2.2. The profile of the broad iron line

The iron K line is intrinsically a rather narrow line. Hence, we can use broadening of the line to study the dynamics of the accretion disk. The line profiles is shaped by the effects of Doppler shifts and gravitational redshifting. Figure 3 demonstrates these effects at work in a schematic way. In a non-relativistic disk, each radius of the disk produces a symmetric double-horned line profile corresponding to emission from material on both the approaching (blue-shifted) and receding (red-shifted) side. The inenr regions of the disk, where the material is moving the fastest, produce the broadest parts of the line. Near a black hole, where the orbital velocities of the disk are mildly relativistic, special relativistic beaming enhances the blue peak of the line from each radius (second panel of Fig. 3). Finally, the comparable influences of the transverse Doppler effect (i.e. ``moving clocks run slowly'') and gravitational redshifting (i.e. ``clocks near black holes run slowly'') shifts the contribution from each radius to a lower energy. Summing the line emission from all radii of the relativistic disk gives a skewed and highly broadened line profile. It has been suggested by Pariev & Bromley (1998) that turbulence in the accretion disk may also significantly broaden the line. While a detailed assessment of this possibility must await future magneto-hydrodynamic disk simulations, it seems unlikely that the turbulent velocity field in a thin accretion disk will be large enough to broaden the line.

Figure 3. The profile of the broad iron line is caused by the interplay of Doppler and transverse-Doppler shifts, relativistic beaming and gravitational redshifting. The upper panel shows the symmetric double peaked profiles from two narrow annuli on a non-relativistic disk. In the second panel the effects of transverse Doppler shifting and relativistic beaming have been included, and in the third panel gravitational redshifting has been included. These give rise to a broad, skewed line profile, such as that show in the lower panel. A more detailed discussion of this figure is given in section 2.2.

Some fully relativistic model line profiles are plotted in Figs. 4 and 5. In Fig. 4, we show the line profile from an accretion disk in orbit around a non-rotating black hole (described by the Schwarzschild metric). The line is assumed to be emitted from an annulus of the disk extending between 6 r_g and 30 r_g from the black hole, where r_g = GM / c² is the standard gravitational radius. It is seen that the high energy ``bluewards'' extent of the line is a strong function of the inclination of the disk. In fact, the blue extent of the line is almost entirely a function of the inclination, thereby providing a robust way to measure the inclination of the disk. On the other hand, the redward extent of the line is a sensitive funcion of the inner radius of the line emitting annulus. In Fig. 5, we show model iron lines from a Schwarzschild black hole and a rapidly rotating black hole (described by a near extremal kerr metric). In this figure, we have made the assumption that the line emission extends down to the innermost stable orbit of the acretion disk. For these purposes, the principal difference between these two space-time geometries is the location of the innermost stable orbit (and hence the inner edge of the line emission) - this critical radius is at 6 r_g in the Schwarzschild case, and r_g in the extremal kerr case.

Figure 4. Relativistic iron line profiles for the case of an accretion disk around a Schwarzschild (non-rotating) black hole. It is assumed that the fluorescing region of the disk extends from 6 rg (i.e. the radius of marginal stability) to 30 rg. Three inclinations are shown: 10°, 30° and 60°. The main effect of increasing the inclination is to broaden the line by increasing its high-energy extent.

Figure 5. Comparison of relativistic iron line profiles from an accretion disk around a Schwarzschild black hole (narrower, peaky line) and a near-extremal Kerr black hole (broader line). The line emission is assumed to extend down to the radius of marginal stability which is 6 rg and 1.25 rg, respectively. The difference in the width and redshift of the line is principally a result of the difference in the position of the radius of marginal stability.

Model line profiles are given in the Schwarzschild case by Fabian et al (1989) and for the maximal Kerr (spinning black hole) case by Laor (1991). Iron lines in extreme kerr metrics are also computed by Bromley, Miller & Pariev (1998), and Martocchia, Karas & Matt (2000). These last two sets of authors have presented diagnostics that can be used by observers who wish to avoid full spectral fitting of complex relativistic models. The formalism of computing relativistic line profiles is also discussed by ] Fanton et al. (1997).