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\title{PIR/corrector+Echidna Engineering Summary}
\author{Masayuki Akiyama (Astronomical Institute, Tohoku University)}
\date{
\begin{itemize}
\item Ver.1.0 : 2008/06/05 : Updated 2007/12 report with 2008/01,2008/05 engineering results
\item Ver.-1.1 : 2008/01/18 : Bug fixed, the FPI coordinate of the distortion center
is updated to be (2.4mm, 4.0mm) after applying the SKY and SPINE camera offset.
Updated result is consistent with the offset between the CMM center and the
rotator axis.
The target was M38 (not M37). 
\item Ver.-1.0 : 2008/01/17 : Previous report circulated AAO/Kyoto/Subaru. \\
\end{itemize}
This is a summary of the Echidna engineering observations for the telesocpe
model construction.
}

\begin{document}
\maketitle

\section{Converting the SKY CCD coordinate to the FPI XY coordinate}

The relation between the SKY camera coordinate and the FPI XY 
coordinates is determined with stars taken with the SKY camera
at multiple FPI XY positions (i.e. stars in the overwrapping regions).
The CCD camera coordinate (local) can be converted to the FPI XY coordinate
(global) with
\begin{eqnarray}
X({\rm FPI}) &=& A \times X({\rm CCD}) + B \times Y({\rm CCD}) + C, \nonumber \\
Y({\rm FPI}) &=& D \times X({\rm CCD}) + E \times Y({\rm CCD}) + F  \nonumber \\
 &C&=({\rm FPI\ encoder\ x\ coordinate} \nonumber \\
 & &+ {\rm X\ correction\ for\ the\ offset\ between\ the\ SKY\ and\ SPINE\ cameras}) \nonumber \\
 &F&=({\rm FPI\ encoder\ y\ coordinate} \nonumber \\
 & &+ {\rm Y\ correction\ for\ the\ offset\ between\ the\ SKY\ and\ SPINE\ cameras}) \nonumber \\
\end{eqnarray}
We use the SKY coordinate definition measured on the FITS images.

\begin{table}[tb]
\begin{center}
\begin{tabular}{cccccccc}
\hline
Date   &    A     &     B    &  C   &    D     &     E    &  F   & Note              \\ \hline
200712 & $-8.103$ & $+0.677$ & $+$3445 & $-0.668$ & $-8.268$ & $+$2876 & 175.3deg rotation \\
200712 &          &          & $+$3556 &          &          & $+$2821 &                   \\ \hline
200805 & $-8.082$ & $-0.251$ & $+$3556 & $+0.388$ & $-8.673$ & $+$2641 &                   \\
200805 &          &          & $+$3550 &          &          & $+$2639 &                   \\ \hline
\end{tabular}
\caption{
The measured sky-camera to FPI coordinate converstion factors.
Between 200712 and 200805, the sky-camera was re-installed.
}
\end{center}
\end{table}

The XY offset between SKY and SPINE cameras are determined by aligning a
guide fibre to a bright star. Two measurements were performed with the same
guide fiber to different stars. There results are shown as parameters $C$ and $F$
in Table 1. Using these values as the offset between the SKY and SPINE cameras 
in relation (1), the coordinates measured with the SKY camera
can be converted to the spine positions measured with the SPINE camera.
(In order to align a star to guide spine No.168, we need to move
telescope by $+878.48^{\prime\prime}$, $-243.09^{\prime\prime}$
(or $+901.76^{\prime\prime}$, $-258.28^{\prime\prime}$). 

The FPI XY directions and the celestial coordinates NE directions are 
aligned well. The relation is summarized in Figure~\ref{FPI_coordinate}.

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.8\linewidth]{FPI_coordinate.eps}
\end{center}
\caption{
The relation between the sky camera CCD coordinate, 
the FPI XY coordinate, and the cerestial coordinate.
It should be noted that the SKY camera Y coordinate is
flipped on the data and the Echidna GUI window).
}
\label{FPI_coordinate}
\end{figure}

\section{Rotator axis on the FPI coordinate}

\begin{table}
\begin{center}
\begin{tabular}{cccc}
\hline
Date     &   EL     &                  & Note  \\ \hline
20071208 & $\sim$80 & ($-702$, $-727$) &       \\ \hline
20080125 & 85       & ($-826$, $-712$) &       \\
20080125 & 27       & ($-826$, $-712$) & No significant shift from El=85deg. \\ \hline
20080513 & $\sim$80 & ($-754$, $-567$) &      \\ \hline
\end{tabular}
\caption{
The measured FPI coordinates of the rotator axis.
Echidna re-installations happen between 200712 and 200801
and between 200805 and 200801. The shifts of the FPI coordinates
reflect the shift of the Echidna against the rotator axis.
}
\end{center}
\end{table}

The FPI coordinates of the rotator axis correspond to the pointing
center of the FoV, because the pointing center of the telescope is 
set at the rotator axis of the telescope.
The axis of the rotator is measured by taking stars with the sky-camera with rotating the
instrument. The measured CCD coordinate were converted to FPI coordinate
using the relation (1). On 20080125, the rotation axis was measured
at EL=85deg and at EL=27deg (At EL=27deg, the position of the FPI was 
keeped at ($0,0$) referring the FPI encoder values, otherwise the 
recorded star track is not on a circle due to the flexure of the FPI gantry. 
The movement of the FPI was $\pm200\mu$m at most at EL=27deg).
No shift was observed. The difference of 
the FPI coordinate of the pointing center between 20071208 and 20080125 
measurements is ($-124, +15$) (in $\mu$m), which represents the accuracy 
of the installation of Echidna unit to PIR enclosure (we remove Echidna  
20080110 and re-install 20080117, it should be noted that this
is not related to the accuracy of the PIR mounting accuracy).

\section{Measuring the center of the distortion pattern}

We observed open cluster fields with tiling the SKY camera 
FoVs in vertical and horizontal strips on the FPI coordinate.
The data are taken with the instrument rotator with SYNC status, i.e.
the instrument is rotated against the corrector - primary mirror system.
The measured coordinates of the stars on the SKY camera
were at first converted to the FPI XY coordinate using the above relation.
Then the FPI XY catalog was cross matched with the RA-DEC
catalog of the stars in the FoVs. Using the (X, Y, RA, DEC) file, 
the distortion pattern of the field is determined by ccmap
command in IRAF. One measurement of the distortion pattern 
is shown in Figure~\ref{FPI_coo_distortion}.
The offset from the plate scale at the center of FoV is shown 
as the optical distortion pattern.
The distortions are determined by ccmap command in IRAF.
The measured distortion is consistent with the ray trace result
shown with green lines and dots. The FPI XY coordinate of 
the distortion center was
estimated to be ($-4.6$mm,$-1.9$mm) for this measurement. 

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.8\linewidth]{FPI_coo_distortion.eps}
\end{center}
\caption{
The measured distortion of the corrector lens in 20071224 with
rotator angle of $-97$deg.
Horizontal axis is the distance from the FPI (0,0) position in arcmin.
Vertical axis is the offset from the $0.01240$ (or $0.01237$) arcsec/$\mu$m
in arcsec. Left) in Y (or Dec) direction, MAP2-Y and MAP3 results are
shown in red and blue, respectively. Ray trace model is shown in green.
The model is offsetted by $-0.39$arcmin ($-1.9$mm) to match the observed distortion curve. 
Right) in X (or RA) direction, MAP2-X and MAP4 results are shown in 
red and blue, respectively. Ray trace model is shown in green. The model
is offsetted by $-0.95$ arcmin ($-4.6$mm). 
}
\label{FPI_coo_distortion}
\end{figure}

In the nights of 20080124 and 20080125, the same measurements are
done with changing the rotator angle. The results are summarized in 
Table 1 and Figure 3. Using the data, the center of the rotation axis
is estimated to be $(-0.7$mm, $-0.7$mm$)$ on the FPI coordinate. 
The position is consistent with the sky camera measurements as
shown in Section 2. The distance between the rotation axis and the
distortion center are summarized in the 7th column in the table.

\begin{table}
\begin{center}
\begin{tabular}{ccccccccc}
\hline
 Date    & EL & Rotater &     CMM        & FAM  &  Distortion Cnt. & Dist & Scale ($\prime\prime$/mm) & \\ \hline
20071224 & 45 &  $-97$   &  ($+4.0, -4.0$) & $+0.60$ & ($-4.60, -1.94$) & 4082 & (12.398, 12.401) & \\ \hline
20080124 & 30 &  $+0$    &  ($+3.7, -3.8$) & $+0.55$ & ($+0.98, -4.63$) & 4314 & (12.405, 12.409) & \\
         & 30 &  $-90$   &  ($+3.7, -3.8$) & $+0.55$ & ($-4.54, -2.72$) & 4222 & (12.411, 12.401) & \\
         & 30 &  $+90$   &  ($+3.7, -3.8$) & $+0.55$ & ($+3.18, +0.97$) & 4344 & (12.409, 12.401) & \\
         & 30 &  $+180$  &  ($+3.7, -3.8$) & $+0.55$ & ($-2.74, +3.26$) & 4409 & (12.420, 12.416) & \\ \hline  
20080125 & 85 &  $+6$    &  ($+3.7, -3.8$) & $+0.60$ & ($+1.68, -4.26$) & 4344 & (12.405, 12.405) & \\
         & 85 &  $-84$   &  ($+3.7, -3.8$) & $+0.60$ & ($-4.67, -3.72$) & 4881 & (12.400, 12.403) & \\
         & 85 &  $-174$  &  ($+3.7, -3.8$) & $+0.60$ & ($-3.35, +3.13$) & 4597 & (12.404, 12.402) & \\
         & 85 &  $+96$   &  ($+3.7, -3.8$) & $+0.60$ & ($+3.00, +1.65$) & 4496 & (12.404, 12.408) & \\ \hline
20080513 &    &          &  ($+4.7, -4.2$) & $-5.00$ &                  &      &                  & \\ \hline
\end{tabular}
\caption{
Summary of the results of the distortion center measurements.
}
\end{center}
\end{table}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.5\linewidth]{plot_dist_cen.eps}
\end{center}
\caption{
Summary of the results of the distortion center measurements.
The numbers in the figure indicate the ROTATOR\_REAL\_ANGLE
during the measurements. The cross mark indicates the measured
position of the rotator center. The measured distortion centers
circles arount the rotator center with radius of 4.3mm.
The direction of the ROTATOR\_REAL\_ANGLE is different from
the OFFSET\_ANGLE.}
\label{plot_dist_cen}
\end{figure}

The estimated positions of the distortion center and the rotator axis, 
and the relation between the CMM coordinate and the FPI XY coordinate
are shown in Figure~\ref{FPI_coordinate3}. 

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.4\linewidth]{FPI_coordinate4.eps}
\end{center}
\caption{
The position of the distortion center and the rotator axis, and
the relation between the CMM coordinate and the FPI XY coordinate,
evaluated based on the CMM position (($+3.7$mm, $-3.8$mm) in the CMM coordinate) 
of the measurements, the rotator angle ($\sim0$deg), 
and the distortion center results.
The zeropoint of the CMM coordinate is estimated to be
($-2.1$mm, $-0.6$mm) on the FPI XY coordinate, 
thus the offset between the CMM center and the rotation center is
($-1.4$mm, $+0.1$mm), which is roughly 
consistent with the measured offset
between the CMM center and the rotation center ($\sim$1mm, $\sim$1mm).
(!!! It need to be confirmed that the CMM coordinate 
is aligned with FPI coordinate !!!).
}
\label{FPI_coordinate3}
\end{figure}

\section{Summary of 1st pass telescope model with sky camera}

In order to refine the telescope model further, 
several 10s stars scattered in FoV are observed with the sky-camera
(Item 6.9: 1st pass telescope model with sky camera).
The offset between the predicted position using a telescope model
and the observed position were measured for each star. Using the
measured offsets, the telescope models were refined further by multiple
parameter fitting. The obtained data were fitted with the telescope 
model with the rotation, the distortion parameters (1st, 3rd, 5th orders), 
and offset of the distortion as free parameters.

The first observations were performed on 20080127 
with M35. 69 11th magnitude stars were observed. The first refined
telescope model was determined with 0.67$^{\prime\prime}$ RMS

On the night of 20080513 and 20080514, multile fields were
observed.  The telescope model fitting 
results are summarized in Table 4. The observed offsets and the residual of
the fitting for the offsets were shown 
in the top left and right panels, respectively,
of Figure~\ref{fields_all} for field5. 
The observed offsets for field5, 6, and 7 were similar, and the fitted results
were sufficiently consistent. 
The residuals of the fitting were 0.43$^{\prime\prime}$ RMS and there remains
systematic pattern in the middle of the field. 

We applied the telescope model determined from field5 and observed field8 and field9.
The observed offsets are shown in the bottom panels of Figure~\ref{fields_all} for
field9. The observed offsets still show a systematic pattern, but the residuals
after fitting the observed offsets look sufficiently random as shown in the right panel.
It need to be noted that the OffsetX and OffsetY parameters were significantly 
different from those determined in field5(field6,field7) and field9(field8). 
We still need to carefully watch these parameters during the further observations. 
The RMS of the residuals is $0.32^{\prime\prime}$. We should use the telescope.model
refined based on the field9 observation. 

\begin{table}
\begin{center}
{\tiny
\begin{tabular}{ccclrrrrrcc}
FieldID & Date     & PA(OFF)       &  Angle  &   Dist1   &   Dist2  &  Dist3   & OffsetX  & OffsetY   & RMS  & No. \\ \hline
field5  & 20080514 & $0(0.52)$     & $ 0.0094$ &  $0.29055$  & $0.04205$  & $0.06522$  & $8284.24$  & $-1031.67$  & $0.43$ & 70  \\
field6  & 20080514 & $180(-118.1)$ & $ 0.0047$ &  $0.29062$  & $0.03769$  & $0.12886$  & $7966.91$  & $-365.99$   & $0.44$ & 69  \\
field7  & 20080514 & $90(-25.76)$  & $ 0.0120$ &  $0.29055$  & $0.04287$  & $0.04349$  & $6558.55$  & $-3925.39$  & $0.44$ & 39  \\
field2  & 20080514 & $0(-51.7)$    & $ 0.0046$ &  $0.29055$F & $0.04205$F & $0.06522$F & $8284.24$F & $-1031.67$F & $0.34$ & 17  \\
field3  & 20080514 & $-90(43.8)$   & $-0.0001$ &  $0.29055$F & $0.04205$F & $0.06522$F & $8284.24$F & $-1031.67$F & $0.55$ & 16  \\
field4  & 20080514 & $-180(136.6)$ & $-0.0095$ &  $0.29055$F & $0.04205$F & $0.06522$F & $8284.24$F & $-1031.67$F & $0.53$ & 17  \\ \hline
field8  & 20080514 & $0(70.3)$     & $-0.0157$ &  $0.29043$  & $0.04558$  & $0.01920$  & $3355.42$  & $2601.85$   & $0.45$ & 47  \\
field9  & 20080514 & $0(71.2)$     & $-0.0190$ &  $0.29065$  & $0.03667$  & $0.11354$  & $2749.29$  & $2877.59$   & $0.32$ & 79  \\ 
\end{tabular}
}
\caption{
The results of the 1st pass telescope model with the sky-camera. Angle, Dist1, Dist2, Dist3, OffsetX, and OffsetY
are the fitting results. For field2, field3, and field4, the distortion parameters were fixed because the
number of available stars were small. The field7 and field8 were observed after appling the telescope model
determined in field5. The resulting RMS values and the number of the stars are shown in the last two columns.
OffsetX and OffsetY are offsets of the distortion pattern from the center of the pointing (the axis of
the rotator). The best OffsetX,Y for field5-7 look not consistent with the measured offset shown in 
Figure 3. The results for field8,9 are roughly consistent (it shold be noted that the CMM positions
were ($+4.7,-4.2$) for 20080514). 
}
\end{center}
\end{table}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.7\linewidth]{fields_all.eps}
\end{center}
\caption{
The measured offsets (left panels) and the residuals after
fit for the measured offsets (right panels). The top panels
are for field5 and the bottm panels are for field9. 
The measurements for field9 were done after appling the 
telescope model determined in the field5.
FPI X+ is to the right, and FPI Y+ is to the top.
}
\label{fields_all}
\end{figure}
 
\section{Acquiring guide stars with guide bandles}

After the determination of the telescope model, we tried to acquire
guide stars with the guide bandles. The required telescope offset to
acquire the guide stars are summarized in Table~\ref{offsets}.
For these observations XERR=$+2355$ and YERR=$+2231$ are used. 
We need to update these values using the observed systematic offsets.

\begin{table}
\begin{center}
\begin{tabular}{ccc}
ID      & RAoffset              & DECoffset \\ \hline
field5  & $-7.0^{\prime\prime}$ & $+16.5^{\prime\prime}$ \\
field9  & $-4.5^{\prime\prime}$ & $+14.8^{\prime\prime}$ \\
field10 & $-5.3^{\prime\prime}$ & $+16.6^{\prime\prime}$ \\
field11 & $-3.0^{\prime\prime}$ & $+14.2^{\prime\prime}$ \\
field12 & $-13.8^{\prime\prime}$ & $+7.0^{\prime\prime}$ \\ \hline
average & $-5.0^{\prime\prime}$  & $+15.5^{\prime\prime}$ \\
        & $-416$($\mu$m) & $-1292$($\mu$m) \\ \hline
\end{tabular}
\end{center}
\caption{
Summary of the required offsets to acquire guide stars in several fields.
After slewing telescope, the telescope needed to be moved in the above
amount in order to acquire guide stars. The last line indicates the required
corrections for the XERR and YERR, i.e. XERR and YERR needs to be 1939 and 939, respectively 
(NEED TO BE CHECKED!) \label{offsets}}
\end{table}

\section{Appendix}

\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\linewidth]{FPI_coordinate2.eps}
\end{center}
\caption{
The relation between the Echidna unit and the PIR/CMM unit at
rotator angle of 0 degree. !!! INCONSISTENT WITH FIGURE 3, NEED TO
BE CHECKED FURTHER !!!
}
\label{FPI_coordinate2}
\end{figure}

\end{document}












