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Astron. Astrophys. 344, 105-110 (1999)

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2. Model determining the thermodynamic conditions

We chose the units of distance, time and mass to be [FORMULA], [FORMULA] and M where, G is the gravitational constant, M is the mass of the black hole, and c is the velocity of light. To keep the problem tractable without sacrificing the salient features, we use a well understood model of the accretion flow close to the black hole. We solve the following equations (Chakrabarti 1996a,b) to obtain the thermodynamic quantities:

(a) The radial momentum equation:

[EQUATION]

(b) The continuity equation:

[EQUATION]

(c) The azimuthal momentum equation:

[EQUATION]

(d) The entropy equation:

[EQUATION]

Here, [FORMULA] and [FORMULA] are the heat gained and lost by the flow, and [FORMULA] is the mass accretion rate in units of the Eddington rate. Here, we have included the possibility of magnetic heating [FORMULA] (due to stochastic fields; Shvartsman 1971; Shapiro, 1973; Bisnovatyi-Kogan, 1998) and nuclear energy release [FORMULA] as well (cf. Taam & Fryxall 1985) while the cooling is provided by bremsstrahlung, Comptonization, and endothermic reactions and neutrino emissions. A strong magnetic heating might equalize ion and electron temperatures (e.g. Bisnovatyi-Kogan 1998) but this would not affect our conclusions. On the right hand side, we wrote [FORMULA] collectively proportional to the cooling term for simplicity (purely on dimensional grounds). We use the standard definitions of [FORMULA] (Cox & Giuli 1968),

[EQUATION]

and [FORMULA] is the ratio of gas pressure to total pressure,

[EQUATION]

Here, [FORMULA] is the Stefan constant, k is the Boltzman constant, [FORMULA] is the mass of the proton, µ is the mean molecular weight. Using the above definitions, Eq. (2d) becomes,

[EQUATION]

In this paper, we shall concentrate on solutions with constant [FORMULA]. Actually, we study in detail only the special cases, [FORMULA] and [FORMULA], so we shall liberally use [FORMULA]. We note here that unlike self-gravitating stars where [FORMULA] causes instability, here this is not a problem. Similarly, we shall consider the case for [FORMULA] = constant, though as is clear, [FORMULA] in the Keplerian disk region and probably much greater than 0 near the black hole depending on the efficiency of cooling (governed by [FORMULA], for instance). We use the Paczynski-Wiita (1980) potential to describe the black hole geometry. Thus, [FORMULA], the Keplerian angular momentum is given by, [FORMULA], exactly same as in general relativity. [FORMULA] is the vertically integrated viscous stress, [FORMULA] is the half-thickness of the disk at radial distance x (both measured in units of [FORMULA]) obtained from vertical equilibrium assumption (Chakrabarti 1989) [FORMULA] is the specific angular momentum, [FORMULA] is the radial velocity, s is the entropy density of the flow. The constant [FORMULA] above is the Shakura-Sunyaev (1973) viscosity parameter modified to include the pressure due to radial motion ([FORMULA], where W and [FORMULA] are the integrated pressure and density respectively; see Chakrabarti & Molteni (1995) in the viscous stress. With this choice, [FORMULA] keeps the specific angular momentum continuous across of the shock.

For a complete run, we supply the basic parameters, namely, the location of the sonic point through which flow must pass just outside the horizon [FORMULA], the specific angular momentum at the inner edge of the flow [FORMULA], the polytropic index [FORMULA], the ratio f of advected heat flux [FORMULA] to heat generation rate [FORMULA], the viscosity parameter [FORMULA] and the accretion rate [FORMULA]. The derived quantities are: [FORMULA] where the Keplerian flow deviates to become sub-Keplerian, the ion temperature [FORMULA], the flow density [FORMULA], the radial velocity [FORMULA] and the azimuthal velocity [FORMULA] of the entire flow from [FORMULA] to the horizon. Temperature of the ions obtained from above equations is further corrected using a cooling factor [FORMULA] obtained from the results of radiative transfer of Chakrabarti & Titarchuk (1995). Electrons cool due to Comptonization, but they cause the ion cooling also since ions and electrons are coupled by Coulomb interaction. [FORMULA], chosen here to be constant in the advective region, is the ratio of the ion temperature computed from hydrodynamic (Chakrabarti 1996b) and radiation-hydrodynamic (Chakrabarti & Titarchuk 1995) considerations.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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