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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