It should be stressed that we have used conservative parameters in our
assessment of these observational constraints. In particular, we assume
that the power-law component of the continuum emission possesses an energy
index of 
(corresponding to a photon index of 
) and a
high energy cutoff of 
. In fact, the overall X-ray spectrum is
harder than this (especially once the Compton reflection component is
accounted for) and the high energy cutoff may well occur at rather higher
energies. Either of these effects will raise the Compton temperature of
the power-law component and require an even cooler black body component in
order to cool the Compton cloud below the 
limit. It should
also be noted that we have ignored any infra-red emission from the
continuum source. Due to the high densities of the matter in the Compton
cloud, IR emissions redwards of 
will be free-free
absorbed and act to heat the cloud rather than Compton cool it.
Again, the neglect of the IR emissions is a conservative assumption for our
purposes.
There is another, independent, problem faced by the Compton cloud model:
it is very difficult to maintain the required ionization state. F95
treated this problem by considering the required cloud size necessary to
acheive some critical ionization parameter 
. According to F95, the AGN spectrum of Mathews & Ferland
(1987), 
can be considered the point at
which a photoionized plasma becomes completely ionized. Using the
observed luminosity of MCG-6-30-15, they deduced that the cloud must
have a size 
in order to achieve at least this critical
ionization parameter. As we will now show, this is a very conservative
argument and, in fact, ionization balance imposes much more severe
limits on the cloud size.
While the formal ionization parameter may be very high, the very soft continuum spectrum postulated by MS99 may still have trouble fully ionizing the iron throughout the whole cloud. To see this, note that all continuum photons capable of ionizing hydrogen like iron (FeXXVI) reside in the power law component of the continuum. The continuum source in MCG-6-30-15 emits FeXXVI ionizing photons at a rate
where 
is the ionization potential of FeXXVI. This evaluates to 
. The radiative recombination rate of the postulated
Compton cloud, on the other hand, is given by
where the coefficients 
and 
are given by Shull
& van Steenberg (1982). For a temperature of 
and
, this gives 
. Thus, there
are just enough ionizing photons present in the entire power law tail to
ionize the hydrogen-like iron. If the temperature of the Compton cloud is
below 0.5keV, or the radius of the cloud is larger
, it will be impossible to
photoionize the cloud. Very large iron edges would then be present in the
observed X-ray spectrum, contrary to observations.
Thus, ionization balance imposes a size limit of 
, independently of continuum variability constraints.
Finally, we address whether there are reasonable modifications that can be
made to the MS99 scenario that will avoid the constraints imposed in this
paper. There are three such modifications that we should consider.
Firstly, if the geometry is such that the X-ray continuum source is viewed
directly (rather than through the Compton cloud), one might imagine that
the size of the Compton cloud and the X-ray continuum variability would be
decoupled thereby relaxing the constraints discussed above. An example of
such a geometry is if the Compton cloud forms a torus around the central
X-ray source. In such a geometry, the X-ray continuum source illuminates
and ionizes the observed face of the Compton cloud and powers iron line
fluorescence from an optical depth of 
into the cloud.
However, in this case, one would expect ionized iron lines (from the
ionized zones that overlay the near-neutral zones in the Compton cloud)
rather than the observed cold iron lines. Also, the illuminated surface of
the Compton cloud, which must be highly ionized so as not to be a strong
narrow iron line emitter, would act as a Compton mirror and smear out the
observed continuum variability, even though the continuum source is viewed
directly.
Of course, any such modification to the basic Comptonization model in
which the Compton cloud is allowed to be bigger than 
must be subject to the ionization problem described above.
Secondly, a large region of parameter space would open up if the Compton cloud experienced a different soft continuum to that observed (i.e. if the soft excess can be `hidden' from view). Noting that the black body component must scatter though the same parts of the Compton cloud that broadens the iron line (in order to Compton cool it), one concludes that the black body photons and broad iron lines photons will follow very similar paths through the system. Hence, it is impossible to hide the soft excess emission from view in a system in which we observe a Compton broadened iron line.
Thirdly, the black body limit can be bypassed if the soft continuum source
is placed outside of the Compton cloud. While it is difficult to construct
rigorous arguments against this case, we consider that placing a powerful
( 
) soft continuum source at large distances from
the central hard X-ray continuum source is an ad-hoc solution.