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Register a new userA non-linear mathematical model for the X-ray variability of the microquasar GRS 1515+105 -- III: Low-frequency Quasi Periodic Oscillations
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The X-ray emission from the microquasar GRS 1515+105 shows, together with a very complex variability on different time scales, the presence of low-frequency quasi periodic oscillations (LFQPO) at frequencies lower than 30 Hz. In this paper, we demonstrate that these oscillations can be consistently and naturally obtained as solutions of a system of two ordinary differential equations that is able to reproduce almost all variability classes of GRS 1515+105. We modified the Hindmarsh-Rose model and obtained a system with two dynamical variables x(t), y(t), where the first one represents the X-ray flux from the source, and an input function J(t), whose mean level J_0 and its time evolution is responsible of the variability class. We found that for values of J_0 around the boundary between the unstable and the stable interval, where the equilibrium points are of spiral type, one obtain an oscillating behaviour in the model light curve similar to the observed ones with a broad Lorentzian feature in the power density spectrum and, occasionally, with one or two harmonics. Rapid fluctuations of J(t), as those originating from turbulence, stabilize the low-frequency quasi periodic oscillations resulting in a slowly amplitude modulated pattern.To validate the model we compared the results with real RXTE data which resulted remarkably similar to those obtained from the mathematical model. Our results allow us to favour an intrinsic hypothesis on the origin of LFQPOs in accretion discs ultimately related to the same mechanism responsible for the spiking limit cycle.
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The complex time evolution in the X-ray light curves of the peculiar black hole binary GRS 1915+105 can be obtained as solutions of a non-linear system of ordinary differential equations derived form the Hindmarsch-Rose model and modified introducing an input function depending on time. In the first paper,assuming a constant input with a superposed white noise, we reproduced light curves of the classes rho, chi, and delta. We use this mathematical model to reproduce light curves, including some interesting details, of other eight GRS 1915+105 variability classes either considering a variable input function or with small changes of the equation parameters. On the basis of this extended model and its equilibrium states, we can arrange most of the classes in three main types: i) stable equilibrium patterns: (classes phi, chi, alpha, theta, xi, and omega) whose light curve modulation follows the same time scale of the input function, because changes occur around stable equilibrium points; ii) unstable equilibrium patterns: characterised by series of spikes (class rho) originated by a limit cycle around an unstable equilibrium point; iii) transition pattern: (classes delta, gamma, lambda, kappa and alpha), in which random changes of the input function induce transitions from stable to unstable regions originating either slow changes or spiking, and the occurrence of dips and red noise. We present a possible physical interpretation of the model based on the similarity between an equilibrium curve and literature results obtained by numerical integrations of a slim disc equations.
The microquasar GRS 1915+105 is known to exhibit a very variable X-ray emission on different time scales and patterns. We propose a system of two ordinary differential equations, adapted from the Hindmarsh-Rose model, with two dynamical variables x(t), y(t) and an input constant parameter J_0, to which we added a random white noise, whose solutions for the x(t) variable reproduce consistently the X-ray light curves of several variability classes as well as the development of low frequency Quasi-Periodic Oscillations (QPO). We show that changing only the value of J_0 the system moves from stable to unstable solutions and the resulting light curves reproduce those of the quiescent classes like phi and chi, the delta class and the spiking rho class. Moreover, we found that increasing the values of J_0 the system induces high frequency oscillations that evolve to QPO when it moves into another stable region. This system of differential equations gives then a unified view of the variability of GRS 1915+105 in term of transitions between stable and unstable states driven by a single input function J_0. We also present the results of a stability analysis of the equilibrium points and some considerations on the existence of periodic solutions.
We report the results of a systematic timing analysis of all archival Rossi X-Ray Timing Explorer (RXTE) observations of the bright black-hole binary GRS 1915+105 in order to detect high-frequency quasi-periodic oscillations (HFQPO). We produced power-density spectra in two energy bands and limited the analysis to the frequency range 30-1000 Hz. We found 51 peaks with a single trial significance larger than 3 sigma. As all but three have centroid frequencies that are distributed between 63 and 71 Hz, we consider most of them significant regardless of the number of trials involved. The average centroid frequency and FWHM are 67.3 +/- 2.0 Hz and 4.4 +/- 2.4 Hz respectively. Their fractional rms varies between 0.4% and 2% (total band detections) and between 0.5% and 3% (hard ban detections). As GRS 1915+105 shows large variability on time scales longer than 1s, we analysed the data in 16s intervals and found that the detections are limited to a specific region in the colour-colour diagram, corresponding to state B of the source, when the energy spectrum is dominated by a bright accretion disk component. However, the rms spectrum of the HFQPO is very hard and does not show a flattening up to 40 keV, where the fractional rms reaches 11%. We discuss our findings in terms of current proposed models and compare them with the results on other black-hole binaries and neutron-star binaries.
The microquasar GRS 1915+105, exhibits a large variety of characteristic states, according to its luminosity, spectral state, and variability. The most interesting one is the so-called rho-state, whose light curve shows recurrent bursts. This paper presents a model based on Fitzhugh-Nagumo equations containing two variables: x, linked to the source photon luminosity L detected by the MECS, and y related to the mean photon energy. We aim at providing a simple mathematical framework composed by non-linear differential equations useful to predict the observed light curve and the energy lags for the rho-state and possibly other classes of the source. We studied the equilibrium state and the stability conditions of this system that includes one external parameter, J, that can be considered a function of the disk accretion rate. Our work is based on observations performed with the MECS on board BeppoSAX when the source was in rho and nu mode, respectively. The evolution of the mean count rate and photon energy were derived from a study of the trajectories in the count rate - photon energy plane. Assuming J constant, we found a solution that reproduces the x profile of the rho class bursts and, unexpectedly, we found that y exhibited a time modulation similar to that of the mean energy. Moreover, assuming a slowly modulated J the solutions for x quite similar to those observed in the nu class light curves is reproduced. According these results, the outer mass accretion rate is probably responsible for the state transitions, but within the rho-class it is constant. This finding makes stronger the heuristic meaning of the non-linear model and suggests a simple relation between the variable x and y. However, how a system of dynamical equations can be derived from the complex mathematical apparatus of accretion disks remains to be furtherly explored.
We report the discovery in the Rossi X-Ray Timing Explorer data of GRS 1915+105 of a second quasi-periodic oscillation at 34 Hz, simultaneous with that observed at 68 Hz in the same observation. The data corresponded to those observations from 2003 where the 68-Hz oscillation was very strong. The significance of the detection is 4.2 sigma. These observations correspond to a very specific position in the colour-colour diagram for GRS 1915+105, corresponding to a harder spectrum compared to those where a 41 Hz oscillation was discovered. We discuss the possible implications of the new pair of frequencies comparing them with the existing theoretical models.
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