approximation, by the relation
M ~ MS[1+0.75(j+j^2)].
2.1 Frequency relations given by the relativistic precession 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 the curves are very similar.
M = 2.5….4 MSUN
![Page 10: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/10.jpg)
It was previously noticed that the RP model fits the data qualitatively well but often with non-negligible residuals (which arise especially on the top part of the correlation). It is often quoted that the model implies a high angular momentum (j>0.25) for which the residuals are somewhat lower (but still significant).
Here we suggests that a fit for the non-rotating neutron star with only free parameter Ms implies a rough mass-angular-momentum relation
M ~ MS[1+0.75(j+j^2)].
related to a “family of best fits” giving comparable chi^2.
We investigate this suggestion for the source 4U 1636-53.
2.2 Fitting the data
![Page 11: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/11.jpg)
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
2.2 Fitting the data
![Page 12: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/12.jpg)
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
2.2 Fitting the data
![Page 13: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/13.jpg)
- spin frequency of the source expected from x-ray bursts: either 290 or 580 Hz
- Hartle-Thorne spacetimes, SKYRME EOS
- RNS, LORENE…
2.3 Realistic configurations
EOS 580Hz
EOS 290Hz
![Page 14: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/14.jpg)
3. Other models, other sources
We checked RP model for several other sources, the relation
M = Ms[1+k(j+j^2)]
With k = 0.75 well indicates the best chi M-j region in any high frequency source. For low-frequency sources the best fits are obtained for somewhat lower values of k=0.5 (Circinus X-1).
We also checked four other orbital models (listed later), for these there are also similar mass-angular momentum relations, in general it is
M = Ms[1+k(j+j^2)], k= 0.5..1
![Page 15: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/15.jpg)
3. Other models, other sources
chi^2 maps [M,j, each 10^6 points]: 4U 1636-53 data
![Page 16: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/16.jpg)
3. Other models, other sources
chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data
![Page 17: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/17.jpg)
![Page 18: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/18.jpg)
6. Non-geodesic corrections ?
- It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources
![Page 19: Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec Mass and spin of NS implied by models of kHz QPOs *Institute of Physics, Faculty of Philosophy](https://reader035.vdocuments.us/reader035/viewer/2022070413/5697bfd31a28abf838cac5d0/html5/thumbnails/19.jpg)
6. Non-geodesic corrections ?
- It is often believed that, e.g., RP model fits well low-frequency sources but not high-frequency sources
Circinus X-1 data 4U 1636-53 X-1 data