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Astron. Astrophys. 362, L41-L44 (2000)

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2. Governing equations, initial and boundary conditions

The set of equations and boundary conditions used here are identical to those in HCb. Further, we use the previous solutions they obtained to initiate the present runs. Using axi-symmetry, HCb solve the radiative hydrodynamical equations consisting of the continuity equation for the density, Euler's equations for three momenta, two equations for the internal energies of the ions and electrons, and one equation for the energy density of the radiation field.

In the radiation density equation, HCb modified the opacity [FORMULA] such that the radiation-matter collisional term [FORMULA] reduces to pure relativistic Bremsstrahlung [FORMULA] in optically thin regions (Stepney & Guilbert 1983) and to the radiative diffusion operator for higher optical depths. Here [FORMULA] and [FORMULA] denote the density, temperature, density of radiation and A and B are constant coefficients. Synchrotron cooling [FORMULA] and Compton cooling [FORMULA] are incorporated as well.

Here, we further include in the ion- and electron-energy equations the corresponding second order conductive operators: [FORMULA] where [FORMULA] and [FORMULA] in cgs units. The subscripts [FORMULA] and [FORMULA] denote ions and electrons respectively (see [FORMULA] & Leer 1994).

Gravity of the central object is described in terms of the quasi-newtonian potential of Paczynski & Wiita (1980). A central [FORMULA] black hole is assumed and an accretion rate of [FORMULA] is set to enter the domain [FORMULA] via a thin disk across the outer boundary. The radii in the following are measured in units of [FORMULA]

In the 2T description (Shapiro et al. 1976), turbulent dissipation heats up the ions only, and subsequently the electrons via Coulomb interaction. For the dynamical viscosity we use [FORMULA] as the turbulent diffusion coefficient, where [FORMULA] and [FORMULA] are used.

The calculations are run till the maximum time-independent residual [FORMULA] has dropped below a certain small value [FORMULA]. This corresponds to several thousand orbital periods of the inner radius.

[FIGURE] Fig. 1. Conductive-2T model. 35 uniformly, logarithmically scaled iso-lines of the electron temperature (solid lines) and of the ion-pressure (dotted lines). Distances are given in units of [FORMULA].These solutions are obtained after 5000 orbital periods (in units of the orbital period at [FORMULA] around a [FORMULA] black hole). A material flux of [FORMULA] is set to enter the domain of integration through the outer boundary. In this model, cooling via relativistic Bremsstrahlung, Compton and synchrotron emission, ion- and electron conduction are incorporated.Apparently, the disk is geometrically thin, optically thick (to scattering) and surrounded by hot electrons from beneath and from above. It truncates at [FORMULA] to form an optically thin non-spherical ion-torus in the inner region.

[FIGURE] Fig. 2. Conductive-2T model. The velocity field across the transition region. The inwards motions close to the equator and the outwards oriented motion in the transition between the corona and the disk are shown also.

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000