Transcript M - Physics.cz

Mass and spin of NS implied by models of kHz QPOs
Gabriel Török, P.Bakala, E. Šrámková, Z. Stuchlík, M. Urbanec
*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 draws
from a collaboration with
D. Barret, J. Miller, J. Horák
1. Data and their models: LMXBs, accretion discs, variability
Artists view of LMXBs
“as seen from a hypothetical planet”
Compact object:
- black hole or neutron star (>10^10gcm^3)
LMXB Accretion T ~ 10^6K
Companion:
• density comparable to the Sun
• mass in units of solar masses
• temperature ~ roughly as the T Sun
• moreless optical wavelengths
disc
>90% of radiation
in X-ray
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).
Figs: space-art, nasa.gov
1.2 Data and their models: pairs of kHz 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 a time
evolution of an oscillator.
power
Sco X-1
• Low frequency QPOs (up to 100Hz)
• hecto-hertz QPOs (100-200Hz)
• kHz QPOs (~200-1500Hz):
Lower and upper QPO mode
forming twin peak QPOs
Fig: nasa.gov
frequency
kHz QPO origin remains questionable,
it is often expected that they are
associated to the orbital motion in the
inner part of the disc.
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.
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.)
2. Relativistic precession model
*
*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 nU(nL) which can be
compared to those observed.
2.1 Frequency relations given by the relativistic precession model
M=1.4M_sun, j=0.3
M=1.4M_sun, j=0
M=2M_sun, j=0
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.
2.1 Frequency relations given by the relativistic precession model
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
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
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)].
2.2 Fitting the data
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
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
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
Best chi^2
chi^2 ~ 300/20dof
chi^2 ~ 400/20dof
2.3 Realistic configurations
- spin frequency of the source expected from x-ray bursts: either 290 or 580 Hz
- Hartle-Thorne spacetimes, SKYRME EOS
- RNS, LORENE…
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
3. Other models, other sources
chi^2 maps [M,j, each 10^6 points]: 4U 1636-53 data
3. Other models, other sources
chi^2 maps [M,j, each 10^6 points]: Circinus X-1 data
6. Non-geodesic corrections ?
- It is often believed that, e.g., RP model fits well low-frequency sources but not
high-frequency sources
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