\begin{quotation}

\subsection{Pileup in the Data Sets}
\label{sec:acis_pileup}

\subsubsection{Pileup Basics}

All the effective area measurements performed at XRCF were primarily
optimized for flux in the higher order in order to get sufficient
statistics. Naturally this will cause pileup in the CCD detector
array at least in zero as well as in some of the lower orders.\par

A recent analysis by the ASC ACIS team (Allen et al. 1998) showed
that pileup affects the data in different ways and also shows
differently in front- and back-illuminated CCDs. Pileup basically
occurs when two or more photons incidently hit the same 3x3 detection cell.
Two photons of the same energy  would be detected as one event
at twice the pulse height of a single photon, three photon at three
times the pulse height and so on. Naturally this process has to be 
flux dependent. At very high fluxes will also allow two or more
photons only partially contained within the 3x3 detection cell,
which will then cause an enhanced background in between the main
and pileup peaks.\par

In this respect it is more appropriate to think in terms of "fluences" rather than
fluxes. Fluence descibes the incident photon density, i.e how many
photons strike the CCD within one frametime per unit area, i.e.
"ph/s/cm$^2$" or "ph/s/pix$^2$". 
 
Higher fluences cause events to overlap more often, which then causes the
detection algorithm to issign a higher number grade. This effect is 
called grade migration and causes a systematic depletion of standard
grades, here the ASCA grade set 0,2,3,4,and 6. This migration is likely
to be only effective at energies above $\sim 3$ or 4 keV, when the sizes
of charge clouds increase significanty.

Grade migration may not be the only effect of overlapping charge clouds.
In the case when a very large charge cloud created by a high energy photon
overlaps with a "normal" single or double pixel event, the 3x3 detection
cell algorthm may not be able to detect any event anymore. In this case
the charge would is lost.

\subsubsection{Demonstration: the Zero-order Data}

Zero-order data sets were not designed for effective area
analysis because in order to get sufficient
statistics in the higher orders, the source flux had to be high,
especially at high energies. Therefore the zero order data will be
entirely dominated by pileup effects in the CCD.  They thus serve
as useful pileup example.

The simplest way to identify pileup is to determine higher order
pileup peaks in the pulse height spectrum. For energies below 3 keV it
is possible to detect higher order pulse heights corresponding to up
to 4 photons hitting the same event detection cell. The corresponding
number of counts is then summed up and added back to the single photon
count rate. The left diagram of figure~\ref{fig:zero_pileup} already
includes that summation. Clearly above about 3 keV this procedure
starts to fail because higher order pulse heights fall beyond the
maximum pulse height channels. Therefore we observe a large drop in
the measured effective area. In a first attempt to estimate the amount
of piled-up photons we fit the ratio of the non-piled-up fraction to
the measured piled-up fraction in the range 0.9 to 2.5 keV. In this
range we were able to recover piled-up photons for up to 4 photons
hitting an single detection cell. This could be done by a power law of
index 0.27. We extrapolated this function into the high energy range
and added that flux to the count rate above 2.5 keV. We also had to
take in account that the source flux above 2.5 keV increased by a
factor of 8 and scaled that function simply by that increase. Since
pile-up is actually a stochastic process we could have also estimated
the missing higher order counts out of a poisson distribution.  In
any case the resulting area was still short of the expectation by up
to 45$\%$.\par 
Following the method to correct for grade migration as descibed by
Allen at al. 1998 the grade distributions of all measurements above 
3 keV were compared to the ones obserbed during subassembly analysis,
i.e. data that were evidently free of migration effects.
This results in another energy dependent correction factor. We thus
calculated the grade migration correction factor for our data sets
applied it to the
data. Again the result above 6 keV was still short by about
20$\%$. Allen et al. 1998 also apply a correction for lost charge and
indetected events.  Above 6 keV simply applied those correction
factors from table 1 in \cite{allen98} for the case of S3.  The result
can be seen in the right handed diagram of
figure~\ref{fig:zero_pileup}.  The area now seem to fit the
expectation, although a clear overcorrection is visible.  However, we
have to emphasize, since we did not in particular detemine the lost
charge correction for our own data sets, the result is merely an
estimation. It however demonstrates that those corrections applied by
\cite{allen98} are indeed necessary and in a reasonable order of
magnitude.  The applied corrections induce additional systematic
errors to the data, which are {\it not} reflected in the error bars in
the right handed diagram of figure~\ref{fig:zero_pileup}.

\begin{figure}[ht]
\psfig{file=zero_effarea.ps,height=10cm}
\caption[Measured and predicted HETGS 0-order effective area
({\tt zero\_effarea.ps})]
{\small Comparison of the measured and expected effective area for the combined
0th order of MEG and HEG as a  function of energy. The left hand diagram shows the
result without, the right hand diagram with pile up correction as described in the text.}
\label{fig:zero_pileup}
\end{figure}

\subsubsection{Pileup in diffracted orders}
The quantitative evaluation of pileup levels in the diffracted orders is still under investigation.
The expected pileup level is low in all orders, however there are severall aspects to consider.
For example, one major obstacle is, that at energies below 2 keV the first orders in HEG and MEG
spatially mix with higher orders of the W lines generated by the DCM. For energies around
1 keV this emission interferes with the first pileup peak in the pulse height spectra.\par  
Another problem is introduced by the design of the measurements at higher energies.
In order to avoid overlaps of MEH and HEG images, the out-of-focus distance of originally
40 mm was gradually reduced with increasing incident photon energy. This hasd the effect
that although flux stays constant, the fluence, i.e flux per unit area, increases.
Therefore grade migration and charge loss effects will appear in the first order images.
In addition that eefect is more likely to affect HEG orders, simply because of the
smaller image size.