*Astron. Astrophys. 326, 885-896 (1997)*
## 4. Accretion disk model used in spectral fits
We assume a standard geometrically thin
-accretion disk around a massive Kerr black hole. A detailed
description of our model is given in Dörrer et al. (1996 ). Here,
we just summarize the basic concepts of the calculations. Model
parameters are the mass *M* of the central black hole, the
accretion rate , the viscosity parameter
, and the specific angular momentum *a* of
the central black hole. In addition to the parameters describing the
physical properties of the accretion disks the inclination angle under
which the observer sees the disk is also a free parameter. The
specific angular momentum *a* of the central black hole was fixed
at in our spectral fits. All relativistic
corrections on the disk structure with respect to a Newtonian model
are included according to Riffert & Herold (1995 ).
The radiative transfer is solved in the Eddington approximation,
and the plasma is assumed to be in a state of local thermodynamic
equilibrium. Multiple Compton scattering is treated in the
Fokker-Planck approximation using the Kompaneets operator. The
absorption cross section contains only free-free processes for a pure
hydrogen atmosphere. Induced contributions to the radiative processes
have been neglected throughout.
For a given radial distance *R* from the central black hole, a
self-consistent solution of the vertical structure and radiation field
of the disk is obtained from the hydrostatic equilibrium equation,
radiative transfer equation, energy balance equation, and equation of
state, when proper boundary conditions are imposed. Note, that we have
not considered convection in our model.
In our calculations the viscosity is assumed to be entirely due to
turbulence. Because the standard -description
(viscosity proportional to the total pressure) leads to diverging
temperature profiles in the upper parts of the disk, we have included
the radiative energy loss of the turbulent elements in the optically
thin regime. For the turbulent viscosity we then have
where is the shear viscosity,
is the mass density, *H* is the
self-consistently calculated height of the disk,
is the upper limit for the velocity of the
largest turbulent elements (see Dörrer et al. 1996 for details on
the determination of ) and
is the viscosity parameter of our model.
We used a finite difference scheme in the vertical direction
*z* and in frequency space to find
solutions of the given set of equations. The vertical structure was
resolved with 100 points on a logarithmic grid, and 64 grid points
were used in frequency space. The resulting set of algebraic
difference equations was then solved by a Newton-Raphson method.
To get the overall structure and emission spectrum of the disk we
calculated the vertical structure and the local emission spectrum at
50 logarithmically spaced radial grid points from the last stable
orbit to the outer disk radius (here , where
is the Schwarzschild radius, *G* is the
gravitational constant, and *c* is the velocity of light). The
whole disk spectrum as seen by a distant observer at an inclination
angle with respect to the disk axis
( for a face-on observer) is then calculated by
integration of the local spectra over the disk surface. All general
relativistic effects on the propagation of photons from the disk
surface to the observer are included, using a program code (Speith et
al. 1995) to obtain values of the Cunningham transfer function
(Cunningham 1975 ) for any set of parameters.
The accretion disk spectrum as determined from the above
calculations extends from the optical to the soft X-ray range
( 1 keV), the maximum of the emission being in
the far UV. Qualitatively, the dependence of the spectral shape on the
model parameters is such that, if the mass accretion rate
is kept at a constant fraction of the Eddington
accretion rate , the central mass *M*
mainly determines the total flux while approximately maintainig the
spectral shape. Increasing the mass accretion rate in terms of the
Eddington accretion rate , on the other hand,
while also increasing the total flux, makes the spectrum harder, i.e.
a larger fraction of the flux is emitted in the X-ray range.
Similarly, an increase of the viscosity parameter
also leads to a hardening of the spectrum. Going
from low (disk seen face on) to high (disk seen edge on) inclination
angles of the accretion disk, the total flux from the disk is reduced
and at the same time a hardening of the spectrum due to Doppler
boosting resulting from the rotation of the disk occures.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998
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