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Astron. Astrophys. 348, 71-76 (1999)

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2. Modeling line profiles

In our model a viscous, geometrically thin, and warped - because of the Bardeen-Petterson effect - accretion disk is irradiated by a point-like central source. We use a cylindrical coordinate system, centered at the black hole (Fig. 1). The irradiating source is located at [FORMULA] above the disk plane, along the black hole axis instead of at the exact center, as this is a more realistic situation. The profiles of HFI emission as a result of the reprocessing of the central hard X-ray radiation are obtained. The disk thickness is neglected in the computations - it does not affect the profile shapes significantly as the semithickness to radius ratio is usually much less than the tilt angle. All relativistic corrections are also neglected, because they are small while our goal is to obtain qualitative results only. The disk is assumed to be optically thick to visual light and X-rays - no emission from the lower surface (probably also illuminated) can reach an observer located above the X-Y plane.

[FIGURE] Fig. 1. The twisted accretion disk - a steady system of inclined rings, each defined by two Eulerian angles and the radial distance (R). The axis of the black hole is aligned with the Z direction. To explore the irradiation, the disk is divided into [FORMULA] elements. Cylindrical coordinates are used to express the position of each element.

2.1. Geometry of nonplanar disks

Since the precession velocity is small compared to the Keplerian orbital velocity [FORMULA], the disk is a stationary structure and can be treated as being composed of concentric rings, laying in different planes. Warp waves can propagate through the disk surface if [FORMULA] (Papaloizou & Pringle 1983), where [FORMULA] is the dimensionless viscosity parameter (Shakura & Sunyaev 1973), but this is not the case for AGN accretion disks, where [FORMULA] is usually assumed to be 0.1-1. Each disk ring is defined by two Eulerian angles [FORMULA] and [FORMULA] (Fig. 1) and its radius R.

For a stationary twisted accretion disk, [FORMULA] and [FORMULA] are slowly varying functions of the radial distance R; [FORMULA], [FORMULA] and [FORMULA], [FORMULA], [FORMULA], as is originally shown by Bardeen & Petterson (1975). In this paper we use the steady solution of Scheuer & Feiler (1996), derived following Pringle (1992). This solution, shown in Fig. 2, can be presented analytically by:


Here the dimensionless parameter [FORMULA] if the semithickness-to-radius ratio of theaccretion disk is about 0.01 (Collin-Souffrin & Dumont 1990), [FORMULA], and R is measured in [FORMULA]. A similar solution has been obtained by Bardeen & Petterson (1975) and Hatchett at al. (1981), althoughthey did not take into account the internal hydrodynamics of the flow. In that case though, [FORMULA] and m are different and depend mostly on the radial velocity and the Kerr parameter (a), and the resulting alignment radius is much larger. Eqs. 1 are valid for [FORMULA] (Kumar & Pringle 1985, Natarajan & Pringle 1998), large radial distances and small initial inclination angles of the flow - [FORMULA]. If the initial tilt is significant, the influence of a significant amount of intercepted ionizing continuum on the disk structure should be taken into account and the disk shape should be studied numerically.

[FIGURE] Fig. 2. Eulerian angles [FORMULA] (solid line) and [FORMULA] (dashed line) as functions of radial distance, according Scheuer & Feiler (1996). Here [FORMULA]. [FORMULA] is normalized to the initial tilt angle - [FORMULA]. [FORMULA] is measured in radians.

2.2. Illumination and line profiles

In order to explore the nonplanar disk illumination and the resulting line profiles we created a simple code. The input parameters are: the initial inclination angle [FORMULA], the angles defining the line of sight, the ionizing luminosity [FORMULA], the source height [FORMULA] and the black hole mass [FORMULA]. For small inclinations of the disk, the deviation angle [FORMULA] of a disk point (P) is also small ([FORMULA]) and the following approximation is valid: [FORMULA]; [FORMULA]; see Fig. 1. The disk is divided into about [FORMULA] elements, each defined by its dimensions [FORMULA], [FORMULA] and coordinates R, [FORMULA]. An element of the grid constructed in this way absorbs an amount of the central hard radiation proportional to its cross section area [FORMULA] (Fig. 1), which is roughly:


For all elements the following value is calculated:


In the calculation of the total HFI luminosity, shadowing of the elements by the inner parts of the disk must be taken into account. This has been done as follows: The contribution of each element to the total HFI luminosity from the disk is equal to [FORMULA] (Eq. (2)) only in case that there does not exist another element with [FORMULA] and is 0 in the opposite case. [FORMULA] - the hard X-ray luminosity, is close to the bolometric luminosity if the continuum extends up to [FORMULA] with a spectral index close to 1. [FORMULA] is the line response to the ionizing flux, i.e. it represents the probability that an absorbed X-photon produces an HFI -photon. It depends on the physical parameters of the emitting layers, taking into account the saturation of the lines (due to the limited number of line-emitting atoms), the absorbed fraction of hard radiation (if the column density of the absorbing gas [FORMULA], not all hard radiation is absorbed), etc. It also may depend slightly on the incident flux - the reprocessing is not necessarily a linear process (Eq. (2)). In our computations we used this dependence as given by Collin-Souffrin & Dumont (1989, 1990b). In their work [FORMULA] is presented in a multifunctional form, taking into account all effects mentioned.

To obtain the profile of the line emission from the whole disk we have to calculate the Doppler shift of each element, i. e. the projection ([FORMULA]) of the Keplerian velocity onto the line of sight. [FORMULA] is given approximately by:


where [FORMULA] and [FORMULA] are the angles defining the direction to the observer. The whole profile can be constructed by adding contributions of all elements, taking into account their Doppler shifts.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999