2. Model determining the thermodynamic conditions
We chose the units of distance, time and mass to be , 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:
(b) The continuity equation:
(c) The azimuthal momentum equation:
(d) The entropy equation:
Here, and are the heat gained and lost by the flow, and is the mass accretion rate in units of the Eddington rate. Here, we have included the possibility of magnetic heating (due to stochastic fields; Shvartsman 1971; Shapiro, 1973; Bisnovatyi-Kogan, 1998) and nuclear energy release 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 collectively proportional to the cooling term for simplicity (purely on dimensional grounds). We use the standard definitions of (Cox & Giuli 1968),
and is the ratio of gas pressure to total pressure,
Here, is the Stefan constant, k is the Boltzman constant, is the mass of the proton, µ is the mean molecular weight. Using the above definitions, Eq. (2d) becomes,
In this paper, we shall concentrate on solutions with constant . Actually, we study in detail only the special cases, and , so we shall liberally use . We note here that unlike self-gravitating stars where causes instability, here this is not a problem. Similarly, we shall consider the case for = constant, though as is clear, in the Keplerian disk region and probably much greater than 0 near the black hole depending on the efficiency of cooling (governed by , for instance). We use the Paczynski-Wiita (1980) potential to describe the black hole geometry. Thus, , the Keplerian angular momentum is given by, , exactly same as in general relativity. is the vertically integrated viscous stress, is the half-thickness of the disk at radial distance x (both measured in units of ) obtained from vertical equilibrium assumption (Chakrabarti 1989) is the specific angular momentum, is the radial velocity, s is the entropy density of the flow. The constant above is the Shakura-Sunyaev (1973) viscosity parameter modified to include the pressure due to radial motion (, where W and are the integrated pressure and density respectively; see Chakrabarti & Molteni (1995) in the viscous stress. With this choice, 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 , the specific angular momentum at the inner edge of the flow , the polytropic index , the ratio f of advected heat flux to heat generation rate , the viscosity parameter and the accretion rate . The derived quantities are: where the Keplerian flow deviates to become sub-Keplerian, the ion temperature , the flow density , the radial velocity and the azimuthal velocity of the entire flow from to the horizon. Temperature of the ions obtained from above equations is further corrected using a cooling factor 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. , 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.
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999