Figure: Constraints diagrams for the Comptonization model
applied to MCG-6-30-15 (left) and NGC 3516 (right). The almost
vertical line corresponds to a Compton temperature of 
, with the region left of the line being excluded
since it would produce too much Compton upscattering of the iron
line photons. The region shaded with lines of negative slope is
forbidden since it would produce a soft excess in the ASCA
(MCG-6-30-15) or BeppoSAX (NGC 3516) bands (which is not
observed). The shaded region is forbidden since the source would
violate the black body limit.
In the situation postulated by the MS99 model, the temperature of
the Compton cloud will be locked to the Compton temperature of
the (local) radiation field. We model the continuum spectrum of
the central source as the superposition of a black body spectrum
(which may represent thermal emission from an accretion disk) and
a power-law spectrum with energy index 
which extends
up to hard X-ray energies (which may be identified as accretion
disk photons that have been subjected to multiple Compton
upscattering by an accretion disk corona).
The flux at the inner edge of the Compton cloud is then given by
where 
is the inner radius of the Compton cloud,
is the luminosity of the black body component,
the Stefan-Boltzmann constant,
is the luminosity in the power-law component,
in the range 
(and zero elsewhere), and 
is given by
Guided by the hard X-ray observations of MCG-6-30-15 (e.g., Lee et al. 1999), the parameters describing the power-law component are fixed to have the following values:
The resulting Compton temperature is given by
where 
is the ratio of the black body luminosity to the
power-law luminosity:
and 
is the Riemann zeta function ( 
). The line corresponding to a Compton temperature of
on the 
-plane is shown on
Fig. 1a, and the forbidden region of parameter space (giving
) is shaded with lines of positive gradient.
For completeness, it should be noted that the above expression
for the Compton temperature is only strictly valid due to the
soft nature of our spectrum. The Compton temperature depends, of
course, on the form of the radiation field inside the cloud.
Ignoring downscattering, this field is greater than the external
radiation field by a factor of 
. For the high-energy
radiation ( 
), has an energy
dependence due to Klein-Nishina corrections, thereby affecting
the Compton temperature. The neglect of
downscattering is also invalid at these energies. However, these
corrections to the Compton temperature have a negligible effect
in our case.
The ASCA observation shows no evidence for a soft excess component in MCG-6-30-15 across the entire well-calibrated spectral range of the solid-state imaging spectrometers (SIS; 0.6-10keV). Thus, we impose the condition that the black-body flux at 0.6keV is less than the power-law flux at the same energy:
The region on the -plane forbidden by this constraint is
shaded with lines of negative gradient in Fig. 1a.
Finally, we make the observation that there is a fundamental limit to the black body luminosity which is imposed by thermodynamics:
where 
is the maximum allowed size of the black body source.
Since the continuum source is hypothesized to be interior to the Compton
cloud, we must have 
. The region of the
-plane forbidden by this constraint is shown in solid-shade
in Fig. 1a.
We see that applying these three constraints eliminates all
regions of the -plane. One must conclude that the
Compton cloud model discussed by Misra & Kembhavi (1998) and
MS99 is not valid in the case of MCG-6-30-15.
NGC 3516 also displays a strong broad iron line that has been observed at
high signal-to-noise with ASCA (Nandra et al. 1999). We have also
examined constraints on the Comptonization model for this iron line.
Continuum variability in this object is observed on timescales down to
(Edelson & Nandra 1998; K. Nandra, private communication),
giving a maximum size of 
for the
Comptonizing cloud, rather larger than the case of MCG-6-30-15. Also,
BeppoSAX observations fail to see a soft excess in the X-ray spectrum
all of the way down to 
(Stirpe et al. 1998). Noting that 
produces the constraint diagram shown in
Fig. 1b. It is seen that these constraints eliminate all but a very small
region of parameter space.
Thus, although the broad line in NGC 3516 could in principle be
explained with the Comptonization model, the amount of fine
tuning necessary for finding the line parameters makes the model
improbable in this case.