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