% draft_error_budget.tex

The LRF of the HETGS can be crudely yet usefully characterized
by the location and FWHM of the LRF core in both the dispersion and
cross-dispersion directions.  The ``resolving power'' of the spectrometer
is defined by:

\begin{equation}
{\rm ResolvingPower} = {\frac{E}{dE_{\rm FWHM}}} = \frac{y'}{dy'_{\rm FWHM}}
\end{equation}

The design of the HETG involved the use of an error budget to sum the
various contributions to the resolving power.  This error budget was
useful for studying the dependence of the resolving power
on the variation of individual error terms.  The error budget results
were verified by performing simplified ray-traces of single and
multiple facets.
%  The dotted curve of Figure~\ref{fig:res_power}
%is based on conservative assumptions and demonstrates the
%expectation that the HETG would meet its specifications
%(triangles).

The error budget presented in Table~\ref{tab:error_budget} includes
all of the important error terms for the flight HETGS resolving power;
a resulting range of ``realistic'' curves for the flight HETGS
resolving power is plotted as the solid and dashed curves in
Figure~\ref{fig:res_power}.

The error budget terms, their equations, and the current flight value
estimates are discussed in the sections below.

\begin{quotation}
{\it To-do}: \\
Describe the ``error budget ray-trace'' software.\\
\end{quotation}


\begin{figure}
\begin{center}
\epsfig{file=res_power.eps,height=18cm}
\caption[HETGS Resolving Power vs. Energy Curves]
{Resolving Power for the HEG and MEG spectra of the HETGS plotted
vs.~energy.  The resolving power at high
energies is dominated by the HRMA blur; at low energies the grating's
effects become important.\\ The solid and dashed curves are the
current range of realistic flight HETGS estimates.  The ``optimistic''
solid curve is calculated from the parameters shown in
Table~\ref{tab:error_budget} and related equations.  The dashed curve
is the same except for a (plausible) degradation in aspect, focus, and
grating $dp/p$ to: $a=0.6, dx=0.2, dp/p=250\times 10^{-6}$.  }
\label{fig:res_power}
\end{center}
\end{figure}


\begin{table} [p]
\begin{center}
\begin{tabular}{lcrlcc}
\hline
Error Term &  \multicolumn{3}{c}{\it Error Parameter}  & 
\multicolumn{2}{c}{Blur Equations (rms $\mu$m)} \\
           &  symbol & value & units & $dy'$,dispersion & $dz'$,cross-disp. \\
\hline
\hline
HRMA PSF & $\sigma_{\rm H}$ & Equ.~\ref{equ:hrma_sigma_meg}[\ref{equ:hrma_sigma_heg}] & rms mm &
$\sigma_{\rm H}$  & $\sigma_{\rm H}$ \\
Astigmatism & -- &  &  & Equ.~\ref{equ:astig_dy} & Equ.~\ref{equ:astig_dz} \\
Period var. & $dp/p$ & 162[146] & $\times 10^{-6}$ rms &
Equ.~\ref{equ:dpop_error} & -- \\
Roll var. & $\gamma$ & 1.5 & arc min. rms & -- & 
Equ.~\ref{equ:roll_error} \\
Aspect & $a$ & 0.34 & arc sec rms &
 Equ.~\ref{equ:aspect_error} & Equ.~\ref{equ:aspect_error} \\
Dither rate & $R_{\rm dither}$ & 0.1 & arc sec./s &
 Equ.~\ref{equ:dither_error} & Equ.~\ref{equ:dither_error} \\
Detector & $\sigma_D $ & $0.29\times 0.024$ & mm rms & 
 $\sigma_D $ &$\sigma_D $  \\
\hline
\hline
 & & & & & \\
\multicolumn{4}{c}{\it Input Parameters} & & \\
Energy & $E$ & as desired & keV & & \\
Defocus & $dx$ & 0.1 & mm & & \\
Readout time  & $t_{\rm D}$ & 3.3 & s & & \\
\hline
Period & $p$ & 4001.[2001.] & \AA & & \\
equiv Radius  & $R_0$ & 470.[330.] & mm & & \\
Rowland spacing  & $X_{\rm RS}$ & 8634. & mm & & \\
Focal length  &  $F$ & 10065.5 & mm & & \\
\hline
Rowland offset & $\Delta X_{\rm Rowland}$ & $\beta^2 X_{\rm RS}$ & mm &  & \\
Diffr. angle & $\beta$ & $\arcsin (\lambda /p)$ & radians &  & \\
Wavelength & $\lambda$ & hc/E & \AA &  & \\
hc & $hc$ & 12.3985 & \AA keV & \\
\hline
\end{tabular}
\caption[Simplified Resolving Power Error Budget Equations]
{Simplified Resolving Power Error Budget Equations.  The major
terms which add blur to the HETG PSF are listed here.  The
effective rms contributions to the dispersion and cross-dispersion
blur are rss'ed together to arrive at a Gaussian approximation
to the PSF core in each direction.  Note that the equations include
unusual factors which convert the values to 1-sigma equivalents,
{\it e.g.}, the rms blur added due to CCD-like pixelization
is 0.29 times the pixel size.  The parameter values listed
here and in the equations are the current best estimates for
the flight HETGS.  Entries that are different for
MEG and HEG are shown as ``MEG\_value[HEG\_value]''.}
\label{tab:error_budget}
\end{center}
\end{table}

\clearpage
\subsection{HRMA PSF}
\label{sec:hrma_psf}

The HRMA on-axis, in focus, 1-dimensional projected PSF
has a central cusp and extended wings.  The result is that the
FWHM of this 1-D profile is not a good indicator of the FWHM
that will result when convolved with other (generally closer to
Gaussian) error terms.  In order to parameterize the HRMA for
inclusion in this error budget is is useful to measure the
``effective HRMA rms blur'', $\sigma_H$.  Define $\sigma_T$ as the total
Gaussian sigma that results from convolving the HRMA 1-D PSF with
a Gaussian of width $\sigma_o$ :

\begin{equation}
  \sigma_T = {\rm sigma}\{~~{\rm HRMA~PSF} ~\otimes~ {\rm Gaussian}(\sigma_o)~~\}
\label{equ:hrma_sigma_t}
\end{equation}

then the effective HRMA rms blur is defined as:

\begin{equation}
  \sigma_H ~\equiv~ \sqrt{\sigma_T^2 - \sigma_o^2}
\label{equ:hrma_rms_blur}
\end{equation}

The effective HRMA blur thus depends on the scale at which it is used,
$\sigma_o$, which is set by the other, non-HRMA, error
terms in the error budget.

This procedure was carried out using the approximation to the flight
HRMA embodied in the \mx software, Version 2.20.  The following 
equations give good approximations to the value of $\sigma_H$
when $\sigma_o \approx 0.0107~mm$.

\begin{equation}
  \sigma_{H,MEG} = 0.00998 ~+~ 0.00014\log_{10}E ~+~ -0.00399\log_{10}^2E
~+~ 0.000505\log_{10}^3E
\label{equ:hrma_sigma_meg}
\end{equation}

\begin{equation}
  \sigma_{H,HEG} = 0.01134 ~+~ 0.00675\log_{10}E ~+~ -0.01426\log_{10}^2E
~+~ 0.01133\log_{10}^3E
\label{equ:hrma_sigma_heg}
\end{equation}

\begin{figure}
\begin{center}
\epsfig{file=marx_hrma_meg.eps,height=9cm}
\caption[Fitting the \mx MEG HRMA Effective Blur]
{Fitting the \mx MEG HRMA Effective Blur}
\label{fig:marx_hrma_meg}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=marx_hrma_heg.eps,height=9cm}
\caption[Fitting the \mx HEG HRMA Effective Blur]
{Fitting the \mx HEG HRMA Effective Blur}
\label{fig:marx_hrma_heg}
\end{center}
\end{figure}

\clearpage
\subsection{Rowland Astigmatism}

Including the effect of a defocus, $dx$, and a factor converting
the peak-to-peak blur into an rms equivalent, we get the following
equations for the Rowland astigmatism contribution to the error
budget:

\begin{equation}
  dy' = 0.354~ {\frac{2R_0}{X_{\rm RS}}} ~dx
\label{equ:astig_dy}
\end{equation}

\begin{equation}
  dz' = 0.354~ {\frac{2R_0}{X_{\rm RS}}} ~(\Delta X_{\rm Rowland} + dx)
\label{equ:astig_dz}
\end{equation}

These equations assume that the detector conforms to the Rowland
circle except for an overall translation by $dx$ (positive towards the
HRMA).  The values of $R_0$ used in the error budget are effective
values -- weighted combinations of the relevant mirror shells.

\subsection{Grating Period and Roll Variations}

There are two main error terms which depend on how well the HETG
is built:  i) period variations within and between facets (``$dp/p$'')
and ii) alignment (``roll'') variations about the normal to the facet
surface within and between facets.  The period variations lead to
an additional blur in the dispersion direction:

\begin{equation}
  dy' \approx \beta X_{\rm RS} {\frac{dp}{p}}
\label{equ:dpop_error}
\end{equation}

\noindent where $dy'$ and $dp/p$ are rms quantities. 
The roll errors lead to additional blur
in the cross-dispersion direction through the equation:

\begin{equation}
  dz' \approx \beta X_{\rm RS} ~\gamma ~({\frac{1}{57.3}})(\frac{1}{60})
\label{equ:roll_error}
\end{equation}

\noindent where $dz'$ and $\gamma$ are rms values in units of mm and 
arc minutes respectively.

\subsection{Aspect and Dither Blurs}

Aspect reconstruction adds a blur that is currently expected to be $a$ =
0.34 arc seconds rms diameter (Aldcroft memo 6/18/97).  The resulting
one-dimensional rms sigma is thus:

\begin{equation}
  dy', dz' = {\frac{1}{2}}{\frac{\sqrt{2}}{2}}~~F~~a~~({\frac{1}{57.3}})
({\frac{1}{3600}})
\label{equ:aspect_error}
\end{equation}

\noindent where $F$ is the HRMA focal length in mm.

An additional blur is added because the arrival time of a photon at
the detector is only known to an accuracy of the detector readout
time, $t_D$.  The product of this readout time and the aspect slew
rate (basically the dither rate) is the size of the aspect error interval.
The rms blur effect is 0.29 times this interval:

\begin{equation}
  dy', dz' = 0.29~~F~~R_{\rm dither}~~t_D~~({\frac{1}{57.3}})
({\frac{1}{3600}})
\label{equ:dither_error}
\end{equation}

\subsection{Detector Blur}

This error term is the spatial error introduced by the detector readout
scheme.

For a pixelated detector like ACIS we assume that the PSF drifts with
respect to the detector pixels and there is a uniform distribution of
the centroid location in pixel phase.  In this case the reported
location of an event is the center of the pixel when in fact the event
may have actually arrived $\pm 0.5$ pixel from the center.  The rms
value of such a uniform distribution is 0.29 times the pixel width:

\begin{equation}
\sigma_D = 0.29\times{\rm pixel~size}
\label{equ:sigma_det}
\end{equation}

For a detector like the HRC, the value $\sigma_D$ is simply
the sigma of the gaussian blur added by the detector.