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Astron. Astrophys. 361, 781-787 (2000)

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2. Basic equations

Recently 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 R only. Within these approximations the mass continuity equation takes form

[EQUATION]

where [FORMULA] is the density of the gas, H is the vertical "half-thickness", and v is the radial velocity. Eq. (1) can be readily integrated to give mass accretion rate,

[EQUATION]

where H is defined in terms of isothermal sound speed as

[EQUATION]

where [FORMULA] is the Keplerian angular velocity. Factor [FORMULA] is chosen in the same way as NKH i.e. in order to obtain correct result for spherical flow [FORMULA].

We assume that the gravitational potential of the central black-hole is described by the pseudo-Newtonian potential (Paczynski & Wiita 1980)

[EQUATION]

where M is the mass of the black hole and [FORMULA] is the gravitational radius. This would allow us to define the Keplerian angular velocity [FORMULA] in the pseudo-Newtonian potential as

[EQUATION]

It should be noted that we use [FORMULA] in place of [FORMULA] 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

[EQUATION]

where [FORMULA] is the angular velocity of the gas. The pressure p can be written in terms of isothermal sound speed as

[EQUATION]

The steady state angular momentum equation in the presence of viscosity can be written as

[EQUATION]

where [FORMULA] is the kinematic viscosity. Here [FORMULA] is assumed to be independent of R. Using Eq. (2) one can integrate the equation for the angular momentum to obtain

[EQUATION]

The integration constant j represent angular momentum per unit mass accreted by the black hole and it is to be determined self-consistently as an eigen-value problem. However, it ought to be noted that the self-similar solution does satisfy Eq. (9) only when [FORMULA].

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 [FORMULA] times the heating rate term. The energy equation can then be written as

[EQUATION]

where [FORMULA] is the ratio of specific heats of the accreting gas. f is in general a function of R, but for an advection dominated flow [FORMULA] i.e. constant. However, [FORMULA] for the cooling dominated flow.

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 [FORMULA] in Eq. (4) and [FORMULA], where [FORMULA] is the Keplerian angular velocity in the presence of of Newtonian gravitational potential. Expressions for the radial velocity v, sound speed [FORMULA], angular velocity [FORMULA] and density [FORMULA] in this limit are given by

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where the dimensional constants [FORMULA] and [FORMULA] are defined as

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where, c is the speed of light, [FORMULA] and [FORMULA]. If we substitute for [FORMULA],[FORMULA], [FORMULA] and [FORMULA] from Eqs. (15-18) into Eqs. (11-14) and use the definition of [FORMULA] one can obtain the self-similar solution of Narayan & Yi (1994).

In order to study how the effects of non-Newtonian potential can affect the self-similar solution in the region far from [FORMULA], we introduce perturbation in the hydrodynamical quantities as [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Here, the `background' quantities [FORMULA] and [FORMULA] describe the self-similar solutions in Eqs. (11-14) and which can satisfy the height integrated hydrodynamical Eqs. (1-10) when [FORMULA] is negligible. The quantities like [FORMULA] 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 [FORMULA], one can write

[EQUATION]

where [FORMULA]. And [FORMULA] can be written from Eq. (19) as

[EQUATION]

In writing Eqs. (19-20) we have neglected the terms of the order [FORMULA] and higher. From the above we have [FORMULA] and [FORMULA].

We linearize the Eqs. (2, 3, 6, 7, 9-10) and retain the terms which are first order in perturbed quantities and [FORMULA],

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

It should be noted that in writing Eq. (25) we have assumed [FORMULA] 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 R for the perturbation equations. It can also be noticed that in order to satisfy Eq. (23) all the terms containing perturbed quantities must have radial dependence like [FORMULA]. This can dictate following choice for the perturbations:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] 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):

[EQUATION]

Thus all the perturbed quantities are well behaved functions of radial coordinate R over the entire region of interest i.e. [FORMULA]. However, one cannot still ascertain that the perturbations are small over the above range of R. This is because of the fact that quantities like [FORMULA], [FORMULA] etc. are functions of [FORMULA], [FORMULA], [FORMULA] and they can, in general, have values larger than unity. One can regard the perturbation theory to be valid till the magnitude of the perturbed quantity is less than that of the corresponding background quantity. From this one can have a criterion for the validity of the perturbation as [FORMULA]. Here [FORMULA] is the radius beyond which the perturbation theory is valid. Thus for [FORMULA] perturbation of the radial velocity becomes larger than the background and therefore perturbed solution of v should be discarded. It should be noted that each quantity has different region of validity for the perturbation.

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:

[EQUATION]

[EQUATION]

[EQUATION]

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

[EQUATION]

where, [FORMULA]. Perturbation coefficient of all other quantities can be written in terms of [FORMULA] as

[EQUATION]

[EQUATION]

The coefficient of density perturbation can be found, using Eqs. (21-22), as

[EQUATION]

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Online publication: October 2, 2000
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