As suggested by the recent studies a class of accretion disk models invoking advection have been successful in explaining low luminosity accreting black hole systems (see Narayan et al. 1998 for a review). In these models viscously generated internal energy is not radiated out immediately - as in the standard thin disk model - but stored in the accreting gas. The stored energy might eventually be lost into the black hole or a considerable portion of it might give rise to wind (Blandford & Begelman 1999). This kind of accretion flows, when there is no out flow, are known as advection dominated accretion flows(ADAFs). By definition ADAFs have very low radiative efficiency and as a consequence they can be considerably hotter than the gas flow in the standard disk models (Narayan & Yi 1995a,b).
ADAFs can occur in two distinct physical regimes. Firstly they can occur when the accreting gas density and consequently its optical depth become higher due to very high mass accretion rates (in super- Eddington limit) (Begelman 1978; Begelman & Meier 1982). In this limit radiation can be trapped in the in-falling gas. Accretion flow in this case was found to be stable against thermal and viscous instabilities (Abramowicz et al. 1988, Kato et al. 1996). Secondly, ADAFs can occur when the infalling gas has low density and low optical depth when the mass accretion rates become very small (Ichimaru 1977, Rees et al. 1982, Narayan & Yi 1994, 1995a,b, Abramowicz et al. 1995, Chen 1995). In this limit radiative time scale becomes lower than the accretion time scale and as a result almost all the internal energy can be lost to the black hole.
Models of optically thin branch of ADAFs comprising two-temperature plasma have recently found many promising applications in explaining various low luminosity accreting black hole systems. Advection models for Sagittarius A* at the center of our galaxy (Rees et al. 1982, Narayan et al. 1995), and for NGC 4258 (Lasota et al. 1996) have been proposed. It has also been suggested that the center of many elliptical galaxies might be accreting in advection dominated mode (Fabian & Rees 1995, Mahadevan 1997). Moreover, ADAF models have been applied to X-ray transients A0620 -00, V404 Cyg and Nova Muscae 1991 in their quiescent states (Narayan et al. 1996). It must be noted that all the above models were based on local self-similar solution of ADAFs (Narayan & Yi 1994, Spruit et al. 1987).
The local self-similar solutions are generally assumed to be valid in the region away from the boundaries. But in their application in spectral calculations their validity is assumed right up to the last marginally stable orbit. It ought to be noted that the self-similar solution does not have a trans-sonic region. However, in realistic situation the radial flow velocity could approach that of light in the vicinity of the black-hole. Due to inefficient radiative processes sound velocity can also increases but the radial velocity can over take it. In other words on might expect trans-sonic region and sonic point in the accretion flows. Keeping this limitation in mind, perhaps, study of global solution of optically thin branch of the ADAFs had been started. First the global solutions were found for a pseudo-Newtonian potential (Narayan et al. 1997 (NKH), Chen et al. 1997) and later for a full Kerr metric (Abramowicz et al. 1996, Peitz & Appl 1997, Gammie & Popham 1998). In the above, exact numerical studies, it was found that the self-similar solutions are good approximations in the region out side the sonic point.
It ought to be noted that the self-similar solution to optically thin branch of ADAFs (Narayan & Yi 1994, Spruit et al. 1987) was found using the Newtonian potential. But the symmetry leading to self-similarity can be destroyed if any non-Newtonian form of the potential is used. The self-similarity can also be destroyed if the specific angular momentum per unit mass accreted by the black hole j is comparable or larger than . Here, is azimuthal velocity of the accreting gas and R is the radial coordinate. it In what follows we assume . Exact numerical solutions of Gammie & Popham 1998 shows that this assumption is justified. However for the case of thin accretion disk this can not be ascertained. We check this assumption for the case of approximate solution in Appendix -A.
Taking the above facts into consideration, we adopt an alternative approach to study the validity of the self similar solution. We regard the self-similar solution as a `background' and introduce effect of pseudo-Newtonian potential in a perturbative fashion. At the distances very far from the central object the perturbations due to the non-Newtonian potential can be extremely small, but as one moves closer to the center they can grow. Of course, with this technique one can not go arbitrarily close to the central object, as the underlying assumption of the perturbation theory may break down. However, if the perturbations can grow and become comparable to the background much before the event horizon is reached, then the point where the perturbation breaks down can be regarded as the point up to which the self-similar solution has its validity. Therefore, this approach can give us a more accurate criterion for the validity of the self similar solution. This method can also provide an appropriate parameter space where the assumption of self-similarity is valid. These features are difficult to be seen in the earlier exact but numerical approach to the global solutions. Therefore, we believe that the present work should be considered to be complimentary to the earlier work on the global solutions.
In Sect. 2 we write the basic set of differential equations for the perturbation theory. Sect. 3 discusses self-similar solutions and the results from perturbation theory. Finally we compare our results with the exact numerical global solutions obtained by NKH. Also the relevance of the perturbation theory results to the earlier work is discussed.
© European Southern Observatory (ESO) 2000
Online publication: October 2, 2000