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Register a new userA non-linear mathematical model for the X-ray variability classes of the microquasar GRS 1915+105 -- I: quiescent, spiking states and QPOs
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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.
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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 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.
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 on the X-ray spectral behavior within the steady states of GRS 1915+105. Our work is based on the full data set on the source obtained using the Proportional Counter Array on the Rossi X-ray Timing Explorer and 15 GHz radio data obtained using the Ryle Telescope. The steady observations within the X-ray data set naturally separated into two regions in the color-color diagram and we refer to them as steady-soft and steady-hard. GRS 1915+105 displays significant curvature in the coronal component in both the soft and hard data within the {it RXTE}/PCA bandpass. A majority of the steady-soft observations displays a roughly constant inner disk radius (R_in), while the steady-hard observations display an evolving disk truncation which is correlated to the mass accretion rate through the disk. The disk flux and coronal flux are strongly correlated in steady-hard observations and very weakly correlated in the steady-soft observations. Within the steady-hard observations we observe two particular circumstances when there are correlations between the coronal X-ray flux and the radio flux with log slopes eta~0.68 +/- 0.35 and eta ~ 1.12 +/- 0.13. They are consistent with the upper and lower tracks of Gallo et al. (2012), respectively. A comparison of model parameters to the state definitions show that almost all steady-soft observations match the criteria of either thermal or steep power law state, while a large portion of the steady-hard observations match the hard state criteria when the disk fraction constraint is neglected.
We report detailed, long term near-infrared (NIR) light curves of GRS 1915+105 in 2007-2008, covering its long soft state for the first time. From our NIR monitoring and the X-ray data of the All Sky Monitor (ASM) onboard Rossi X-ray Timing Explorer (RXTE), we discovered that the NIR flux dropped by > 1 mag during short X-ray flares with a time-scale of days. With the termination of the soft state, the H-Ks color reddened and the anti-correlation pattern was broken. The observed H-Ks color variation suggests that the dominant NIR source was an accretion disk during the soft state. The short X-ray flares during the soft state were associated with spectral hardening in X-rays and increasing radio emission indicating jet ejection. The temporal NIR fading during the X-ray flares, hence, implies a sudden decrease of the contribution of the accretion disk when the jet is ejected.
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