There are two distinct ingredients involved in determining the absolute
emission rate of fluorescent iron line photons. The most obvious one is
the number of illuminating photons with energy 
, where
is the photoelectric
threshold for neutral iron. Only these incident photons can eject one of
the K-shell electrons from iron and, thus, initiate the radiative cascade
within the atom that results in a K 
line photon being emitted. For
a fixed illuminating spectrum, the iron line strength is simply
proportional to the normalization of that spectrum.
The second ingredient in determining the absolute iron line emission is the
geometry of the illumination (Basko 1978; George & Fabian 1991). Consider
normally incident photons with energy . On average, such
photons traverse a distance corresponding to unity optical depth prior to
being photoelectrically absorbed. The resulting iron fluorescence photons
have to travel through at least the same optical depth of material in order to
escape the slab. Some fraction of these iron line photons will be
absorbed in this process (either by K-shell photoionization of
low-Z metals or L-shell photoionization of iron). Now, consider photons that
are incident on the slab with a large inclination (i.e. grazing
collision). Again, these photons are absorbed after a unity optical depth,
but this now corresponds to a significantly smaller vertical depth in the
slab. Thus, the resultant iron line photons can escape significantly
less impeded by subsequent absorption. The net result is that the
effective fluorescent yield increases with increasing inclination (see
Fig. 1 of George & Fabian 1991).
This geometrical effect only becomes important when most of the incident flux strikes the disk at a high inclination. In the scenario under discussion here, that corresponds to irradiation by sources which are rapidly moving in a direction parallel to the disk plane. Given this fact, and that analytical descriptions of the geometrical dependence are somewhat cumbersome (and approximate; e.g. Basko 1978), we shall ignore this effect and study just the Doppler-shifting phenomenon.
Figure 1: Enhancement in the absolute fluorescent line emission from the
stationary slab as a function of source velocity. Shown here are the cases
(source motion directly towards slab; dashed line),
(source motion parallel to the slab; solid line), and
(source motion directly away from slab; dotted line).
With this restriction, and the assumption of a power-law primary spectrum,
the problem simply amounts to determining how the normalization of the
illuminating spectrum at some given energy ( 
, say),
integrated over the surface of the slab, is affected by the source motion.
Making use of the phase space invariant 
, we see that the
SR enhancement in absolute iron line production is
where F(r) is the illuminating flux striking a unit area of the slab in the absence of any relativistic effects,
and 
is the beaming parameter,
Simple expressions can be obtained for 
in three cases:
):
):
):
In Fig. 1, we plot 
for these cases. As intuitively
expected, the former two cases enhance the absolute iron emission. For
motion along the slab normal ( 
), the beaming can influence
the absolute line production by a factor of two for velocities that are
only mildly relativistic ( 
).