The study of the spectral and temporal evolution of GRS 1915+105 during the periods of flaring activity has shown that bursting behavior of the source can be reduced to a sequence of varying hard and soft episodes. The hard episodes are distinguished by the presence of the prominent QPO peak in the source power density spectrum. Combined time-resolved spectral and timing analyses revealed (i) a strong correlation between the QPO frequency and parameters of the soft component of the energy spectrum (presumably emitted by the optically thick part of the disk) on a wide range of time scales (Trudolyubov et al. 1999a; Markwardt et al. 1999; Muno et al. 1999); (ii) the dependence of the duration of the hard episode upon the corresponding characteristic radius of the optically thick disk (derived through the spectral fitting with multicolor disk black body model) is similar to that expected if the duration of the hard episode is proportional to the viscous time scale of the radiation pressure dominated disk (Belloni et al. 1997b). Given (i) and (ii) some sort of correlation between the QPO frequency and the duration of the hard episode is naturally expected.
To explain the observational properties of black hole and neutron star binaries in the hard state a number of models involving the hot comptonization region near the compact object surrounded by the optically thick accretion disk was proposed. It is often assumed that the QPO phenomenon is caused by interaction between these two distinct parts of the accretion flow occurring on the local dynamical time scale at the boundary of these regions (Molteni et al. 1996; Titarchuk, Lapidus & Muslimov 1998). In the following analysis we will also assume that the QPO frequency is proportional to the Keplerian frequency at the inner boundary of the accretion disk.
In the standard accretion disk theory (Shakura & Sunyaev 1973) the viscous time of the radiation pressure dominated disk is , where - viscosity parameter, m - mass of the compact object in solar masses, - disk accretion rate in units of critical Eddington rate, r - distance to the compact object in units of 3 gravitational radii. Introducing the local Keplerian frequency, Hz, we obtain:
This dependence, , reproduces well the relation between the duration of a hard episode and the associated minimal QPO frequency for the first group of observations (Fig. 2, left panel). Moreover, for several observations from this group the observed relation between the duration of a hard episode and the maximal value of the characteristic inner radius of the disk, determined via spectral fitting, matches remarkably well the expected dependence of the viscous time scale on the radius for the radiation pressure dominated disk (Belloni et al. 1997b).
Assuming that the QPO frequency is proportional to the Keplerian frequency () and duration of the hard episode is proportional to the viscous time (), one can estimate the required mass accretion rate using Eq. (1):
In Eq. (2) compact object mass was normalized to the value of 33 , implied by the interpretation of the 67 Hz QPO feature as a signature of a Keplerian oscillations near the last marginally stable orbit around a Schwarzschild black hole (Morgan et al. 1997). The inferred values of the accretion rate for the observations from the first group are in the range (Fig. 2). This conclusion is supported by the results of the spectral analysis (Trudolyubov et al. 1999b) where the bolometric luminosity of the soft spectral component corresponding to a given value of QPO frequency was used as a measure of the mass accretion rate.
If the QPO frequency is associated with the selected region in the accretion disk (e.g. inner edge of the optically thick disk) which is moving radially on the viscous time scale, then we expect correlation of the QPO frequency and the rate of the frequency change. Assuming that rise of the QPO frequency is caused by the inward motion of the inner edge of the radiation pressure dominated accretion disk one can write:
where - radial velocity of the matter in the disk at radius r (Shakura & Sunyaev 1973). Integrating Eq. (3), we obtain:
where . In Fig. 3 this dependence is shown in comparison with the observed QPO evolution for two observations from the first (18/06/1997) and the second (25/10/1996) groups. For the first group both initial decrease and following rise of the QPO frequency during the hard episodes are generally described by Eq. (4) (Fig. 3, upper panel) but with different values of coefficient A. For the rise phase the value of coefficient A remains practically the same on a time scale of an individual observation (). For the second group of observations only rise of the QPO frequency is generally described by Eq. (4), while the initial decay often has a more complicated structure (Fig. 3, lower panel). In addition, an extended plateau in the time dependence of the QPO frequency near its minimum is sometimes present. This might explain the deviation of the dependence of on the QPO frequency from the law for the second group of observations.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999