Formalisms for computing line profiles

Since the spectrum is emitted from an accretion disk close to the black hole, an observer at infinity will see the reflected X-ray spectrum of Fig. 7 distorted by relativistic effects, namely the Doppler effect, relativistic aberration, and gravitational light-bending/red-shift.

The specific flux $F_{\nu_0}$ at frequency $\nu_0$ as seen by an observer at infinity is defined as the (weighted) sum of the observed specific intensities $I_{\nu_0}$ from all parts of the accretion-disk,

$$F_{\nu_0} = \int_{\Omega} I_{\nu_0} \cos\theta\,d\Omega,$$ 30

where $\Omega$ is the solid angle subtended by the accretion disk as seen from the observer and $\theta$ is the angle between the direction to the disk and the direction of the observed photon. Since the black hole is assumed to be very far away from the observer, we can safely set $\cos\theta=1$. Thus, we have to compute the specific intensity $I_{\nu_0}$ at infinity from the spectrum emitted on the surface of the accretion disk, $I_{\nu_e}$. In an axisymmetric accretion disk, $I_{\nu_e}$ is a function of the radial distance of the point of emission from the black hole, $r_{\rm e}$, and of the inclination angle, $i_{\rm e}$, of the emitted photon, measured with respect to the normal of the accretion disk.

The emitted and observed frequencies differ due to Doppler boosting and gravitational red-shift. We define the frequency shift factor $g$, relating the observed frequency $\nu_0$ and the emitted frequency $\nu_e$,

$$g = \frac{\nu_0}{\nu_e} = \frac{1}{1+z},$$ 31

where $z$ is the red-shift of the photon. According to Liouville's theorem, the phase-space density of photons, proportional to $I(\nu)/\nu^3$, is constant along the photon path. It is therefore possible to express eqn. (40) in terms of the emitted specific flux on the accretion disk:

$$F_{\nu_0}=\int_{\Omega} \frac{I_{\nu_0}}{\nu_0^3} \nu_0^3\,d... ...a =\int_{\Omega} g^3 I_{\nu_e}(r_{\rm e},i_{\rm e})\,d\Omega.$$ 32

In other words, the computation of the emerging spectrum breaks down to the computation of $g$. In the weak field limit, when $r\gg 3$, and in the Schwarzschild metric, $g$ and therefore the line profile emitted by the accretion disk, can be evaluated analytically. Profiles computed this way have been presented, e.g., by Andy Fabian and collaborators [189] for the Schwarzschild case, and by Chen and Halpern [190] in the weak field limit. In general, however, the computation has to be done in the strong-field Kerr metric.

The brute-force approach to the computation of $g$ in the Kerr metric is the direct integration of the trajectory of the photon in the Kerr metric [191,192]. This approach allows the computation of exact line profiles even in the case of very complicated geometries, such as geometrically-thick or warped accretion disks [193] or disks with non-axisymmetric (e.g., spiral wave) structures [194,195]. The second way to compute the observed flux was first used by Cunningham in 1975 [196] who noted that the observed flux from can be expressed as

$$F_{\nu_0} = \int T(i_{\rm e},r_{\rm e},g) I_{\nu_e}(r_{\rm e},i_{\rm e})\,dg\,r_{\rm e}\,dr_{\rm e},$$ 33

where the integration is carried out over all possible values of $g$ and over the whole surface of the accretion disk. This form is well suited for fast numerical evaluation. All relativistic effects are contained in the transfer-function $T$ which can be evaluated once for a given black hole spin parameter. We refer to [196,197,198], and the references in these works for the technical details.