## 2. Basic equationsRecently it was demonstrated by Narayan & Yi (1995a) that the
steady state self-similar regime of advection dominated accretion
flows operates in the region far off the boundaries. Therefore
height-integrated set of fluid equations can well describe the
accretion flow at least in the self-similar regime. Later it was shown
by NKH that correction introduced by the non-sphericity remains under
control even when the effect of boundary near the sonic point is
considered. We consider height-integrated set of equations describing
steady state axisymmetric equations (see NKH and references cited
therein). In steady state and axisymmetry, the height-integrated
hydrodynamical equations allow one to describe all the physical
quantities as function of the cylindrical radius where is the density of the gas,
where where is the Keplerian angular velocity. Factor is chosen in the same way as NKH i.e. in order to obtain correct result for spherical flow . We assume that the gravitational potential of the central black-hole is described by the pseudo-Newtonian potential (Paczynski & Wiita 1980) where It should be noted that we use in place of in Eq. (3) in what follows as in the earlier work by Chen et al. (1997) and NKH. Radial momentum equation of the accreting gas is given by where is the angular velocity of
the gas. The pressure The steady state angular momentum equation in the presence of viscosity can be written as where is the kinematic viscosity.
Here is assumed to be independent of
The integration constant Finally we consider the energy balance between the local viscous heating and local radiative cooling giving rise to overall heat transport (advection). We write this as an entropy equation. The entropy rate is determined by the local viscous heating rate minus local cooling rate. However, in the case when advection dominates the cooling rate is negligible compared to the viscous heating rate. But for generality we retain the cooling term and write it as factor times the heating rate term. The energy equation can then be written as where is the ratio of specific
heats of the accreting gas. One of the aims of this >is to incorporate the effect of
pseudo-Newtonian gravitational potential perturbatively and thereby
increase the accuracy of the self-similar solution. Therefore, we
first briefly review the self-similar solutions (Narayan & Yi
1994). Self-similar solution of Eqs. (1-10) was obtained by
setting in Eq. (4) and
, where
is the Keplerian angular velocity in
the presence of of Newtonian gravitational potential. Expressions for
the radial velocity where the dimensional constants and are defined as where, In order to study how the effects of non-Newtonian potential can affect the self-similar solution in the region far from , we introduce perturbation in the hydrodynamical quantities as , , and . Here, the `background' quantities and describe the self-similar solutions in Eqs. (11-14) and which can satisfy the height integrated hydrodynamical Eqs. (1-10) when is negligible. The quantities like represent the perturbation introduced by the pseudo-Newtonian potential. Next, we consider the perturbations in Keplerian velocity due to pseudo- Newtonian gravitational potential. From Eq. (5), when , one can write where . And can be written from Eq. (19) as In writing Eqs. (19-20) we have neglected the terms of the order and higher. From the above we have and . We linearize the Eqs. (2, 3, 6, 7, 9-10) and retain the terms which are first order in perturbed quantities and , It should be noted that in writing Eq. (25) we have assumed
as in the background self-similar
solution of Narayan & Yi (1994). Eqs. (21-26) have
inhomogeneous terms introduced by the perturbations in the
gravitational potential. In the spirit of the self-similar solution,
we look for the solution in powers of where and are unknown dimensional constants and have dimensions of velocity, angular velocity and density as suggested from their standard notations. These constants are yet to be determined (below). It can be seen from Eqs. (11-14, 27-30): Thus all the perturbed quantities are well behaved functions of
radial coordinate If we substitute for all the perturbed quantities in Eqs. (21, 23, 25-26) from Eqs. (27-30) one can check in a straightforward algebra that the radial dependence can be factored out if the explicit form of the self-similar solution is substituted. We can write equations for the unknown coefficients of the perturbed quantities as: In order to determine the unknown coefficients of the perturbations we need to solve the set of simultaneous Eqs. (31-33). The solution can be found in a rather straight forward way and it can be written as where, . Perturbation coefficient of all other quantities can be written in terms of as The coefficient of density perturbation can be found, using Eqs. (21-22), as © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |