Non-linear oscillator models for the X-ray bursting of the microquasar GRS 1915+105

Abstract

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 ρ-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 emission rate 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 ρ-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 ρ and ν 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 ρ 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 ν class light curves is reproduced. According these results, the outer mass accretion rate is probably responsible for the state transitions, but within the ρ-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. © 2014 Springer Science+Business Media Dordrecht

Lyapunov functions for a non-linear model of the X-ray bursting of the microquasar GRS 1915+105

“…These equations, however, are not the unique for this goal: for instance, we verified that the use the exponent 5 instead of 3 in the non linear dissipation term gives also similar solutions. Moreover, as we showed in [11], the introduction of a slowly time variable J(t), that requires at least an additional equation, makes possible to obtain solutions very similar to other variablity classes. The system (1) should therefore be considered as one of the simplest tool for mathematical description of the ρ class bursting, whose properties depends upon the parameter J.…”

“…On this basis Massaro et al [11] were able to describe this cycle by means of a modified FitzHugh-Nagumo (FitzHugh [12], Nagumo et al [13]) model and found that it reproduces the mean burst shape and some other observational properties of the ρ class. In the same paper [11] we also demonstrated that a limit cycle can exist and gave the conditions for the parameters to have this solution. It is given by the following system in the two dimensionless variables x(t) and y(t):…”

Time properties of the theρ-class burst of the microquasar GRS 1915+105 observed withBeppoSAX in April 1999

“…However, a dynamical approach to reproduce the observed general structures has not been developed so far. In a first attempt, Massaro et al (2014) applied the well-known FitzHugh-Nagumo dynamical equations in two non-dimensional variables x and y, which was found to be proportional to the photon flux and disk temperature, respectively. Their time evolution is given by two differential equations, one of which contains a "cooling" term in x 3 , which is necessary to achieve an unstable equilibrium point around which a limit cycle can be established: dx/dt = −ρx 3 + χx − γy − J dy/dt = x − y.…”

“…These equations have four parameters, one of which is a forcing J that Massaro et al (2014) proposed to be related to the mass accretion rate in the disk. They demonstrated that changes A56, page 4 of 5 of its value affect both the recurrence time of the bursts and the mean level of intensity.…”

“…6 for the two data series P3b (black curves) and P16 (red curves), which correspond to points in the decreasing and increasing branch that have similar BL rates but average T rec equal to 46.0 s and 51.9 s, respectively. We computed these mean curves by means of the algorithm applied to the CR-E trajectories; this is explained in detail in Massaro et al (2014). The definition of HXD used in our works (see Paper III) is graphically explained in Fig.…”