%+
% Name:
%   memo_lsf_ha_1.0.tex
%
% Usage:
%    latex memo_lsf_ha_1.0.tex
%
% Description:
%    This document describes the quality of LSFPARM file for 
%       TG-unrandomized event datasets. 
%
% Input:
%
% Output:
%
% Notes:
%
% History:
%   21 April 04, created (v1.0) 
%-

\documentclass{article}

\usepackage{cxo-memo-logo}
\usepackage[dvips]{graphics,graphicx}
\usepackage{gea}
\usepackage{psfig}

\newcommand{\lsf}{LSFPARM}
\newcommand{\tgres}{tg\_resolve\_events}
\newcommand{\hpe}{hrc\_process\_events}
\newcommand{\apeb}{acis\_process\_events}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 0. Header

\memobasic{Dale Graessle, CALDB lead}{
   Bish K. Ishibashi, SDS }{
   Evaluation on the quality of HETG/ACIS-S LSFPARM file for unrandomized event datasets}{
   1.0 }{http://space.mit.edu/CXC/docs/memo\_lsf\_ha\_1.0.ps}{
   /nfs/cxc/h2/bish/CXC\ SDS/LSFDIR/HO\_LSF\_HA/memo\_lsf\_ha\_1.0.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 1. Theme

\section{Unrandomized LSFPARM files for HETG/ACIS-S Configuration}

\noindent
In Repro {\em{III}}, decisions have been made to turn off data randomization 
on TG R/D. This action necessitates rapid development of line spread function
parameter files (LSFPARM) without built-in data randomization. By applying the
similar LSF processing scheme\footnote{See an example in https://icxc.harvard.edu/calco/ard\_updates/ECR/Approved\_ECRs/ECR\_2002\_016.html}, 
new LSFPARM files for HETG+ACIS-S configuration have been developed. This document
describes the quality of the new LSFPARM products. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 2. Procedure to build LSFPARM 

\section{ Building LSFPARM }

\noindent
Procedures for building a LSFPARM file is described at: \\

\hspace{10mm}{http://space.mit.edu/CXC/LSFX/LSF\_valid2.html} \\

\vspace{3mm}
\noindent
For rapid development, several parts of IDL parameterization routines 
are S-Langified for better and faster convergence to optimum fitting 
parameters through automated processes. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 3. Products 

\section{ Data products }

\noindent
Four LSFPARM products are developed: HEG and MEG $\pm$ 1$^{st}$ orders 
configuration with HETG+ACIS-S without data randomization in TG coordinate. 
Evaluation of the four products are documented in the following section. 

\subsection{ Header Keyword for Data Randomization }

Two new FITS header keywords -- RAND\_TG and CBD60001 -- are included 
in each LSFPARM product in order to specify whether LSFPARM products is 
for randomized or not. For unrandomized dataset, these header 
keywords are set to the values as listed in Table 1. These keywords 
would enable to form proper CALDB query expression and to search for 
unrandomized LSFPARM products. 

\begin{table}
  \begin{center}
    \begin{tabular}{ccl}
      \multicolumn{3}{c}{Table~1. New header keywords for LSFPARM} \\
      \hline
      \hline
        Keyword         & Value            & Comment \\
        \hline
        RAND\_TG        & 0.0              & No data randomization in TG values \\
        CBD60001        & RAND\_TG(0.0)    & CALDB query expression \\
      \hline
    \end{tabular}
  \end{center}
\end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 4. Testing 

\section{ Testing LSFPARM products}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 4.1 Procedure for testing

\subsection{ Procedures }

\noindent
Testing of the LSFPARM products are performed with MARX simulated and 
real Capella (ObsID 1103) datasets. Two key points of our interests
are (1) line profile and (2) encircled energy fractions (hereafter
EEFRAC). 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 4.2 Testing with Simulated Datasets

\subsection{ Evaluation I. MARX-Simulated Dataset }

\noindent
Mono-energetic beam simulations are performed with MARX at various
energy values and concatenated together to make a L1 event file. 
The L1 file has gone through nominal CIAO processing, except that
pixel randomization in TG space being turned off at {\em{\tgres}}. 
Corresponding grating RMF files are generated via {\em{mkgrmf}} command 
with the new LSFPARM files and used as a fitting kernel in {\em{ISIS}} 
for evaluation. Cash statistics is applied throughout this work. 

\subsubsection{ Line Profile Analysis }

\noindent
If the LSFPARM products are valid, simulated line profiles can be 
described with a Kronecker delta function folded over grating RMFs. 
Figures 1 through 4 show several line profiles fitted with a delta
function. For brevity, selected four of 21 line samples are shown in 
each figure for inspection: lines at E = 800, 1400, 1800, and 6500 eV. 
These are representative of the results of other line profile fittings, 
resulting in the value of goodness of fit generally less than 
1.1 for HEG and 1.5 for MEG (these are not meant to be necessarily 
the best fitting results). 

\vspace{3mm}
\noindent
Despite unusually high S/N ratio in these line profile simulations, 
the model fitting has resulted in the low ${\chi}^{2}$ values and 
hence the results validate that the LSF parameterization (used earlier 
in recent LSF works) works here just as well as before. 


\subsubsection{Encircled Energy Fraction}

\noindent
The second key component of LSFPARM products is the value of 
encircled energy fraction as a function of energy. 
The value EEFRAC is derived from cross-dispersion line 
profile of a mono-energetic beam, integrated over arbitrary 
extraction width centered around the line peak (or in Chandra's
term, at TG\_D = 0 degree). If derived correctly, EEFRAC will
allow users to correct measured count rates (or fluxes) for 
the aperture of the extraction. 

\vspace{3mm}
\noindent
The following tests are performed such that the simulated line spectra 
(samples shown in Figures 1 -- 4) are extracted at five different aperture 
widths; then the delta-function model is fitted to 22 mono-energetic lines 
in order to derive model line fluxes $F_m$ in counts~s$^{-1}$. ISIS is also 
enabled to provide cumulative counts in each line, which is used to derive
a ``observed'' line flux $F_d$. In an ideal condition, both the model 
and observed line fluxes should be identical, i.e., 

$$ {F_d \over{F_m}} \approx 1. $$

\vspace{3mm}
\noindent
All four configurations (HEG and MEG $\pm$ 1st) are tested and verified
that the new LSFPARM products meet this criterion within a few percentage 
(see Tables 2 -- 5), presumably better than our knowledge in effective area 
of Chandra's HRMA system.  

\vspace{3mm}
\noindent
Note that the ratio value is consistently equal to or slightly higher 
than 1 for all configurations. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 4.3 Testing with Real Datasets

\subsection{ Evaluation II. Chandra Dataset }

\noindent
The same four LSFPARM products are tested against real Chandra 
observation (Capella, ObsID = 1103) in the similar manner as 
performed on the simulated datasets. 

\subsubsection{ Line Profile Analysis }

\noindent
Instead of using a delta function, thermal X-ray APED model and simple 
Gaussian function are fitted on the emission spectrum of Capella. 
Five to eight bright emission lines -- Si~XIII, Mg~XI and Mg~XII, 
Ne~X, Fe~XVII, O~VII and O~VIII, and N~VII -- are selected for testing. 
Upon fitting the parameter for line width $dv$ (converted to velocity 
scale in km~s$^{-1}$) is allowed to vary. Ideally the parameter $dv$
should be $\approx$ 0 km~s$^{-1}$ for thermal APED modeling; however,
for Gaussian fitting, the value $dv$ is expected to be $ge$ 0~km~s$^{-1}$
as most of these lines are either H-like doublet lines or blended with
other weak atomic lines. The results of fitting with the APED model 
is tabulated in Tables 6 -- 9; likewise, the results with the Gaussian 
model in Tables 10 -- 13. Corresponding figures for line profile fit
are shown in Figures 5 -- 8 for the APED model and in Figures 9 -- 12 
for the Gaussian model . These are, once again, not meant to be the best 
fitting parameters. 

\vspace{3mm}
\noindent
Upon inspection of Tables 6 -- 9, it is noted that the measured line
width is consistent to 0 km~s$^{-1}$ for the most of lines. A notable
exception is a strong Fe~XVII line $\lambda$ 15.014, which consistently 
shows positive measured widths in all HETG configurations. It is not 
clear if this is a failure in the LSFPARM products or is it a detection
of small line broadening in Capella's system. Moreover, it is possible
that the model is not adequate for this line. 

\vspace{3mm}
\noindent
The same practice with the Gaussian model is repeated and their results 
show some positive measured line widths, just as expected from blended 
lines. The positive line widths are more notable with HEG configuration, 
implying that some doublet lines -- separated by 0.0050\AA -- are marginally
resolved. To verify this point, I have explored the parameter space 
in Gaussian width and wavelength to see how significant the detection 
of non-zero line width is. Figure 13 illustrates that all H-like doublet
lines have the measurable widths of $\approx$ 0.003 -- 0.004\AA. 

\vspace{3mm}
\noindent
All of the line plots (Figures 5 -- 12) show a reasonably good fit 
to their spectral line profiles. Some lines are more susceptible to 
line blends which are not carefully dealt with in these exercise. 


\subsubsection{Encircled Energy Fraction}

\noindent
Lastly the model EEFRAC values need to be compared with the observed ones. 
Here we first compute the theoretical EEFRAC values at three different aperture 
widths based on the new LSFPARM products. Then at the same aperture widths
extract spectra from observed event datasets. Two targets are chosen for this
exercise: Capella\footnote{All the Capella datasets are combined for this
purpose.} and X-ray binary X1543-62. Since absolute spectral energy 
distributions of these targets are not known, the ratio of two entities 
are compared instead. Here in this exercise, the ratio of the observed 
spectra ``Extraction B'' (at a width equivalent to EEFRAC = 80% at E = 1.497keV) 
to ``Extraction A'' (similarly at EEFRAC = 98.5%, or nearly at CIAO default 
extraction width) is compared with the theoretical counterpart. The same
exercise is repeated for ``Extraction C'' (at EEFRAC = 60%) as well. 

\vspace{3mm}
\noindent
If the EEFRAC values in the LSFPARM products are derived properly, 
the ratio (as a function of wavelength) should be consistent to each other. 
Figure 14 shows the ratio plots for every HETG configuration derived 
with the Capella dataset. Figure 15 shows the same for X1543-62. 
As shown in these figures, the model trend agrees well within a few percentages
with the observed trend for all cases. Small deviation between the theoretical
and observed ratios is noted in the MEG configuration for X1543-62, although
the difference is only about 5%; and a number of possible sources for
uncertainties may explain such difference. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 5. 

\section{ Summary }

\noindent
All the testings show that the quality of these new LSFPARM products
for unrandomized datasets are equally as good as the earlier LSFPARM
products produced through the same LSF generation scheme. While some 
anomalous (but minute) differences are found in both line profile 
and EEFRAC analyses, the difference may originate due to some unidentified
calibration issues not included in the current MARX distribution. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
\begin{table}
  \begin{center}
    \begin{tabular}{lccccc}
      \multicolumn{6}{c}{Table~2. HEG +1st: the Data-to-Model flux ratio} \\
      \hline
      \hline
      \
Energy &         &        & Widths &        &        \\
(keV)  &         &        &(radian)&        &        \\
       & 2.84e-4 &4.19e-4 &5.35e-4 &7.09e-1 &1.41e-3 \\
      \hline
0.8    & 1.11 &1.10 &1.02 &1.01 &1.00 \\
1.0    & 1.00 &1.03 &1.00 &1.04 &1.02 \\
1.2    & 1.00 &1.01 &1.01 &1.02 &1.02 \\
1.4    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.6    & 1.02 &1.01 &1.01 &1.01 &1.01 \\
1.8    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
2.0    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
2.2    & 1.02 &1.02 &1.01 &1.01 &1.01 \\
2.4    & 1.03 &1.03 &1.03 &1.02 &1.02 \\
2.6    & 1.03 &1.02 &1.03 &1.03 &1.03 \\
2.8    & 1.03 &1.02 &1.02 &1.01 &1.02 \\
3.0    & 1.01 &1.02 &1.02 &1.00 &1.01 \\
3.2    & 1.02 &1.01 &1.01 &1.01 &1.02 \\
3.4    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
3.6    & 1.02 &1.01 &1.01 &1.01 &1.01 \\
3.8    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
4.0    & 1.03 &1.02 &1.02 &1.02 &1.02 \\
4.5    & 1.02 &1.01 &1.01 &1.01 &1.01 \\
5.0    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
6.0    & 1.02 &1.02 &1.03 &1.02 &1.03 \\
6.5    & 1.00 &1.02 &1.00 &1.00 &1.00 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}
  \begin{center}
    \begin{tabular}{lccccc}
      \multicolumn{6}{c}{Table~3. HEG -1st: the Data-to-Model flux ratio} \\
      \hline
      \hline
      \
Energy &         &        & Widths &        &        \\
(keV)  &         &        & (radian) &        &        \\
       & 2.84e-4 &4.19e-4 &5.35e-4 &7.09e-1 &1.41e-3 \\
      \hline
0.8    & 1.00 &1.03 &1.03 &1.03 &1.03 \\
1.0    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.2    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.4    & 1.02 &1.02 &1.01 &1.01 &1.01 \\
1.6    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.8    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
2.0    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
2.2    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
2.4    & 1.03 &1.03 &1.03 &1.03 &1.03 \\
2.6    & 1.03 &1.02 &1.02 &1.02 &1.02 \\
2.8    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
3.0    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
3.2    & 1.01 &1.01 &1.01 &1.01 &1.02 \\
3.4    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
3.6    & 1.02 &1.02 &1.02 &1.03 &1.02 \\
3.8    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
4.0    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
4.5    & 1.03 &1.02 &1.02 &1.02 &1.02 \\
5.0    & 1.02 &1.01 &1.01 &1.01 &1.04 \\
6.0    & 1.00 &1.01 &1.01 &1.01 &1.01 \\
6.5    & 1.00 &1.01 &1.00 &1.01 &1.00 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\clearpage

\begin{table}
  \begin{center}
    \begin{tabular}{lccccc}
      \multicolumn{6}{c}{Table~4. MEG +1st: the Data-to-Model flux ratio} \\
      \hline
      \hline
      \
Energy &         &        & Widths &        &        \\
(keV)  &         &        & (radian) &        &        \\
       & 2.84e-4 &4.19e-4 &5.35e-4 &7.09e-1 &1.41e-3 \\ 
      \hline
0.8    & 1.11 &1.08 &1.05 &1.05 &1.03 \\ 
1.0    & 1.01 &1.01 &1.01 &1.01 &1.00 \\ 
1.2    & 1.03 &1.02 &1.02 &1.02 &1.02 \\ 
1.4    & 1.02 &1.02 &1.02 &1.02 &1.02 \\ 
1.6    & 1.02 &1.01 &1.01 &1.01 &1.01 \\ 
1.8    & 1.02 &1.02 &1.02 &1.02 &1.02 \\ 
2.0    & 1.01 &1.01 &1.01 &1.01 &1.01 \\ 
2.2    & 1.02 &1.02 &1.02 &1.02 &1.02 \\ 
2.4    & 1.04 &1.03 &1.03 &1.03 &1.03 \\ 
2.6    & 1.04 &1.04 &1.03 &1.03 &1.03 \\ 
2.8    & 1.02 &1.02 &1.01 &1.01 &1.01 \\ 
3.0    & 1.02 &1.02 &1.01 &1.01 &1.01 \\ 
3.2    & 1.03 &1.02 &1.02 &1.02 &1.02 \\ 
3.4    & 1.01 &1.01 &1.01 &1.01 &1.01 \\ 
3.6    & 1.02 &1.01 &1.02 &1.02 &1.02 \\ 
3.8    & 1.02 &1.02 &1.02 &1.01 &1.01 \\ 
4.0    & 1.02 &1.02 &1.02 &1.02 &1.02 \\ 
4.5    & 1.02 &1.02 &1.02 &1.02 &1.02 \\ 
5.0    & 1.01 &1.01 &1.01 &1.01 &1.01 \\ 
6.0    & 1.03 &1.02 &1.02 &1.02 &1.02 \\ 
6.5    & 1.03 &1.03 &1.03 &1.03 &1.03 \\ 
      \hline
    \end{tabular}
  \end{center}
\end{table}

\begin{table}
  \begin{center}
    \begin{tabular}{lccccc}
      \multicolumn{6}{c}{Table~5. MEG -1st: the Data-to-Model flux ratio} \\
      \hline
      \hline
      \
Energy &         &        & Widths &        &        \\
(keV)  &         &        & (radian) &        &        \\
       & 2.84e-4 &4.19e-4 &5.35e-4 &7.09e-1 &1.41e-3  \\
      \hline
0.8    & 1.01 &1.02 &1.01 &1.01 &1.02 \\
1.0    & 1.02 &1.01 &1.01 &1.01 &1.01 \\
1.2    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.4    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.6    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
1.8    & 1.01 &1.01 &1.01 &1.01 &1.01 \\
2.0    & 1.02 &1.02 &1.01 &1.02 &1.01 \\
2.2    & 1.03 &1.02 &1.02 &1.02 &1.02 \\
2.4    & 1.03 &1.03 &1.02 &1.02 &1.02 \\
2.6    & 1.03 &1.03 &1.03 &1.03 &1.03 \\
2.8    & 1.02 &1.02 &1.03 &1.03 &1.02 \\
3.0    & 1.03 &1.03 &1.03 &1.03 &1.02 \\
3.2    & 1.03 &1.02 &1.02 &1.02 &1.02 \\
3.4    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
3.6    & 1.02 &1.01 &1.01 &1.02 &1.01 \\
3.8    & 1.03 &1.03 &1.03 &1.03 &1.03 \\
4.0    & 1.02 &1.02 &1.02 &1.02 &1.02 \\
4.5    & 1.04 &1.03 &1.03 &1.03 &1.03 \\
5.0    & 1.03 &1.03 &1.02 &1.03 &1.02 \\
6.0    & 1.09 &1.09 &1.08 &1.08 &1.07 \\
6.5    & 1.10 &1.04 &1.04 &1.04 &1.04 \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\clearpage

\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~6. Capella HEG +1st: thermal APED model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 0.56 & 0 -- 124      & 1.70        & He-like triplet lines \\
             Mg~XII & 8.4192   & 0.29 & 0 -- 125      & 1.69        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 0.06 & 0 -- 75.9     & 1.54        & He-like triplet lines \\
             Ne~X   & 12.1321  & 0.01 & 0 -- 128      & 1.52        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 54.7 & 20.8 -- 119   & 2.04        & Expected thermal width $\approx 38$km/s \\
      \hline
    \end{tabular}
  \end{center}
\end{table}


\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~7. Capella HEG -1st: thermal APED model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 1.03 & 0 -- 149      & 1.72        & He-like triplet lines \\
             Mg~XII & 8.4192   & 0.31 & 0 -- 143      & 1.23        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 78.0 & 0 -- 127      & 2.02        & He-like triplet lines \\
             Ne~X   & 12.1321  & 95.7 & 31.2 -- 172   & 1.16        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 76.0 & 52.9 -- 122   & 2.02        & Expected thermal width $\approx 38$km/s \\
      \hline
    \end{tabular}
  \end{center}
\end{table}



\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~8. Capella MEG +1st: thermal APED model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 17.2 & 0 -- 151      & 3.37        & He-like triplet lines \\
             Mg~XII & 8.4192   & 3.03 & 0 -- 124      & 1.24        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 0.0  & 0 -- 92.0     & 3.28        & He-like triplet lines  \\
             Ne~X   & 12.1321  & 0.08 & 0 -- 67.7     & 2.23        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 54.8 & 0 -- 121      & 4.48        & Poor modeling on the neighboring lines \\
             O~VIII & 18.9671  & 1.40 & 0 -- 90.3     & 1.86        & H-like doublet lines \\
             O~VII  & 21.6015  & 10.0 & 0 -- 206      & 0.801       & Weak line \\
             N~VII  & 24.7792  & 5.85 & 0 -- 212      & 0.953       & Weak line \\
      \hline
    \end{tabular}
  \end{center}
\end{table}



\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~9. Capella MEG -1st: thermal APED model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
       \hline
            Si~XIII& 6.6479   & 0.01 & 0 -- 120      & 2.45        & He-like triplet lines \\
             Mg~XII & 8.4192   & 0.0  & 0 -- 184      & 1.26        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 0.0  & 0 -- 47.4     & 2.59        & He-like triplet lines  \\
             Ne~X   & 12.1321  & 0.0  & 0 -- 139      & 2.23        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 93.8 & 71.5 -- 131   & 8.64        & Poor modeling on the neighboring lines \\
             O~VIII & 18.9671  & 50.5 & 0 -- 110      & 2.38        & H-like doublet lines \\
             O~VII  & 21.6015  & 72.1 & 0 -- 141      & 1.34        & Weak line \\
             N~VII  & 24.7792  & 2.56 & 0 -- 116      & 1.42        & Weak line \\
      \hline
    \end{tabular}
  \end{center}
\end{table}


\clearpage



\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~10. Capella HEG +1st: Simple Gaussian model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 87.6 & 0 -- 196      & 2.94        & blend \\
             Mg~XII & 8.4192   & 83.2 & 40.9 -- 165   & 1.81        & doublet lines blended \\
             Mg~XI  & 9.1687   & 41.7 & 0 -- 103      & 1.26        & He-like triplet lines \\
             Ne~X   & 12.1321  & 151  & 127 -- 181    & 1.80        & Strong Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 47.8 & 13.8 -- 76.7  & 1.92        & Expected thermal width $\approx 38$km/s \\
      \hline
    \end{tabular}
  \end{center}
\end{table}




\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~11. Capella HEG -1st: Simple Gaussian model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 110 & 78.7 -- 110    & 0.677       & blend \\
             Mg~XII & 8.4192   & 106 & 64.4 -- 158    & 1.32        & doublet lines blended \\
             Mg~XI  & 9.1687   & 104 & 77.7 -- 162    & 2.25        & He-like triplet lines \\
             Ne~X   & 12.1321  & 161 & 141  -- 202    & 1.58        & Strong Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 47.1& 16.6 -- 75.0   & 1.88        & Expected thermal width $\approx 38$km/s \\
      \hline
    \end{tabular}
  \end{center}
\end{table}



\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~12. Capella MEG +1st: Simple Gaussian model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 0.0  & 0 -- 0.01     & 2.26        & He-like triplet lines \\
             Mg~XII & 8.4192   & 30.0 & 0 -- 107      & 1.09        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 19.0 & 0 -- 93.6     & 0.972       & He-like triplet lines  \\
             Ne~X   & 12.1321  & 123  & 83.5 -- 157   & 2.21        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 29.2 & 1.12 -- 58.9  & 2.29        & Poor modeling on the neighboring lines \\
             O~VIII & 18.9671  & 47.8 & 0.285 -- 96.1 & 1.67        & H-like doublet lines \\
             O~VII  & 21.6015  & 5.42 & 0 -- 192      & 1.26        & Weak line \\
             N~VII  & 24.7792  & 35.8 & 0 -- 144      & 0.926       & Weak line \\
      \hline
    \end{tabular}
  \end{center}
\end{table}


\begin{table}
  \begin{center}
    \begin{tabular}{lccccl}
      \multicolumn{6}{c}{Table~13. Capella MEG -1st: Simple Gaussian model. } \\
      \hline
      \hline
      \
      Element& $\lambda_{lab}$ & $dv$   & $dv$ 3$\sigma$ range  & ${\chi}^{2}$ & Comment \\
             & (\AA)           & (km/s) &             &             &         \\
      \hline
             Si~XIII& 6.6479   & 0.92 & 0 -- 134      & 1.40        & He-like triplet lines \\
             Mg~XII & 8.4192   & 119  & 0 -- 196      & 1.47        & H-like doublet lines \\
             Mg~XI  & 9.1687   & 24.9 & 0 -- 75.0     & 1.07        & He-like triplet lines  \\
             Ne~X   & 12.1321  & 148  & 116 -- 173    & 1.53        & Ne~X and Fe XVII blend \\
             Fe~XVII& 15.0140  & 20.1 & 0.09 -- 56.6  & 3.75        & Poor modeling on the neighboring lines \\
             O~VIII & 18.9671  & 77.8 & 61.4 -- 111   & 2.06        & H-like doublet lines \\
             O~VII  & 21.6015  & 0.20 & 0 -- 91.8     & 4.03        & Weak line \\
             N~VII  & 24.7792  & 0.108& 0 -- 80.4     & 1.35        & Weak line \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\clearpage





\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hp1_800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hp1_1400ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hp1_1800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hp1_6500ev.ps}
\caption{HEG +1st-order line profiles, fitted with a Kronecker delta function (red) folded
over HEG +1st order grating RMF: (top right) line profile at E = 800~eV, with a goodness of fit 
${\chi}^2$ = 0.23; (top left) E = 1400~eV, ${\chi}^2$ = 0.63; (bottom right) E = 1800~eV,
${\chi}^2$ =1.0; and (bottom left) E = 6500~eV, ${\chi}^2$ = 0.88. The delta function model
describes simulated line profiles fairly well, which validates my scheme for LSF parameterization. 
\label{fig1}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hm1_800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hm1_1400ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hm1_1800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/heg_hp9800_hm1_6500ev.ps}
\caption{HEG -1st-order line profiles: (top right) line profile at E = 800~eV, 
with a goodness of fit ${\chi}^2$ = 0.30; (top left) E = 1400~eV, ${\chi}^2$ = 0.67; 
(bottom right) E = 1800~eV,${\chi}^2$ =1.0; and (bottom left) E = 6500~eV, ${\chi}^2$ = 1.0.
\label{fig2}}
\hfil
\end{figure}



\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mp1_800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mp1_1400ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mp1_1800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mp1_6500ev.ps}
\caption{MEG +1st-order line profiles: (top right) line profile at E = 800~eV, 
with a goodness of fit ${\chi}^2$ = 1.1; (top left) E = 1400~eV, ${\chi}^2$ = 1.5; 
(bottom right) E = 1800~eV,${\chi}^2$ =1.6; and (bottom left) E = 6500~eV, ${\chi}^2$ = 0.86.
In some cases the model fitting has resulted in goodness of fit slightly larger than unity. 
\label{fig3}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mm1_800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mm1_1400ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mm1_1800ev.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/SIML/meg_hp9800_mm1_6500ev.ps}
\caption{MEG -1st-order line profiles: (top right) line profile at E = 800~eV, 
with a goodness of fit ${\chi}^2$ = 1.1; (top left) E = 1400~eV, ${\chi}^2$ = 1.2; 
(bottom right) E = 1800~eV,${\chi}^2$ =1.7; and (bottom left) E = 6500~eV, ${\chi}^2$ = 0.79.
In some cases the model fitting has resulted in goodness of fit slightly larger than unity. 
\label{fig4}}
\hfil
\end{figure}

\clearpage

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_hp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_hp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_hp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_hp1.ps}
\caption{HEG +1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, fitted with an APED thermal model (red) and its goodness of fit ${\chi}^2$ = 1.69; (top left)
Mg XII H-like doublet line, with  ${\chi}^2$ = 1.69; (bottom right) Mg XI He-like triplet
lines, with ${\chi}^2$ = 1.54; and (bottom left) Fe XVII doublet line, with ${\chi}^2$ = 2.04.
This simplified APED model (not necessarily the best fit) can adequately describe the Capella
line spectra, though some neighboring lines are occasionally missing, which lead to higher
 ${\chi}^2$ values. 
\label{fig5}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_hm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_hm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_hm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_hm1.ps}
\caption{HEG -1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, with goodness of fit ${\chi}^2$ = 1.72; (top left)
Mg XII H-like doublet line, with  ${\chi}^2$ = 1.23; (bottom right) Mg XI He-like triplet
lines, with ${\chi}^2$ = 2.02; and (bottom left) Fe XVII doublet line, with ${\chi}^2$ = 2.02.
\label{fig6}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_mp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_mp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_mp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/nex_mp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_mp1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/oviii_mp1.ps}
\caption{MEG +1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, with goodness of fit ${\chi}^2$ = 3.37; (top left)
Mg XII H-like doublet line, with  ${\chi}^2$ = 1.24; (middle right) Mg XI He-like triplet
lines, with ${\chi}^2$ = 3.28; (middle left) Ne X and Fe XVII blend, with with ${\chi}^2$ = 2.23;
(bottom right) Fe XVII doublet line, with ${\chi}^2$ = 4.48; and (bottom left) O VIII line, with 
${\chi}^2$ = 1.86. The high  ${\chi}^2$ values results from a poorly defined APED thermal 
model (i.e., missing neighboring lines). 
\label{fig7}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_mm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_mm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_mm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/nex_mm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_mm1.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/oviii_mm1.ps}
\caption{MEG -1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, with goodness of fit ${\chi}^2$ = 2.45; (top left)
Mg XII H-like doublet line, with  ${\chi}^2$ = 1.26; (middle right) Mg XI He-like triplet
lines, with ${\chi}^2$ = 2.59; (middle left) Ne X and Fe XVII blend, with with ${\chi}^2$ = 2.23;
(bottom right) Fe XVII doublet line, with ${\chi}^2$ = 8.64; and (bottom right) O VIII line, 
with ${\chi}^2$ = 2.38. Again, the high  ${\chi}^2$ values results from a poorly defined APED thermal model.
\label{fig8}}
\hfil
\end{figure}






\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_hp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_hp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_hp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_hp1_gau.ps}
\caption{HEG +1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, fitted with a single Gaussian function (red; velocity width $dv \approx$ 88km~s$^{-1}$) 
and its goodness of fit ${\chi}^2$ = 2.94; (top left) Mg XII H-like doublet line 
($dv \approx$ 83km~s$^{-1}$), with  ${\chi}^2$ = 1.81; (bottom right) Mg XI He-like triplet
lines ($dv \approx$ 42km~s$^{-1}$), with ${\chi}^2$ = 1.26; and (bottom left) Fe XVII doublet 
line ($dv \approx$ 48km~s$^{-1}$), with ${\chi}^2$ = 1.92.
\label{fig9}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_hm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_hm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_hm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_hm1_gau.ps}
\caption{HEG -1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, fitted with a single Gaussian function (red; velocity width $dv \approx$ 110km~s$^{-1}$) 
and its goodness of fit ${\chi}^2$ = 0.68; (top left) Mg XII H-like doublet line 
($dv \approx$ 106km~s$^{-1}$), with  ${\chi}^2$ = 1.31; (bottom right) Mg XI He-like triplet
lines ($dv \approx$ 104km~s$^{-1}$), with ${\chi}^2$ = 2.25; and (bottom left) Fe XVII doublet 
line ($dv \approx$ 47km~s$^{-1}$), with ${\chi}^2$ = 1.88.
\label{fig10}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_mp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_mp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_mp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_mp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/oviii_mp1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/nvii_mp1_gau.ps}
\caption{MEG +1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, fitted with a single Gaussian function (red; velocity width $dv \approx$ 0km~s$^{-1}$) 
and its goodness of fit ${\chi}^2$ = 2.26; (top left) Mg XII H-like doublet line 
($dv \approx$ 30km~s$^{-1}$), with  ${\chi}^2$ = 1.09; (middle right) Mg XI He-like triplet
lines ($dv \approx$ 19km~s$^{-1}$), with ${\chi}^2$ = 0.97; (middle left) Fe XVII doublet 
line ($dv \approx$ 29km~s$^{-1}$), with ${\chi}^2$ = 2.29; (bottom right) O VIII line
($dv \approx$ 48km~s$^{-1}$), with ${\chi}^2$ = 1.67; and (bottom left) N VII line 
($dv \approx$ 36km~s$^{-1}$), with ${\chi}^2$ = 0.93.
\label{fig11}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/sixiii_mm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxi_mm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_mm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fevii_mm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/oviii_mm1_gau.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/nvii_mm1_gau.ps}
\caption{MEG -1st-order Capella line profiles; (top right) Si XIII He-like triplet
lines, fitted with a single Gaussian function (red; velocity width $dv \approx$ 0.92km~s$^{-1}$) 
and its goodness of fit ${\chi}^2$ = 1.40; (top left) Mg XII H-like doublet line 
($dv \approx$ 119km~s$^{-1}$), with  ${\chi}^2$ = 1.47; (middle right) Mg XI He-like triplet
lines ($dv \approx$ 25km~s$^{-1}$), with ${\chi}^2$ = 1.07; (middle left) Fe XVII doublet 
line ($dv \approx$ 20km~s$^{-1}$), with ${\chi}^2$ = 3.75; (bottom right) O VIII line
($dv \approx$ 78km~s$^{-1}$), with ${\chi}^2$ = 2.06; and (bottom left) N VII line 
($dv \approx$ 0.11km~s$^{-1}$), with ${\chi}^2$ = 1.35.
\label{fig12}}
\hfil
\end{figure}





\begin{figure}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_confcntr_heg.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/mgxii_confcntr_meg.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fexvii_confcntr_heg.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/fexvii_confcntr_meg.ps}
\includegraphics[keepaspectratio=true,angle=-90,width=8cm]{PS/REAL/CAPELLA/oviii_confcntr_meg.ps}
\caption{Confidence contour plots in Gaussian width vs. wavelength parameter space. Mg~XI, Fe~XVII, 
and O~VIII lines (doublet lines) are selected for this exercise. The solid contour curves are for 
+1st order spectrum; and the dashed curves for -1st order. Note that, due to some calibration 
issues involved in MEG configuration, the $\pm$1st contour levels do not overlap in the MEG datasets. 
\label{fig13}}
\hfil
\end{figure}





\begin{figure}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/CAPELLA/eef_ratio_val_heg1_80to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/CAPELLA/eef_ratio_val_heg1_60to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/CAPELLA/eef_ratio_val_meg1_80to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/CAPELLA/eef_ratio_val_meg1_60to985.ps}
\caption{Ratio plots of theoretical (blue) and Capella's observed (black) EEFRAC values: (top right) HEG $\pm$1st, 
the ratio of Extraction B to A; (top left) HEG $\pm$1st, the ratio of Extraction C to A; (bottom right)
MEG $\pm$1st, Extraction B to A; (bottom left) MEG $\pm$1st, Extraction C to A.
\label{fig14}}
\hfil
\end{figure}

\begin{figure}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/X1543/eef_ratio_val_heg1_80to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/X1543/eef_ratio_val_heg1_60to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/X1543/eef_ratio_val_meg1_80to985.ps}
\includegraphics[keepaspectratio=true,angle=90,width=9cm]{PS/REAL/X1543/eef_ratio_val_meg1_60to985.ps}
\caption{Ratio plots of theoretical (blue) and X1543-62's observed (black) EEFRAC values: (top right) 
HEG $\pm$1st, the ratio of Extraction B to A; (top left) HEG $\pm$1st, the ratio of Extraction C to A; (bottom right)
MEG $\pm$1st, Extraction B to A; (bottom left) MEG $\pm$1st, Extraction C to A.
\label{fig15}}
\hfil
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

