American Astronomical Society, High-Energy AstrophysicsDivison, 14th Divisional Meeting (Chicago, IL), 17–21 August 2014 Poster #122.03, ‘XRBs and Population Surveys’

E-mail: vladimir.karas@cuni.cz

ConclusionsThe signatures of spot motion can be revealed by the harmoniccontent of the observable signal.We have studied the comparison between spot and torus scenarios serving as fiducial represen-tations of two very specific kinematic models. Obviously, any general validity of this discussionis limited. For instance, a consideration of resonance driven effects or the role of torus geomet-rical thickness could also give rise to some harmonic content in the signal from the oscillatingtori. Despite these uncertainties, the elaborated comparison indicates clearly that the increasedsensitivity of the proposed LOFT mission can be crucial for resolving the nature of QPOs instrong gravity.

See Bakala et al. (2014) and Karas et al. (2014) for additional details.

References

Bakala P., Torok G., Karas V., Dovciak M., Wildner M., Wzientek D., Sramkova E., Abramowicz M., Golu-chova K., Mazur G. P., & Vincent F. H. 2014, MNRAS, 439, 1933-1939 (arXiv:1401.4468)

Bursa, M., Abramowicz, M. A., Karas, V., & Kluzniak, W. 2004,ApJ, 617, L45

Karas V. 1999a, ApJ, 526, 953

Karas V. 1999b, PASJ, 51, 317

Karas V., Bakala P., Torok G., Dovciak M., Wildner M., Wzientek D., Sramkova E., Abramowicz M., Golu-chova K., Mazur G. P., & Vincent F. H. 2014, Acta Polytechnica, 54,No. 3, 191-196 (arXiv:1402.5353)

Pechacek, T., Goosmann, R. W., Karas, V., Czerny, B., & Dovciak, M. 2013, A&A, 556, id. A77

Pechacek, T., Karas, V., & Czerny, B. 2008, A&A, 487, 815

Stella, L., & Vietri, M. 1999, Physical Review Letters, 82, 17

Torok, G., Abramowicz, M. A., Kluzniak, W., & Stuchlık, Z. 2005, A&A, 436, 1

Witzel, G., Eckart, A., Bremer, M., Zamaninasab, M., Shahzamanian, B., Valencia-S., M., Schodel, R., Karas, V.,Lenzen, R., Marchili, N., Sabha, N., Garcıa-Marın, M., Buchholz, R. M., Kunneriath, D., & Straubmeier, C.2012, ApJ, 203, id. 18

Fig. 5: Expected flux measured by a distant observer for different inclination angles. Left: Amplitude spectrum. Right: Time dependent energy spectra drawn for the distant observer.

Fig. 4: The same as in Fig. 3 but for i = 30. We note that RXTE PDS includes a barely significant excess of power at 160Hz only for the highest displayed

value of n. For the same value, the LOFT PDS is ex[ected to reveal the first two harmonics and also the radial epicyclic frequency (53Hz in this particularexample). For large inclination angles, the LOFT observations could easily reveal the Keplerian frequency of the spot together with its first and secondharmonics when the strongest (yet weak) signal is around the limits of RXTE detectability.

Fig. 3: The model PDS obtained for different levels of the signal fraction n assuming inclination i = 80 with eccentric orbits of spots in the disk plane. Left: A

simulation considering the RXTE response. Right: Outputs considering the assumed LOFT capabilities. One should note that the lowest displayed values ofn corresponding to gray and yellow colours do not indicate any significant features within the RXTE PDS. On the other hand, the LOFT PDS already revealKeplerian frequency (gray and yellow) respectively its first two harmonics (yellow).

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Results: PDS predictions from the spot vs. the torus modelsIn our model we assume a spot orbiting near above ISCO (the innermost stable circular orbit), with constant angular velocity ofthe Keplerian valueΩK. The spot trajectory can deviate (slightly) from the circular shape due to the radial (epicyclic) oscillation,having a small amplitudeE > 0.

Figure 3 shows the PDS resulting from the RXTE and LOFT simulations assuming various levels ofn and the inclinationi = 80

(i.e., nearly equatorial view). Because the signal from spots depends strongly on the source inclination, inFigure 4 we show theresults also for the view close to the vertical axis,i = 30. Different levels of signal-to-noise fraction (n) are explored, as detailedin Bakala et al. (2014) and Karas et al. (2014).

In Figure 5 (upper panels) we include amplitude spectra and time dependent energy spectra of the net spot signal calculated forthe distant observer ati = 80. The spot signal is dominated by the Keplerian frequency andits harmonics amplified by relativisticeffects, which is well illustrated by the amplitude spectrum on the upper left panel of Fig. 5. The eccentricity corresponding to theamplitude of the radial epicyclic oscillationE = 0.1M causes only negligible modulation at the radial and precession frequencies.

An increased eccentricity corresponding to the amplitude of the radial epicyclic oscillation (E = 1M ) can be well recognized inthe amplitude spectra, but the signal is still dominated by the Keplerian frequency and its harmonics. The time dependent energyspectra of the spot are depicted in the upper right panel; thesignatures of relativistic redshift effects are clearly revealed.

Likewise, to define the torus kinematics, we assume the radial and vertical oscillations withm = 0 and comparable intrinsicamplitudes. This provides us with a well-defined scheme for which predictions can be derived.

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Fig. 2: Comparing multiple-spot and oscillating-torus PDS obtained for the two instruments. Superimposed red curves indicate various multi-Lorentzian models. Wecan find that the RXTE PDS obtained for the given setup of the two models are rather undistinguishable from each other. On the other hand, the LOFT PDS reveals thepresence/absence of the harmonics additional to the 3:2 peaks, representing the signature of spot motion. Further details: Bakala et al. (2014); Karas et al. (2014).

Fig. 1: Left: Sketch of the QPO model based on spots orbiting close to two preferred radii, producingthe 3:2 ratio of observed frequencies (Karas, 1999b). Right: Illustration of the role of lensing and Dopplereffects in the visual appearance of a torus located near the resonant orbit r3:2 (Bakala et al., 2014).

Spotsdrdr

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The contribution of the two individualoscillations to the variations of the ob-served flux strongly depends on the in-clination angle. Here we seti = 65,where the fractions of the power in thetwo observed peaks are comparable. Weset the black hole massM = 5.65M⊙

and a = 0 (r3:2 = 10.8M), implyingthat the two oscillatory frequencies areνθ(r3:2) = 160Hz andνr(r3:2)

.= 110Hz.

The resulting PDS drawn for the signalfractionn = 10% is shown inFigure 2.

IntroductionMathematical properties of Fourier power spectra, as predicted by the hot-spot scenario in the context of HFQPOs (Stella &Vietri, 1999), have been studied in detail by many authors (Pechacek & Karas, 2008, 2013). In Karas (1999a,b), examples of aquantitative study of the expected PDS were presented for a particular case of orbital motion of the spots; their distribution withina narrow range of radii was assumed and explored. The adoptedphenomenological description of the source employs spots orclumps orbiting in the Schwarzschild and Kerr metrics as an approximation to more realistic models of inhomogeneous disc–type accretion flows around black holes and neutron stars. Toreproduce the quality factor of the observed QPOs, it was foundnecessary that the spots must be distributed in a zone of onlyseveral gravitational radii from the central black hole, with theirobserved luminosity influenced by Doppler and lensing effects. Furthermore, we found that the expected values of the qualityfactor of the oscillations can reach about several hundred (typically, Q ≈ 3 × 102). However, the assumption of strictly circularorbits was clearly a simplification which calls for further discussion in the present paper. Furthermore, the effect of backgroundnoise has to be taken properly into account (Witzel et al., 2012).

Set-up of the modelThe left panel ofFigure 1 illustrates the spot scenario which we investigate. The radial distribution and drifting of the spots canclearly result in various levels of signal coherence. Nevertheless, small circular spots related to a single preferredradius do notreproduce well the often observed 3:2 frequency ratio. We also consider a more elaborate scheme where the multiple spotsarecreated and drifted around radii close to two preferred orbits with Keplerian frequencies roughly in the 3:2 ratio. We compare thePDS obtained for this setup to the PDS resulting from the model of an oscillating, optically thin torus, slowly drifting throughthe resonant radiusr3:2. To define the torus kinematics, we assume them = 0 radial and vertical oscillations with equal intrinsicamplitudes. The possible QPO origin in the resonances between this and similar disc oscillation modes has been extensivelydiscussed; here we adopt the concept previously investigated by Bursa et al. (2004) and Torok et al. (2005), who focused onoptically thin tori in slender geometry. The visual appearance of a torus influenced by lensing and Doppler effects is illustrated inthe right panel of Fig. 1. Within the adopted concept the periodic changes of the observed luminosity are partially governed by theradial oscillations due to changes of the torus volume, while the vertical oscillations modulate the flux just due to lensing effectsin the strong gravitational field.

Different theoretical schemes have been proposed to explain the origin of high-frequency (kilohertz) quasi-periodicoscillations (HFQPOs) from accreting neutron stars in low-mass X-ray binaries and stellar-massaccreting black-holes. In the case of twin-peak sources, Fourier power-spectral density exhibits two dominant oscillation modes, often in the approximate ratio of small integers (in particular, 3:2). Despite therich phenomenology, base frequencies alone do not allow us to distinguish in a unique way among the most popular models. We discuss the harmonic content predicted by two competing scenarios, namely, theorbiting spot model and the oscillating torus model. By employing a ray-tracing code, we study the relativistic regime where the emerging radiation signal is influenced by effects of strong gravity (energy shiftsand light bending). To produce quantitative predictions, we have to assume specific (simplified) models which, at the same time, are able to reproduce the basic facts from observations.

We consider spots moving on slightly non-circular trajectories in an accretion disk, and tori oscillating with fundamental modes. The harmonic content of the observed signal can allow us to reveal the ellipticityof the orbits and discriminate between the scheme of orbiting spots and the case of an oscillating torus. On a practical side, we estimate the required signal-to-noise ratio of the model light curve and we discusswhat improvement would be needed in comparison with RXTE, depending on the source brightness. We find that when a weak signal corresponding to the hotspot Keplerian frequency is around the limits of theRXTE detectability, the LOFT observations can clearly reveal its first and second harmonics.

Vladimır KarasAstronomical Institute, Academy of Sciences, Prague, Czech Republic

Pavel Bakala, Gabriel Torok, Martin Wildner, & Kate rina GoluchovaInstitute of Physics, Faculty of Philosophy and Science, Silesian University, Opava, Czech Republic

Harmonic Content of High-Frequency QPOsfrom the Relativistic Orbiting-Spot vs. Oscillating-Torus Models