The emerging line profile

Figure 9: Iron line profiles from relativistic accretion disk models as a function of disk inclination (as measured by the angle between the normal to the disk plane and the observers line of sight). The black hole is assumed to be rapidly-rotating ($a=0.998$), and the disk is assumed to possess a line emissivity index of $\beta =0.5$ down to the radius of marginal stability $r\approx 1.23\,GM/c^2$.

Figure 10: Iron line profiles from relativistic accretion disk models as a function of emissivity index, $\beta$. The black hole's spin parameter is assumed to be $a=0.5$ and, again, it is assumed that the disk extends to the radius of marginal stability.

Figure 11: Iron line profiles from relativistic accretion disk models as a function of black hole spin parameter. The disk is assumed to extend to the radius of marginal stability with an emissivity index of $\beta =3$.

In Figs. 9 to 11 we illustrate the relativistic effects on the emerging line profile. All line profiles have been computed with the code of Roland Speith [198]. In our computations we assumed a geometrically thin, but optically thick Keplerian accretion disk to be the source of the line radiation. The local emissivity of the line on the disk was parameterized as

$$I_{\nu_e}(r_{\rm e},i_{\rm e})\propto f(i_{\rm e})\,r_{\rm e}^{-\beta},$$ 34

where $f(i_{\rm e})$ is a function that can be chosen to model various limb-darkening scenarios. This parameterization is sufficient for most practical work [199]. For optically thick material, $f={\rm constant}$.

In all line profiles shown here, the emitting part of the disk is assumed to extend from the radius of marginal stability to an outer radius of $r_{\rm out}=50GM/c^2$. For $\beta>2$, the location of this outer radius is unimportant to the line profile since the line emission is dominated by the inner regions. However, for $\beta<2$, this choice of outer radius is of some importance since the bulk of the line emission comes from these outer regions. In the latter case, the separation between the red and blue peaks of the iron profile diminishes as the outer radius is increased (due to the diminishing line of sight velocity difference between the approaching and receding sides parts of the disk).

Common to all line profiles is a characteristic double-horned shape (Fig. 9). To see the origin of this, consider the emission line profile that arises from a narrow annulus of the disk $r\rightarrow r+dr$. By assumption, the emitting matter in this annulus has a single orbital speed but, clearly, the observed Doppler shift from different parts of the annulus will differ due to the changing angle between the observer's line-of-sight and the velocity vector of the matter (which is undergoing near-circular motion). Line emitting material on the portion of the orbit moving away from the observer will contribute lower-frequency emission than material on the approaching portion of the orbit. Considering this simple case, it can be seen that the line flux in a given frequency range $d\nu$ will be at a maximum when the Doppler shifts are at an extremum, i.e., for the minimum and maximum Doppler shifts from the annulus. These ``red'' and ``blue'' peaks¹⁰ in the line profile of the ring-element correspond to extrema in the ``pathlength'' through material at a given line-of-sight velocity -- these extrema are found on the receding and approaching parts of the annulus. Once these horns are formed in the line profile, special relativistic aberration (i.e., the beaming of emission in the direction of motion) strongly enhances the blue horn over the red. This effect is clearly visible in Fig. 9.

In addition to the line-of-sight Doppler boosting, the line emission is also red-shifted due to the gravitational potential and the transverse Doppler effect (i.e., special relativistic time dilation). The influence of all of these effects on the line profile depends on the observer's inclination angle $i_0$: For a disk seen almost face on (i.e. $i_0$ close to $0^\circ$), the gravitational/time-dilation redshifts dominate. Even with a face-on disk, integrating the observed emission across all radii including the gravitational/time-dilation effects can still produce a rather broad emission line, unlike the case of a face-on Newtonian disk which would display a very narrow line. With progressively larger inclinations $i_0$, line-of-sight Doppler effects become dominant and the line becomes broader.

The broadest parts of the profile are due to material emitted very close to the black hole, as is evident from Fig. 10, where line profiles for different emissivity coefficients $\beta$ are shown. For large $\beta$, most of the line emission takes place close to the last stable orbit, so that these profiles are the broadest. Note that for values of $\beta<2$ the red wing of the profile gets weaker until it is almost undetectable.

If one assumes that the line emitting disk extends down to, and is then truncated, at the radius of marginal stability, the strong dependence of $r_{\rm ms}$ on the black hole spin translates into a strong dependence of line profile on black hole spin. In particular, the red-wing of the iron line can extend to much lower energies in the case of a rapidly-rotating black hole as compared with a non-rotating black hole. This is illustrated in Fig. 11. However, as we discuss in more detail in §6.1, it is unclear that the effects of the radius of marginal stability can be modeled so trivially, leading to many subtleties to using these line profiles to probe black hole spin.