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\begin{document}

\title { {\sc Coordinate Systems for Analysis of On-Orbit Chandra Data
\\ Paper I: Imaging}}

\author{\sc Jonathan McDowell \\ Harvard-Smithsonian Center for Astrophysics}

\date{2001 May 13}
%\date{\today}

\maketitle

\section{Introduction}

The Chandra X-ray Observatory, launched in July 1999, provides
X-ray imaging and spectral data of unprecedented, subarcsecond, resolution.
The geometry of Chandra's detectors is more complicated than
those of previous missions; this, coupled with the higher
accuracy requirements, means that more care must be taken
to derive accurate coordinates and to distinguish clearly between
different coordinate systems.

This revision includes the values current in the processing
system as of that date. Revised values are currently being evaluated
by the CXC calibration group.


There are important operational differences between imaging performed by
X-ray telescopes and those performed by optical telescopes, even
spaceborne ones such as the Hubble Space Telescope. An optical telescope
will typically track its target, and accumulate counts in an image. Any
variation in the telescope's pointing direction will degrade the final
image quality. In contrast, an X-ray telescope's pointing direction
varies slightly with time, deliberately smearing the target source's
image over different parts of the detector. The image is then
reconstructed by  adjusting the positions of each photon by taking into
account the exact pointing direction of the spacecraft at the photon's
arrival time (``aspect correction''). This image reconstruction is
possible because the fundamental X-ray data product is not an image but
an `event list' in which the positions and arrival times of individual
photons are recorded. 

In the first part of this paper I discuss the general problem of
obtaining celestial coordinates from the telemetered data.  In the
second section, I define the coordinate systems precisely, presenting
the transformation equations used to convert between them.
The final part of the paper gives specific values for in-flight
data from the Chandra Observatory.

\subsection{Fundamentals}

The Chandra X-ray Observatory has a single Wolter I imaging telescope,
the HRMA (High Resolution Mirror Array) with four nested mirror pairs.
The HRMA focusses X-rays onto a detector in the ISIM (Integrated Science
Instrument Module); one of two transmission grating assemblies may be
placed in the beam to disperse the image for spectroscopic studies.
In this paper I consider only non-dispersed images.
There are two focal plane instruments in the ISIM, the ACIS (Advanced
CCD Imaging Spectrometer) and the HRC (High Resolution Camera). The HRC
is made up of two microchannel plane detectors, the imaging HRC-I and
the long HRC-S array used mostly as a spectroscopy detector.  ACIS also
has imaging and spectroscopic arrays, but for the purposes of data
analysis we consider ACIS to be a single detector, since unlike HRC its
chips may be read out in any combination.  The detectors are mounted on
a movable optical bench, the SIM (Science Instrument Module). The SIM
can move in both the focus (X) direction and along the axis (Z)
separating the detectors; this axis is perpendicular to the dispersion
axis (Y). In initial operations there are four standard Z positions for
the SIM, but non-standard positions may also be commanded.

In the idealization adopted in this paper, each of the three detectors
HRC-S, HRC-I and ACIS consists of a number of finite rectangular
detector planes or `logical chips' (the correct term for HRC is `segment' but we
will use the term chip generically to refer to a semiconductor chip, a
microchannel plate segment, a proportional counter,  a photographic
plate or any other finite rectanglular plane). In Chandra, these chips
are oriented approximately along the best focal surface, in contrast to
previous X-ray telescopes in which the chips were normal to the optical
axis. The chips are considered to consist of a rectangular array of
identical rectangular (in practice square) chip pixels. For ACIS, these
chip pixels are the physical pixels of the CCD, while for HRC they
represent a conventional, artificial pixelization employed by the
instrument's position calculation algorithm. In this paper we take the
chip coordinates as given and assume all nonlinearities and detector
effects have already been removed. Typically the chip coordinates are
integers, but this is not required (grade-based event finding algorithms
could give fractional chip positions even for the physically real pixels
of ACIS). The logical chip plane may be larger than the active
area of the detector (for the ROSAT PSPC, the logical plane is a rectangle
containg the actual detector, which is circular).


\subsection{The chip, detector and sky coordinate systems}

There are then three fundamental coordinate systems which must be
considered in analysing such an event list. In the `chip' coordinate
system, we consider the position of the photon in the chip plane,
corrected for all detector effects. In the `focal plane' or `detector'
space, the position of the photon in the tangent plane to the optical axis is
considered; and in the `sky' coordinate system, we record the position
of the photon in the fictitious tangent plane to a nominal fixed
celestial pointing direction. These three systems are useful for
considering different effects: in the chip system, stars will be smeared
out but bad pixels and chip boundaries will be well defined, while in
the sky system stars will be pointlike and bad pixels will be smeared
out. The position of the photons in the detector system tells you the
off-axis angle, allowing you to calculate mirror-based effects such as
point spread function and vignetting. The name `detector' for this
coordinate system is a misnomer, but is retained for compatibility with
earlier instruments such as ROSAT in which chip and detector coordinates
were identical.

In principle we can distinguish between the focal plane system (tangent
plane to the optical axis at the detector end of the telescope) and an
incoming tangent plane system (tangent plane at the sky end of the
mirrors); these differ because the mirror is not a perfect point lens.
We choose instead to identify the incoming tangent plane and focal plane
systems, and handle mirror effects by using the point spread function
formalism. In particular, any variations in angular deflection across
the field are assumed to be absorbed in an offset of the point spread
function's origin relative to its centroid. In practice for Chandra ray
trace simulations demonstrate that this effect can be neglected. For the
calculation of coordinates, the fact that the telescope consists of
multiple optical assemblies with different optical axes and focal
lengths is entirely ignored; we adopt a single nominal optical axis and
plate scale, and absorb the complexities of the actual optical system
into the point spread function. 


To convert from chip to detector coordinates, we must correct for the
position of the chips in three-dimensional space on the SIM (optical
bench), for the translation position of the SIM  in X and Z, and for any
misalignment of the SIM Z axis with the aspect camera axis. We also
allow six degrees of freedom for time-dependent warping of the telescope
structure due to thermal effects, moving the SIM frame relative to the
mirror axes. In practice, only three of these degrees of freedom are
measured (and contribute motions of the order of 1 arcsecond during an
observation) and the remaining three are assumed to have a negligible
effect. We must also know the origins of the instrument coordinate
systems accurately with respect to the telescope optical axis; there are
still significant uncertainties in these values, which will be
improved with continuing analysis of on-orbit data.

To convert from detector to sky coordinates, we simply apply the time
dependent `aspect solution', that is, the RA and Dec of the optical axis
and the roll angle of the spacecraft axes with respect to the celestial
meridian versus time. Applying the aspect solution involves
reprojecting the photons from the tangent plane of the instantaneous
pointing direction to the tangent plane of the nominal pointing
direction. In previous missions the calculation has been simplified by
using fixed translational offsets for the whole field, and this
approximation is also used in early Chandra data processing. However it
is not quite good enough for full accuracy, and will in future be
replaced by the full calculation.

The Chandra observatory is `dithered' with a deliberate periodic
Lissajous oscillation tracing out an approximately square region
on the sky with an amplitude of 15 arcseconds. A single star tracker
camera is used to determine the pointing direction as a function of time.
The offset between the star tracker and the telescope optical axis
must be calibrated with on-orbit observations. This procedure is discussed
in Paper 2.



\begin{figure}[h]
\framebox{\rotatebox{0}{\includegraphics*[height=3.0in]{det.ps} }}
\caption{An image in detector coordinates, showing a dithered source.}
\end{figure}

\begin{figure}[h]
\framebox{\rotatebox{0}{\includegraphics*[height=3.0in]{sky.ps} }}
\caption{The same image in sky coordinates, with aspect applied.
The source is PKS 0637-752, a quasar with a newly discovered X-ray jet.}
\end{figure}

\begin{figure}[h]
\framebox{\rotatebox{0}{\includegraphics*[height=3.0in]{tdx.ps} }}
\caption{The same image showing DETX versus time along the x axis}
\end{figure}

\begin{figure}[h]
\framebox{\rotatebox{0}{\includegraphics*[height=3.0in]{tx.ps} }}
\caption{The same image showing the X coordinate on the y axis
versus time along the x axis. The jet is visible at larger X values.}
\end{figure}


\clearpage

Because of the tilting of the individual chips and the dithering of the
optical axis, and the possibility of changing focal length, individual
physical pixels in chip space do not correspond directly to pixels in
detector and sky space, and we are free to choose entirely different
units for these coordinate frames. We retain the concept of `pixels'
since it is convenient and conventional to have a standard pixelization
for  image analysis. However, while the sizes of chip pixels are defined
in millimeters, the sizes of detector and sky pixels are defined to be a
fixed number of arcseconds. In practice we adopt conventional pixel
sizes for the detector and sky coordinate systems which have the same
size as chip pixels when the detector is at the nominal focus. In the
event lists, the detector and sky coordinate values are retained as
floating point values in pixel units. To avoid aliasing effects when
making images, the calculation of detector coordinates in standard
processing introduces a uniform randomization of $\pm0.5$ pixel in the
integral chip coordinate value, representing the fact that uniformity is
a reasonable prior for the probability distribution  of the exact
location the photon fell within the chip pixel.

\begin{figure}[h]
\framebox{\rotatebox{-90}{\includegraphics*[height=4.0in]{ps/adass.eps} }}
\caption{The different pixel plane coordinate systems. Distorting
effects are highly exaggerated.}
\end{figure}






\begin{figure}[h]
\rotatebox{-90}{\includegraphics*[height=3.0in]{ps/sky.eps} }
\caption{Imaging the sky in LSI coordinates}
\end{figure}



In Chandra FITS files, the chip, detector and sky pixel
systems are stored in the (CHIPX,CHIPY), (DETX,DETY) and (X,Y)
columns respectively. For a given detector, there may be more
than one chip, each with its own chip coordinate system.
We therefore also define a `tiled detector coordinate system',
(TDETX,TDETY) which combines chip coordinate and chip number
tto provide a single system covering all the chips. Unlike DET,
the TDET system does not retain the true 3D spatial relationship
between the chips, but does retain the identity of indiviudal
chip pixels. It may therefore be an appropiate system to use
when filtering out bad pixels.

\subsection{Pixel convention}

In all cases where we use discrete {\bf pixel numbers}, 
the corresponding real-valued, continuous
{\bf pixel coordinates} are defined to be equal
to the pixel number at the center of the pixel. We further recommend
that for finite detector planes, one corner be designated
as the lower left corner, LL. Then the pixel which has LL as one
of its corners (i.e. the lower left pixel) shall be numbered
(1,1) so that its center has coordinates (1.0, 1.0). The
coordinates of the LL point itself are (0.5, 0.5). If the
detector is rectangular with sides of length XMAX, YMAX the pixel coordinates
then run from (0.5, 0.5) in the lower left corner (LL) to (XMAX+0.5, YMAX+0.5)
in the upper right corner (UR) while the pixel numbers in each axis
run from 1 to XMAX, 1 to YMAX. 
These conventions are consistent with established practice in archival
X-ray FITS event lists.

\begin{figure}[ht]
\centerline{\rotatebox{-90}{\includegraphics*[height=2.8in]{ps/pixel.eps} }}
\caption{Pixel convention.}
\end{figure}


\section{Transformation equations and intermediate coordinate systems}


\subsection{Sky coordinates to celestial coordinates}

The sky coordinate system X,Y is defined by a reference pixel X0,Y0,
the world coordinates at that reference point $\alpha_0,\delta_0$ and
their first derivatives at that point $\Delta_X,\Delta_Y$. We choose square
pixels so that
\[ -\Delta_X  = + \Delta_Y = \Delta \]
The minus sign on the X derivative arises because we are on the inside
of the celestial sphere looking out, so that right ascension increases
to the left, which is -X on the image.

To convert from tangent plane pixels X,Y to actual celestial
coordinates $\alpha,\delta$, first calculate scaled coordinates
(note that $\Delta$ must be expressed in radians per pixel here)
 \[  \cvecp{x}{y} = \cvecp{ -\Delta ( X - X0 ) }{ \Delta (Y-Y0) } \]
and then the local angular coordinates.
 \[  \phi = \arg( x, -y ) \]
and 
 \[  \theta = \arg( 1.0, \sqrt{ x^2 + y^2 } ) \]
Finally convert local angular coordinates to world coordinates
by rotating through Euler angles $(\alpha_0, \pi/2 - \delta_0, \pi )$
as follows:
\[
 \alpha = \alpha_0 + \arg( -\cos\theta\sin\phi, \sin\theta\cos\delta_0 - \cos\theta\sin\delta_0\cos\phi)
\]
and
\[
 \sin\delta = \sin \theta * \sin \delta_0 + \cos\theta \cos\delta_0 \cos\phi 
\]

The inverse calculation:
Rotate the world coordinates forward through the same Euler angles:
\[
\phi = \pi + \arg( -\cos\delta \sin(\alpha-\alpha_0), \sin\delta\cos\delta_0 - \cos\delta\sin\delta_0\cos(\alpha-\alpha_0) )
\] and \[
\sin\theta = \sin\delta \cos\delta_0 + \cos\delta\sin\delta_0\cos(\alpha-\alpha_0)
\]
Then let 
\[
\cvecp{x}{y} = \cot\theta \cvecp{ \sin\phi }{ -\cos\phi }
\]
and
\[
\cvecp{X}{Y} = \cvecp{ X0 - x / \Delta }{ Y0 + y / \Delta }
\]


\subsection{Detector coordinates to celestial coordinates}

The procedure for mapping detector to celestial coordinates is
the same with the addition of a rotation through the spacecraft
roll angle $\gamma$. In this case $\alpha_0,\delta_0$ are the coordinates
of the instantaneous pointing direction.




To convert from tangent plane pixels DETX,DETY to actual celestial
coordinates $\alpha,\delta$, first calculate scaled coordinates
(note that $\Delta$ must be expressed in radians per pixel here)
 \[  \cvecp{x}{y} = \Delta 
\cvecp{-( X - X0 )\cos\gamma - (Y-Y0)\sin\gamma }{
-(X-X0)\sin\gamma +(Y-Y0)\cos\gamma } \]
and then the local angular coordinates.
 \[  \phi = \arg( x, -y ) \]
and 
 \[  \theta = \arg( 1.0, \sqrt{ x^2 + y^2 } ) \]
Finally convert local angular coordinates to world coordinates
by rotating through Euler angles $(\alpha_0, \pi/2 - \delta_0, \pi )$
as follows:
\[
 \alpha = \alpha_0 + \arg( -\cos\theta\sin\phi, \sin\theta\cos\delta_0 - \cos\theta\sin\delta_0\cos\phi)
\]
and
\[
 \sin\delta = \sin \theta * \sin \delta_0 + \cos\theta \cos\delta_0 \cos\phi 
\]

\subsection{Chip coordinates to detector coordinates}

Recall that chip pixel coordinates run from (0.5,0.5) to (XMAX+0.5,YMAX+0.5)
on the chip.
We introduce several intermediate coordinate systems in three dimensions
(and specified to be evaluated in millimeters)
\begin{enumerate}
\item the Chip Physical Coordinate (CPC) system, whose Z axis 
is normal to the chip, and whose origin coincides with chip coordinates
(0.5,0.5).
\item  the Local Science Instrument (LSI) system whose
origin is at a conventional point on the detector and whose
axes are approximately aligned with the spacecraft axes.

The four corners of each chip are specified in three dimensions
in the LSI frame. The CPC to LSI rotation matrix is made up
from the unit vectors in LSI coordinates of the CPC axes.

Let us denote the position vectors of the four chip corners
as \mb{LL},\mb{UL},\mb{UR},\mb{LR}. Then the unit vectors of the CPC
origin and axes in LSI coordinates are
\be \mb{p_0} = \mb{LL},
 \mb{e}_X = \mb{LR}-\mb{LL}, \mb{e}_Y = \mb{UL}-\mb{LL} \ee
and 
\be \mb{e}_Z = \mb{e}_X \wedge \mb{e}_Y. \ee

\begin{figure}[h]
\rotatebox{-90}{\includegraphics*[height=3.0in]{ps/xrcf14.eps} }
\caption{The relationship between CHIP, LSI and STT coordinates.}
\end{figure}

\begin{figure}[h]
\framebox{\rotatebox{-90}{\includegraphics*[height=4.0in]{ps/adass.fig3.eps} }}
\caption{The instrument compartment (dashed line) may be misaligned
with the telescope mirrors. At calibration, this misalignment 
(highly exaggerated here) may
be significant and variable as the mirror is tilted with respect
to the instruments. The instrument bench moves with respect to
the instrument compartment, as indicated by the arrow.}
\end{figure}



\begin{figure}[h]
\rotatebox{-90}{\includegraphics*[height=3.0in]{ps/xrcf13.eps} }
\caption[The relationship between STT and STF coordinates.]
{The relationship between STT and STF coordinates, AXAF example.
The instrument (SIM) table has moved so that HRC is at the focus.}
\end{figure}


\[
\cvect{X_{LSI}}{Y_{LSI}}{Z_{LSI}} = 
\cvect{X_{LL}}{Y_{LL}}{Z_{LL}}
\left( \begin{array}{lll}
(\mb{e_X})_X & (\mb{e_Y})_X  & (\mb{e_Z})_X \\
(\mb{e_X})_Y & (\mb{e_Y})_Y  & (\mb{e_Z})_Y \\
(\mb{e_X})_Z & (\mb{e_Y})_Z  & (\mb{e_Z})_Z \\
\end{array}\right)
\cvect{X_{CPC}}{Y_{CPC}}{Z_{CPC}}
\]

\item The  LSI coordinate system must now be related to the
SIM Translation Table (STT) and  SIM Translation Frame (STF) coordinate
system, which respectively take into account the position of the
instrument on the optical bench and the current position to which the
bench has been moved. The STT coordinate system is fixed in the optical
bench, while the STF coordinate system is fixed in the ISIM module. We
record the position of each instrument's LSI origin OLSI on the optical
bench by giving the position of OLSI in STT coordinates. We also
allow for a possible angular misalignment of the LSI system through
a small angle $\lambda_i$ by defining `corrected LSI coordinates'
\[
\cvect{X'_{LSI}}{Y'_{LSI}}{Z'_{LSI}} = 
\left( \begin{array}{lll}
1  & 0 & 0 \\
0  & \cos\lambda_i & -\sin\lambda_i \\
0  & \sin\lambda_i &  \cos\lambda_i \\
\end{array}\right)
\cvect{X_{LSI}}{Y_{LSI}}{Z_{LSI}} 
\]
chosen so that the corrected LSI axes of all the detectors line up.
In practice we plan to recalculate the chip corners in this
corrected LSI space once the alignment angles are known, and omit
use of the `raw' LSI values.

The coordinates of a pixel in STF coordinates are obtained by adding the
instrument origin coordinates OLSI and the current SIM translation
position to the LSI coordinates. OLSI also gives the STF coordinates of
the instrument's LSI origin when the SIM(X,Y,Z) value is zero; in other
words, the origins of STT and STF are defined to coincide when the SIM
is at its zero position. 

\[ \cvect{X_{STF}}{Y_{STF}}{Z_{STF}} =
\cvect{X'_{LSI}}{Y'_{LSI}}{Z'_{LSI}} +  \cvect{SIM_X}{SIM_Y}{SIM_Z} +
\cvect{OLSI_X}{OLSI_Y}{OLSI_Z}
\]

\item The focal coordinate (FC) system is fixed in the HRMA and
centered on the telescope's nominal focus. The origin of FC and STF
coordinates is nominally the same, but time-dependent shifts in
the spacecraft structure require corrections:

\[
\cvect{FCX}{FCY}{FCZ} = 
\cvect{dX_{STF}}{dY_{STF}}{dZ_{STF}} 
R_E(\psi_X,\psi_Y,\psi_Z)\cvect{X_{STF}}{Y_{STF}}{Z_{STF}} 
\]
where $R_E$ is a rotation matrix. The angles $\psi_Y$ and $\psi_Z$
and the focal shift $dX$ have only a tiny effect on the
celestial coordinates, and in the case of Chandra they are both not measured
and neglected. The angle $\psi_X$ (an additional roll of the detector
module relative to the mirrors) and the translational shifts dY, dZ
are measured by observations of the fiducial lights mounted on the 
instruments. In the case of Chandra, they are reported on a time
resolution of about one second and have a typical magnitude of
a few tenths of an arcsecond.

\item We now correct to the nominal mirror node, adopting
Mirror Nodal Coordinates (MNC)
\[
\cvect{ MNX}{MNY}{MNZ} = \cvect{FCX-f}{FCY}{FCZ}
\]
where f is the nominal focal length. Chandra has four mirror
shells, each of which has a different optical node and focal length;
we adopt a nominal mean node and focal length and absorb deviations
from this into the energy and position dependent mirror point spread function.
Note that MNX is large and negative for a position near the focus.

\item finally, the detector (focal plane pixel) coordinates
are calculated from the focal coordinates by
\[
\cvecp{ DETX}{DETY} = \cvecp{X0 - \Delta MNY/MNX}{Y0+ \Delta MNZ/MNX}
\]

\end{enumerate}

\subsection{Detector coordinates and off-axis angles}

The optical axis of the telescope is defined to lie at
the reference pixel X0, Y0 in detector coordinates, chosen
to be the center of the image. (However, users are warned
that in early data processing the position of the optical axis is not 
known, and so coordinates in this system will be revised).

We define Mirror Spherical Coordinates $(r,\theta_H, \phi_H)$ in
terms of mirror nodal Cartesian coordinates as follows:
 \be  \vect{X_N}{Y_N}{Z_N} = \vect{
 r cos\theta_H }{ r\sin\theta_H\cos\phi_H}{ r\sin\theta_H\sin\phi_H}  \ee 
The angle $\theta_H$ is the {\bf MSC off-axis angle} and $\phi_H$
is the {\bf MSC Azimuth}.
The inverse is
\begin{eqnarray} 
r &=& \sqrt{X_N^2+Y_N^2+Z_N^2}\\
\theta_H &=& \cos^{-1} (X_N/r) \nonumber \\
\phi_H   &=& \arg( Y_N, Z_N )\nonumber
\end{eqnarray}

\subsection{Dithered-corrected chip (DC) and chip-aligned source-centered sky (CSC) coordinates}


The systems described in this subsection have
not yet been implemented in the CXC data processing system.

In ACIS `continuous clocking' mode, only the CHIPX coordinate
is available; CHIPY is unknown. In fact, if $t_r$ is the
time tag at readout and $t$ is the actual event time,
\[
 t_r = t + CHIPY * dt
\]
where dt is the time to clock out one row. To get accurate
times and positions we must know the actual sky position of the
source. The best we can do is to obtain a 1-dimensional profile
of the source perpendicular to the readout direction.
Because the ACIS-I and ACIS-S chips are at a variety
of orientations, we can't define a single 1-D coordinate in the
sky frame which is well defined over multiple chips. Instead,
we define new coordinates, the dither corrected chip system and
the 1-D sky system. It is not possible to obtain corrected times
and positions for photons not associated with a known source.

Let A(t) be the aspect solution at time t (more precisely,
the operator mapping chip to sky coordinates, which also includes
the time dependent boresight and the SIM information).
Define the reference aspect solution $A_0$ to be the nominal
aspect solution (actually, you can use the time averaged
aspect solution 
the aspect solution at some specific time instead;
the time average would make the result more stable and would center
the resulting image, but is more work
to calculate). 

Let $(x_s,y_s)$
be the sky coordinates of a source. Then the chip coordinates
of the source as a function of time are
\[ ( cx_{ref}(t), cy_{ref}(t) ) = A^{-1}(t) (x_s, y_s, E(cx)) \]
where E(cx) reflects the dependence on predicted chip position
of the photon energy for grating data.
The reference chip coordinates are
\[ ( cx_s, cy_s ) = A0^{-1} (x_s, y_s) \]
We define the dither corrected chip coordinates for an event detected
at cx, cy to be
\[  (dcx,dcy) = (cx,cy) + (cx_s,cy_s) - (cx_{ref}(t),cy_{ref}(t)) \]
This definition is meaningful for both TE and CC mode data for 
chips with the same orientation as the chip on which the source lies.
It may be evaluated for other chips but the result will not be
useful. In CC mode data, only the dcx coordinate can be evaluated,
and we assume cy is at the expected position for the 
source $cy = cy_{ref}(t)$ so that $dcy = cy_s$.

The disadvantage of the dither corrected chip coordinates is that
the sky angular positions are not preserved accurately. 
We define a source-centered, readout oriented, coordinate system,
chip-aligned source sky coordinates (CSCX,CSCY) as follows.
The positive CSCX axis lies in the direction from $(x_s,y_s)$ to
the point $(x_s+dx_s,x_s+dy_s)$ obtained by calculating 
the sky coordinates 
\[ (x,y) = A0 (cx_s+\Delta,cy_s) \]
where $\Delta$ is an arbitrary positive increment of CHIPX. In words,
we take the reference aspect and use it to project a line from
the source parallel to CHIPX onto the sky plane.  The source position
is (CSCX0,CSCY0), chosen to have the same values as the reference
position of the sky coordinate system (e.g. (4096.5,4096.5) for ACIS).

Then for each event, 
we apply the aspect solution to obtain sky coordinates
\[ (x,y) = A(t) (cx,cy) \]
and calculate (CSCX,CSCY) with the appropriate rotation matrix.
For CC mode and events on the target chip, CSCY is zero and
\[
CSCX = CSCX0 + \sqrt{ ( x - x_s )^2 + (y - y_s )^2}
\]

The (CSCX,CSCY) system is useful for a single
chip (or for a set of chips which have exactly aligned readout directions,
such as the ACIS-S array) and for a constant roll angle.

So in processing CC mode data, we:
\begin{itemize}
\item Calculate the reference aspect 
\item Calculate the sky position of the target 
\item Calculate the predicted chip position of the target at the reference aspect,
 $(cx_s,cy_s)$
\item For each event at observed raw time $t_r$, chip id $c$ and chip position (cx,cy), 
calculate the predicted source chip position
\[ ( cx_{ref}(t), cy_{ref}(t) ) = A^{-1}(t) (x_s, y_s) \]
and the dither correction 
$(\Delta cx,\Delta cy) = (cx_{ref}(t),cy_{ref}(t)) - (cx_s,cy_s)$
for the chip $c$, and calculate (dcx,dcy) if required.
\item Calculate the corrected time assuming the photon is from the source
\[ DCTIME = TIME - dt * cy_{ref}(t) \]
Recall there's no way of knowing the correct time for non-source photons.
\item Optionally, recalculate $cy_{ref}(t), \Delta cx, \Delta cy$ and then (dcx,dcy),
DCTIME using the aspect for DCTIME instead of TIME (necessary if the aspect
solution changes measurably on a timescale DCTIME-TIME; one iteration
should be sufficient).
\item Calculate the predicted sky coordinates (x,y) from the observed
cx and the calculated $cy_{ref}(t)$
\item Calculate the final (CSCX,CSCY).
\end{itemize}

For grating data in CC mode, a more careful calculation should
be done of the predicted dispersed photon chip Y coordinate as a function
of chip X. We replace the equation

\subsection{Source-centered coordinates}

Another useful coordinate system is source-centered coordinates (SCX,SCY).
Similar to CSCX,CSCY, the source-centered coordinates share the pixel
size and reference position of the sky coordinate system, but
are centered on a particular source and aligned with a specified source
axis direction. The celestial coordinates of the source and direction
of its axis may be specified as a function of time; the system is particularly
intended for use with solar system targets. A world coordinate system
giving position angle and angular distance from source center should
be applied to the source-centered coordinates.

This system has not yet been implemented in the CXC data processing system.

\clearpage

\section{In-flight numerical values for Chandra}


\subsection{ACIS and HRC chip locations}

The ACIS instrument has 10 CCD chips.
In the event list data, each is identified by an integer from 0 to 9.
Four of the chips, the imaging set, are arranged in a rough square (but individually
tilted). Six are the spectroscopic array, arranged in a line.

\begin{longtable}{ll}
Chip Name & CHIP ID\\
\hline
ACIS-I0   &  0\\
ACIS-I1   &  1\\
ACIS-I2   & 2 \\
ACIS-I3   & 3\\
ACIS-S0   & 4\\
ACIS-S1   &  5\\
ACIS-S2   & 6\\
ACIS-S3   & 7\\
ACIS-S4   & 8\\
ACIS-S5   & 9\\
\end{longtable}

The CHIP ID is also called CCD ID for consistency with ASCA.

\newpage
{\small
\begin{longtable}{llll}
\caption{ACIS Chip corner locations in ACIS-I LSI coordinates}\\
\hline
Chip & Corner&  CPC coords & ACIS-I LSI coords \\
\hline
I0  &LL & (0.0, 0.0, 0.0)   &(2.361, -26.484, 23.088)\\
    &LR & (24.58, 0.0, 0.0)& (1.130, -26.546, -1.458)\\
    &UR & (24.58, 24.58, 0.0) &(-0.100, -2.001, -1.458)\\
    &UL & (0.0, 24.58, 0.0) &(1.130, -1.939, 23.088)\\
\\
I1 
    &LL & (0.0, 0.0, 0.0) & (1.130, 23.086, -1.458)\\
    &LR & (24.58, 0.0, 0.0)& (2.360, 23.024, 23.088)\\
    &UR &(24.58, 24.58, 0.0) & (1.130, -1.521, 23.088)\\
    &UL &(0.0, 24.58, 0.0) & (-0.100, -1.459, -1.458)\\
\\
I2 
    &LL &  (0.0, 0.0, 0.0)& (1.130, -26.546, -1.997)\\
    &LR &(24.58, 0.0, 0.0) & (2.361, -26.484, -26.543)\\
    &UR &(24.58, 24.58, 0.0) & (1.130, -1.939, -26.543)\\
    &UL &(0.0, 24.58, 0.0) & (-0.100, -2.001, -1.997)\\
\\
I3
   &LL  &  (0.0, 0.0, 0.0)& (2.361,  23.024, -26.543)\\
   &LR  & (24.58, 0.0, 0.0)& (1.131,  23.086, -1.997)\\
   &UR  & (24.58, 24.58, 0.0)& (-0.100, -1.459, -1.997)\\
   &UL  & (0.0, 24.58, 0.0)& (1.130, -1.521, -26.543)\\
\\
S0
   &LL  &  (0.0, 0.0, 0.0)& (0.744, -81.170, -59.170)\\
   &LR  & (24.58, 0.0, 0.0)& (0.353, -56.597, -59.170)\\
   &UR  & (24.58, 24.58, 0.0)& (0.353, -56.597, -34.594)\\
   &UL  & (0.0, 24.58, 0.0)& (0.744, -81.170, -34.594)\\
\\
S1
   &LL  &  (0.0, 0.0, 0.0)& (0.348, -56.133, -59.170)\\
   &LR  & (24.58, 0.0, 0.0)& (0.099, -31.559, -59.170)\\
   &UR  & (24.58, 24.58, 0.0)& (0.099, -31.559, -34.594)\\
   &UL  & (0.0, 24.58, 0.0)& (0.348, -56.133, -34.594)\\
\\
S2
   &LL  &  (0.0, 0.0, 0.0)& (0.096, -31.100, -59.170)\\
   &LR  & (24.58, 0.0, 0.0)& (-0.011, -6.524, -59.170)\\
   &UR  & (24.58, 24.58, 0.0)& (-0.011, -6.524, -34.594)\\
   &UL  &(0.0, 24.58, 0.0) & (0.096, -31.100, -34.594)\\
\\
S3
   &LL  &  (0.0, 0.0, 0.0)& (-0.011, -6.035, -59.170)\\
   &LR  &(24.58, 0.0, 0.0) & (0.024, 18.541, -59.170)\\
   &UR  &(24.58, 24.58, 0.0) & (0.024, 18.541, -34.594)\\
   &UL &(0.0, 24.58, 0.0) & (-0.011, -6.035, -34.594)\\
\\
S4
   &LL  &  (0.0, 0.0, 0.0)&(0.026, 18.970, -59.170)\\
   &LR  & (24.58, 0.0, 0.0)&(0.208, 43.545, -59.170)\\
   &UR  &(24.58, 24.58, 0.0) &(0.208, 43.545, -34.594)\\
   &UL  &(0.0, 24.58, 0.0) &(0.026, 18.970, -34.594)\\
\\
S5
   &LL  &  (0.0, 0.0, 0.0)& (0.208, 43.986, -59.170)\\
   &LR  & (24.58, 0.0, 0.0)& (0.528, 68.560, -59.170)\\
   &UR  & (24.58, 24.58, 0.0)& (0.528, 68.560, -34.594)\\
   &UL  & (0.0, 24.58, 0.0)& (0.208, 43.986, -34.594)\\
\\
\hline
\end{longtable}


The HRC instrument has two detectors, the HRC-S and the HRC-I. The HRC-I
has a single `chip' or segment/microchannel plate pair (CHIP ID = 0, name HRC-I)
while the HRC-S has three (CHIP ID = 1,2,3; name HRC-S1,S2,S3; HRC team
segment designation +1, 0, -1).  We define an HRC-I chip plane which is 16384 pixels square;
(not all of these pixels correspond to actual readable values);
the HRC-S chips are 4096 x 16456 pixels. The pixel size is 0.0064294 mm.

\begin{longtable}{lllll}
Name  & ID & CHIP size (pix)& Pixel size ($\mu$) & CHIP size (mm)\\ 
HRC-I & 0  & 16384 x 16384 &  6.429 x 6.429 &  105.333 x 105.333 \\
HRC-S1& 1  & 4096 x 16456  &  6.250 x 6.429 &   25.600 x 105.802 \\
HRC-S2& 2  & 4096 x 16456  &  6.250 x 6.429 &   25.600 x 105.802 \\
HRC-S3& 3  & 4096 x 16456  &  6.250 x 6.429 &   25.600 x 105.802 \\
\end{longtable}

The corners of these logical chip planes are located in the LSI
coordinate system as follows; these are calculated by placing
the HRC-S origin (default aimpoint)
at 4.0mm to +LSI Y of the overall detector center and using the chip
sizes calculated above together with the designed S1 and S3 chip
tilts in the X,Y plane. Note that S2 is no longer centered around the
detector center, since its edges are determined by the measured gap
locations.

\begin{longtable}{llll}
\caption{HRC chip (i.e. grid) corner locations in LSI coordinates}\\
\hline
Chip & Corner& CPC coords & HRC-I,S LSI coords \\
\hline
HRC-I   &LL  &(     0.000  ,     0.000  ,     0.000  )&(     0.000  ,     0.000  ,    74.482  )\\
HRC-I   &LR  &(   105.339  ,     0.000  ,     0.000  )&(     0.000  ,    74.482  ,     0.000  )\\
HRC-I   &UR  &(   105.339  ,   105.339  ,     0.000  )&(     0.000  ,     0.000  ,   -74.482  )\\
HRC-I   &UL  &(     0.000  ,   105.339  ,     0.000  )&(     0.000  ,   -74.482  ,     0.000  )\\
\\ 
HRC-S1  &LL  &(     0.000  ,     0.000  ,     0.000  )&(     2.644  ,   161.949  ,   -13.167  )\\
HRC-S1  &LR  &(    26.334  ,     0.000  ,     0.000  )&(     2.644  ,   161.949  ,    13.167  )\\
HRC-S1  &UR  &(    26.334  ,   105.802  ,     0.000  )&(     0.000  ,    56.180  ,    13.167  )\\
HRC-S1  &UL  &(     0.000  ,   105.802  ,     0.000  )&(     0.000  ,    56.180  ,   -13.167  )\\
\\ 
HRC-S2  &LL  &(     0.000  ,     0.000  ,     0.000  )&(     0.000  ,    56.180  ,   -13.167  )\\
HRC-S2  &LR  &(    26.334  ,     0.000  ,     0.000  )&(     0.000  ,    56.180  ,    13.167  )\\
HRC-S2  &UR  &(    26.334  ,   105.802  ,     0.000  )&(     0.000  ,   -49.622  ,    13.167  )\\
HRC-S2  &UL  &(     0.000  ,   105.802  ,     0.000  )&(     0.000  ,   -49.622  ,   -13.167  )\\
\\ 
HRC-S3  &LL  &(     0.000  ,     0.000  ,     0.000  )&(     0.000  ,   -49.622  ,   -13.167  )\\
HRC-S3  &LR  &(    26.334  ,     0.000  ,     0.000  )&(     0.000  ,   -49.622  ,    13.167  )\\
HRC-S3  &UR  &(    26.334  ,   105.802  ,     0.000  )&(     2.253  ,  -155.400  ,    13.167  )\\
HRC-S3  &UL  &(     0.000  ,   105.802  ,     0.000  )&(     2.253  ,  -155.400  ,   -13.167  )\\
\hline
\\
\end{longtable} }

\subsection{Tiled detector (TDET) coordinates)}

The unphysical tiled coordinate system is defined
here. Each system has a label used to distinguish
it from previous choices of definition.

\be
\left(\begin{array}{l}
TDETX \\
TDETY 
\end{array}\right)
=  \Delta_i \left( \begin{array}{ll}
1 & 0 \\
0 & H_i \\
\end{array}\right)  \left( \begin{array}{ll}
\cos \theta_i  & \sin \theta_i \\
-\sin \theta_i  & \cos \theta_i \\
\end{array}\right)
\left(\begin{array}{l} CHIPX - 0.5 \\ CHIPY - 0.5 \end{array} \right)
+\left(\begin{array}{l} X0_i+0.5 \\ Y0_i+0.5
\end{array}\right)
\ee
where the values of $H_i$, $\Delta_i$ and $\theta_i$ are different
for each chip. $H_i$ gives the handedness of the planar rotation and
has values $+1$ or $-1$,
$\Delta_i$ gives the sub-pixel resolution factor which is
always 1 in the current implementation, and $\theta_i$ gives
the rotation angle of the chip axes with respect to the detector coordinate
axes.

\begin{figure}[h]
\rotatebox{-90}{\includegraphics*[height=3.0in]{ps/xrcf15.eps} }
\caption{The relationship between CHIP and Tiled Detector coordinates.}
\end{figure}


For HRC-S, the tiled system coincides with the
continuous (degapped, discontinuity-corrected)
U,V system used by the instrument team. There are electronic
discontinuities in V between the HRC segments, so that the
degapped instrument coordinates $vpos_d$ are corrected to continuous
instrument coordinates $vpos$ and then to CHIPY and TDETY.
The relationship for HRC-I is CHIPX=$upos_d$+1, CHIPY=$vpos_d$+1 while
for HRC-S, CHIPX=$upos_d$+1, and CHIPY as tabulated below.
Users who are not members of the HRC instrument team probably
don't need to worry about the $upos,vpos$ coordinates.

\begin{longtable}{llll}
\caption{HRC-S discontinuities}\\
$vpos_d$  & $vpos$ & CHIPY & TDETY\\
\\
0-16444.5      & $vpos_d$   & $vpos_d$ + 12.5   & CHIPY - 12.0 \\
16444.5-16459.9& $vpos_d$   & 16456.4           & \\
16459.9-32916.4& $vpos_d-16$& $vpos_d$ -16459.5 & CHIPY + 16444.0\\
32916.4-32930.0& $vpos_d-16$& 0.5               &\\
32930.0+       & $vpos_d-29$& $vpos_d$ - 32929.5& CHIPY + 32900.0\\
\end{longtable}



\begin{longtable}{llll}
\caption{Tiled Detector Plane systems}\\
System & Size & X Center, Y Center & Use\\
\\
\hline
AXAF-ACIS-2.2 & 8192 x 8192  & (4096.5, 4096.5)   & Standard\\
AXAF-HRC-2.3I & 16384 x 16384& (8192.5, 8192.5)   & HRC-I\\
AXAF-HRC-2.6S & 49368 x 4096 & (24684.5, 2048.5)  & HRC-S\\
\hline
\end{longtable}

\begin{longtable}{lllllll}
\caption{Parameters of Tiled Detector Coordinate definitions}\\
\hline
Tiled System & Chip & $\theta_i$ & $\Delta_i$  & $X0_i$ & $Y0_i$ & $H_i$ \\
\hline
AXAF-ACIS-2.2 & ACIS-I0  & 90 & 1& 3061.0 & 5131.0&1\\
AXAF-ACIS-2.2 & ACIS-I1  & 270 & 1 & 5131.0 & 4107.0&1\\
AXAF-ACIS-2.2 & ACIS-I2  & 90 & 1 & 3061.0 & 4085.0&1\\
AXAF-ACIS-2.2 & ACIS-I3  & 270 & 1 & 5131.0 & 3061.0&1\\
AXAF-ACIS-2.2 & ACIS-S0  & 0& 1 &  791.0 & 1702.0&1\\
AXAF-ACIS-2.2 & ACIS-S1  & 0& 1 & 1833.0 & 1702.0&1\\
AXAF-ACIS-2.2 & ACIS-S2  & 0& 1 & 2875.0 & 1702.0&1\\
AXAF-ACIS-2.2 & ACIS-S3  & 0& 1 & 3917.0 & 1702.0&1\\
AXAF-ACIS-2.2 & ACIS-S4  & 0& 1 & 4959.0 & 1702.0&1\\
AXAF-ACIS-2.2 & ACIS-S5  & 0& 1 & 6001.0 & 1702.0&1\\
\\
\multicolumn{7}{c}{HRC: old convention}\\
\\
AXAF-HRC-2.3I & HRC-I   & 90 & 1 & 0.0 & 0.0 & -1\\
\\
AXAF-HRC-2.6S & HRC-S1  & 270 & 1 & 49368.0 & 0.0 & 1\\
AXAF-HRC-2.6S & HRC-S2  & 270 & 1 & 32912.0 & 0.0&1\\
AXAF-HRC-2.6S & HRC-S3  & 270 & 1 & 16456.0 & 0.0&1\\
\\
\multicolumn{7}{c}{HRC: new convention}\\
\\
AXAF-HRC-2.4I & HRC-I   & 0 & 1 & -1.0 & 0.0 & 1\\
\\
AXAF-HRC-2.7S & HRC-S1  & 0 & 1 & 0.0 & -12.0 &  1\\
AXAF-HRC-2.7S & HRC-S2  & 0 & 1 & 0.0 &  16444.0&1\\
AXAF-HRC-2.7S & HRC-S3  & 0 & 1 & 0.0 & 32900.0 &1\\
\\
\hline
\end{longtable}



\subsection{Relative positions of instruments}

The location of the origins of the LSI system for each
instrument are given in STT coordinates
in the table below. The in-flight calibration
will be refined when angular misalignments in the
system are measured.

\begin{longtable}{lll}
\caption{Location of instrument origin on Translation Table}\\
\hline
\multicolumn{3}{c}{Values of LSI origins OLSI in STT system, prelaunch}\\
\\
\hline    
Instrument      &   Prelaunch estimate &   Dec 1999 calibration \\
ACIS origin     & (0.0, 0.0, 237.4)    &  (0.684, 0.750, 236.552 )\\
HRC-I origin    & (0.15, 0.0, -126.6)  &  (1.040, 0.978, -132.028)\\
HRC-S origin    & (0.10, 0.0, -250.1)  &  (1.533, 1.530, -251.437)\\
\hline
\end{longtable}

\begin{figure}[h]
\rotatebox{0}{\includegraphics*[height=6.0in]{ps/stt.eps} }
\caption{The AXAF SIM Translation Table, showing the flight focal plane
instruments to scale. Distances are in mm. Coordinate system is
AXAF-STT-1.0.}
\end{figure}

\subsection{Aimpoints}


Unless the observer specifies an offset, the SIM will
be placed at one of four default locations.
The table below gives the SIM values (STF coordinates
of the instrument table origin) for each of these aimpoints.
Recall that in general the SIM can also be moved in X and Z 
to other aimpoints.

The SIM Z motion is commanded in `TSC motor steps' which are
2.5143153 microns per Z step. The focus stage, FA, mapping from motor step
to SIM X is a degree 6 polynomial.

\[
 SIMZ (mm) = -0.00251431530156 \mbox{TSC}
\]
and
\[
 SIMX (mm) = \sum_{n=0}^6 c_n \left(\mbox{FA} \over 10^4 \right)^n
\]
where

\begin{longtable}{ll}
\caption{Coefficients of FA stage motor step to SIM X position}\\
n     & $c_n$ \\
\hline
0     & 0.0 \\
1     & 14.7906994 \\
2     & 3.5723322 \\
3     & -1.08492544\\
4     & 0.39803832\\
5     & 0.529336\\
6     & 0.1020064\\
\hline
\end{longtable}

\begin{longtable}{lll}
\caption{SIM hard limits}\\
\hline & Min & Max\\
SIM X&  -28.965 & +16.783\\
SIM Z& -262.200 & +263.359\\
\end{longtable}

The X coordinates of the offsets may be regarded as estimates
of the instrument origin X positions, and may be added to the instrument
origin values in later revisions of the processing system.
For the ACIS origin, we note that a SIM X position of -0.684mm
is used for the ACIS-S3 chip aimpoint which is within 0.002 mm
of the ACIS LSI (PSU) X origin, and we therefore adopt this as
the origin of the ACIS system. The fact that the ACIS-I aimpoint
is 0.1mm different in X is taken into account by the different
X values of the corners of the ACIS-I chips relative to ACIS-S.

\begin{longtable}{lllll}
\caption{SIM position offsets for nominal focus positions}\\
\hline
\multicolumn{2}{c}{Values of $P_{STF}(\Sigma)$}
\\
\hline
 &    ODB offset  & & CHIP & TDET\\
 &        Motor steps  & mm   & pixel & \\
AI2 ACIS-I offset & -536, 92905 & (-0.782, 0.0, -233.592) &ACIS-I3 (984.4, 994.8) &  4137.2, 4045.4\\
AS1 ACIS-S offset & -468, 75620 & (-0.684, 0.0, -190.133) &ACIS-S3 (220.7, 531.8)  & 4137.7, 2233.8\\
HI1 HRC-I offset  & -716, -50505 & (-1.040, 0.0, 126.985) &HRC-I (7529.9, 7745.0)  & 7528.9, 7744.0\\
LETG HRC-S offset  & -991, -99612 & (-1.430, 0.0, 250.456) &HRC-S2 (2201.0, 8976.5) & 2200.0, 25420.5\\
HS1 HRC-S offset  & -1065, -99612 & (-1.533, 0.0, 250.456) &HRC-S2 (2201.0, 8976.5) & 2200.0, 25420.5\\
\hline
\end{longtable}

%         -0.8384           -1.494
%         -0.73623 -1.267
% SIM X = -0.78091   scaled -1.267
%         -0.68282          -1.174  defoc = 0.00858
% -1.01603 -1.758
% -0.8154 -1.406

\subsection{Detector and sky pixel plane systems}

In this section we specify the pixel systems used to report detector and
sky coordinates. We decided to use the mean actual instrument pixel
sizes for these systems. Thus, the size of a physical HRC-I pixel at the
nominal focal distance of \ffoc~mm is 0.132 arcseconds, so 0.132
arcseconds is defined as the FP-2.1 pixel size (even if the detector is
moved well off focus). The HRC-S detector has a maximum linear extent of
over 48500 pixels. When making sky coordinates for such a detector, we
must define a square pixel plane (to allow for the roll angle). We
choose a 65536 pixel sided plane, defined as FP-2.3. This plane has a
total of 4.3 gigapixels, slightly more than can fit in a 4-byte signed
integer. A 2-byte integer counts image would then have a file size of
8.0 gigabytes. We therefore recommend not making full resolution full
frame image files in this coordinate system; the CXC data analysis
system may be used to make full resolution images of small areas
and low resolution images of the full field.


The adopted focal length on orbit, based on ray trace simulations
optimized to fit ground calibration data (P. Zhao, private communication)
was reported to be
\begin{quote}
 10061.65mm
\end{quote}
This value is currently under review.
The measured plate scale gives a provisional effective focal length of around 10070 mm.

Here are the current flight pixel systems:
\begin{longtable}{llllll}
System & Pixel size & Pixel size &  X0, Y0 & Full size & Purpose \\
& (mm) & (arcsec) \\
\\
AXAF-FP-1.1 &0.0240  & 0.49200  &  4096.5& 8192 x 8192   &Flight ACIS\\
AXAF-FP-2.1 &0.006427& 0.13175  & 16384.5& 32768 x 32768 &Flight HRC-I\\
AXAF-FP-2.3 &0.006427& 0.13175  & 32768.5& 65536 x 65536 &Flight HRC-S\\
\\
AXAF-FP-2.1 &0.006429& 0.13180  & 16384.5& 32768 x 32768 &Flight HRC-I\\
AXAF-FP-2.3 &0.006429& 0.13180  & 32768.5& 65536 x 65536 &Flight HRC-S\\
\end{longtable}
\clearpage

\subsection{Remaining issues}

There is a remaining uncalibrated rotation of the HRC-I of order 0.3 degree;
and an uncertainty in the plate scale and effective focal length of order 0.2 percent.

\section{Summary}

We have discussed the derivation of celestial coordinates for Chandra
data and presented coordinate systems suitable for description of
instrumental, telescope-related and celestial effects. The formalism is
sufficiently general that it should be applicable to other missions by
appropriately changing the numerical values in section 3. In Paper 2, we
discuss derivation of the alignments and the aspect solution. In Paper
3, coordinate systems useful for analysing dispersed grating data are
presented.

We thank Ian Evans, Tom Aldcroft, Maxim Markevitch, Kenny Glotfelty, Janet DePonte,
Steve Murray, Brian McNamara and Martin Elvis for useful discussions,
and Helen He for implementing the software routines used to apply
the coordinate transforms in the Chandra data processing system.

\clearpage
\tableofcontents

\listoftables

\listoffigures


\end{document}


Issues:

HRC-I and -S TDET systems
Plate scale
Rotations
HRC pixel size
0.49115 sky 0.2 percent
     10
0.13165
      5
0.13200 hrcs
0.13160 hrcs
10073 mm
10079 mm

10069

10077
10081


      ACIS-I                  HRC-I                   HRC-S
scale 0.49105 0.49115 0.49125 0.13160 0.13165 0.13170 0.13200
f     10081     10079   10077  10077   10073   10069    10047
