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2 Coordinate Systems


2.1 Overview of Coordinate Systems and Mappings

To discuss calibrations, we first need to define several coordinate systems, enumerate potential variables and dependencies for each one, and describe how we will map from one to the next (see Figure 2.1). A mapping from one (two-dimensional) coordinate system to another consists of the following elementsgif:

- zeropoint offset in x and y;

- rotation;

- scale; and

- distortion terms.



Figure 2.1: Coordinate Systems - Overview

This section briefly sketches how each of these elements will be determined, with more details to follow in Section 2.2 and Chapters 5, 7 and 8.

Fundamental Sky Coordinate System, FKCS:
The fundamental celestial coordinates of targets. These depend on epoch (to account for proper motion, aberration of starlight) and include an equinox in their description.

Refracted Sky Coordinate System, RSCS:
The apparent celestial coordinates as seen by the telescope. Required for slitmask design (forward mapping) and astrometry from direct images (backwards mapping). The mapping FKCS tex2html_wrap_inline5883 RSCS depends on atmospheric pressure and temperature, observed wavelength, and zenith distance (neglected dependencies include humidity) and will be provided by standard models of refraction.

Mask Form Coordinate System, MFCS:
Closely related to the Slitmask Coordinate System, SMCS (next), MFCS is a coordinate system close to the telescope image surface that is rigidly tied to the spectrograph. It is a rectilinear coordinate system defined on the cylindrically curved surface of the slitmask form with the x direction following the curved surface of the form, and the y direction perpendicular to that (see Figure 2.2). Its zeropoint and orientation are defined by two focal plane fiducials (FPFs) that are attached to the slitmask form near the focal surface. These project onto the fiducial CCD during direct imaging (see below) and are also visible in the TV Guide Camera, tying all these systems together. The MFCS coordinate system is used to align downstream coordinate systems such as the fiducial CCD and to derive the mapping from DEIMOS' front image surface to the camera focal plane. There is a slight dependence on spectrograph temperature, so the MFCS is defined at DEIMOS nominal operating temperature).

Slitmask Coordinate System, SMCS:
The coordinate system tied to the physical slitmask itself. Ideally, this system is identical to the MFCS (above), but in practice the two may differ by small translations and rotations owing to slitmask insertion errors (typically tex2html_wrap_inline5895 m). For most purposes, these two systems can be considered identical and will be referred to as MF/SMCS (Figure 2.2). The SMCS is a rectilinear x,y system defined on the flat slitmask stock. Its fiducials are the edges of the mask that rest against mounting stops on the mill and on the slitmask form. There is a dependency on slitmask temperature, so the SMCS is defined for the nominal operating temperature. The mapping RSCS tex2html_wrap_inline5883 MF/SMCS is determined to first order by the known mapping of the refracted sky onto the telescope Nasmyth focal plane, and thence by projection of the focal plane onto the tilted cylindrical slitmask. This first-order mapping is tuned by observations of astrometric standard star fields through astrometric masks.



Figure 2.2: Mask Form / Slitmask Coordinate System, (MF/SMCS)

Image Coordinate System, ICS:
This system lies in the focal plane of the spectrograph camera, coplanar with the adopted plane of best focus. Its axes are parallel to the imaged axes of the MFCS, as set by images of the focal plane fiducials (FPFs) (see Figure 2.3). The ICS may be thought of as the pixel system of a single, idealized 8K tex2html_wrap_inline5639 8K CCD with a square pixel grid having the same scale as the real CCDs. [Zeropoint ?? -- probably the projected zeropoint of the mask form, or the (nearby) optical axis of the camera. The later is not clearly measurable, but doesn't need to be, and it conceptually simplifies discussion of mappings.] In practice, the mapping MF/SMCS tex2html_wrap_inline5883 ICS is determined by measuring the individual mappings MF/SMCS tex2html_wrap_inline5883 CCDi (see below), corrected back to the ICS using the mappings CCDi tex2html_wrap_inline5883 ICS (see below).

There are two types of MF/SMCS tex2html_wrap_inline5883 ICS mappings, one for direct imaging and one for spectroscopy. Each filter and (grating+tilt) has its own mapping. The direct imaging mappings consist of scale and distortion terms only (since the orientation and zeropoint of the ICS were defined above). These are determined by direct images of a special slitmask with a finely and accurately milled grid of small holes (grid-of-holes mask, GOH). The direct imaging mappings involve only CCDs 1-4. The spectroscopy mappings involve a wavelength function that maps pixel coordinates onto tex2html_wrap_inline5715 and a spatial function that maps pixel coordinates onto position along the slit. These functions are determined by arc spectra taken through a mask with a line of small holes spaced along the x (slit) direction (line-of-holes mask, LOH). These mappings may depend on camera temperature, camera focus, flexure, order-separating filter (for spectroscopy mappings), and the current flexure compensation system (FCS) zeropoint. A nominal mapping is therefore defined for standard T, rotator angle (RPA), focus, order-separating filter, and FCS zeropoint, with families of differential mappings as functions of T, PA, other order-separating filters, and FCS operation.



Figure 2.3: Image Coordinate System, (ICS)

CCD Coordinate Systems, CCDCSi:
Pixel coordinates on the individual CCDs. The fiducial CCD is No. 3 in the array -- images of the focal plane fiducials (FPFs) are projected onto it during direct imaging (see Figure 2.3), and it will be the first CCD installed into the array if only one is available. The mappings ICS tex2html_wrap_inline5883 CCDCSi consist of the rotation and displacement of each CCD from the idealized focal plane coordinate system, ICS. These quantities are determined for CCDs 1-4 from direct images of the above-mentioned grid-of-holes mask. To extend this information to CCDs 5-8, an arc spectrum is taken through the line of holes (LOH) mask. The known rotations and displacements of CCDs 1-4 from part A are corrected to create the lower half of the idealized, seamless ICS focal plane. The spectral traces (x-heights) and tex2html_wrap_inline5715 solutions from this half are then continued onto CCDs 5-8. Ensuring continuity in both x-height and wavelength serves to determine the offsets and rotations of CCDs 5-8 relative to the ICS system determined by CCDs 1-4.

Flexure Compensation CCD Coordinate Systems, FCSj:
Pixel coordinate systems on the two flexure compensation (FC) CCDs, at opposite ends of the slitmasks (see Figures 2.2 and ??). These are used internally to hold the image position steady and/or repeat a previous spectrograph setting, so absolute mappings to the other coordinate systems are not needed. However, calibrations are needed to relate image motion on the FC CCDs to image motion on the main detector (the two are not the same due to optical distortions). Separate mappings tex2html_wrap_inline5933 FCSj tex2html_wrap_inline5883 tex2html_wrap_inline5933 ICS are needed for each direct filter and grating+tilt combination and will be determined from tests of the flexure compensation system.

TV Guider Coordinate System, TVCS:
Pixel coordinates on the TV Guider CCD. The guider has two regions, (i) the offset guiding area, fed by a fixed offset guide mirror in the focal plane that is always visible, and (ii) the slitmask area, which is empty for direct imaging but contains a reflective slitmask or longslit for spectroscopy, which images the sky (see Figure ??). For direct imaging, the slitmask is out and the FPFs are visible. Two additional fiducial marks, called TV fiducials, or TVFs, are etched onto the surface of the offset guiding mirror. The FPFs and the TVFs are all rigidly connected to the DEIMOS front focal surface and form parallel lines with a known offset. At least one pair of these fiducials is visible in the TV at all times.

Two TV mappings are needed, TVCS tex2html_wrap_inline5883 MF/SMCS and TVCS tex2html_wrap_inline5883 RSCS, but, since RSCS tex2html_wrap_inline5883 MF/SMCS is already known, the single mapping MF/SMCS tex2html_wrap_inline5883 TVCS suffices. The zeropoint and orientation of this mapping are provided by the visible pair of fiducials, whose measured pixel locations, continuously updated, allow constant correction for TV flexure. Guiding thus maintains constant guidestar position relative to DEIMOS' focal plane rather than relative to the TV pixels. Scale and higher-order distortion terms in MF/SMCS tex2html_wrap_inline5883 TVCS are provided by calibration images of astrometric standard star fields. [...if needed.] Separate mappings MF/SMCS tex2html_wrap_inline5883 TVCS are needed for each TV filter. They may also depend on temperature.

[a lot changed above - must be reviewed for consistency later on. Also, TV, FCS figures must be created. And two more figures below.]

2.2 Mathematical Expressions for Mappings


For most of these coordinate systems, the definition of the cardinal axes corresponds to the global coordinate axes of the DEIMOS design (see Figures ?? and ?? for the global and individual DEIMOS' coordinate systems). All x-axes point in a set direction (``up'' in Figure ??), normal to the light path, the local z-axis is always in the direction of photon travel, and the y-axes generally point ``outboard''. However, this last convention is inverted for the ICS and CCDCSi coordinate systems, whose y-axes point inboard (see Figure 2.3). The ICS is the coordinate system that the observer sees on the quick-look display, where it has (0,0) in the lower left corner with x increasing to the right and y upwards. It is also defined so that positive y points towards increasing tex2html_wrap_inline5715 along the spectrum and the display shows a direct view of the sky for direct imaging (see Figure 2.3). These constraints require that the y-axis reverse direction in the ICS and CCDCSi coordinate systems.

2.2.1 FKCS tex2html_wrap_inline5620 RSCS:

The fundamental sky coordinates can be described in any equinox and epoch. To update and refract to observed coordinates on the sky, four transforms are needed, to account for (1) proper motion; (2) precession; (3) aberration of starlightgif; and (4) atmospheric refraction:





where tex2html_wrap_inline6001 are the updated, refracted coordinates of the fundamental sky tex2html_wrap_inline6003 at the zenith distance z, wavelength tex2html_wrap_inline5715 , and atmospheric temperature and pressure, tex2html_wrap_inline6009 .

These are all standard transformations, and will be taken from the Starlink SLAlib library.

2.2.2 RSCS tex2html_wrap_inline5620 MF/SMCS:

This mapping is performed in three stages: first, a gnomonic projection is made of the [refracted] sky; second, an adjustment is made in the radial direction for distortion; and lastly a projection onto the tilted, curved surface of the maskgif. Due to the format of the distortion mapping from ray-tracing programs, these take the form:




where tex2html_wrap_inline6029 are the physical coords projected onto the tilted, cylindrical surface of the mask (described by a tilt, tex2html_wrap_inline6031 , and a radius of curvature, tex2html_wrap_inline6033 ); tex2html_wrap_inline6035 are the standard coordinates produced by a gnomonic projection of tex2html_wrap_inline6037 at position angle tex2html_wrap_inline6039 , converted to off-axis angles on the sky. The position angle of the mask, tex2html_wrap_inline6039 , is defined as the position angle of the x-axis on the sky.

The gnomonic projection is:



These are tangents; rotation so that tex2html_wrap_inline6049 is parallel to the y-axis and converting to angles:



where tex2html_wrap_inline6039 is the position angle of the x-axis on the sky.

The mapping from the off-axis angles to the projected physical coordinates, tex2html_wrap_inline6059 , is determined via ray-tracinggif. It is best determined using image centroids rather than principle rays.

The mapping from the projected physical coordinates to the tilted, cylindrical surface is:




The value 135mm above is a zeropoint offset, that is, the value of tex2html_wrap_inline6071 where tex2html_wrap_inline6073 and tex2html_wrap_inline6075 .

In addition, the tilted and curved surface produces a modification to position angle tex2html_wrap_inline6077 with respect to the x-axis ( tex2html_wrap_inline6079 ), given by


2.2.3 MF/SMCS tex2html_wrap_inline5620 ICS:

This mapping is performed in three stages: first, a projection from the tilted, curved slitmask to a plane; secondly, conversion into input angles at the grating/mirror; then mapping with the camera distortions to the camera focus.




where tex2html_wrap_inline6091 and tex2html_wrap_inline6093 are various fixed pupil distances and radii (see Figure 2.4).



Figure 2.4: Schematic of rays, telescope through collimator

The first mapping is a reverse of the earlier projection:




The second mapping to input angles depends on the axial radii and heights, both at the image surface (slitmask) and the collimator.




where tex2html_wrap_inline6107 and tex2html_wrap_inline6109 are the heights of the image and collimator surfaces:



For rough estimates, tex2html_wrap_inline6107 and tex2html_wrap_inline6109 can be ignored. In practice, the approximation for tex2html_wrap_inline6109 is probably sufficiently adequate, although for highest precision and computational speed, it is probably best to provide a table of precise tex2html_wrap_inline6121 values and interpolate for specific tex2html_wrap_inline6123 .

The axial angles tex2html_wrap_inline6125 are converted into incident angles at the grating by



where tex2html_wrap_inline6131 is the tent mirror angle and tex2html_wrap_inline6133 the angle of the grating normal, both with respect to the optical axis.

The third mapping is straightforward, although it includes factors for camera distortion which are available from ray-tracing (but may need to be determined empirically?). In the following, the angles and their sense are given in Figure 2.5.



Figure 2.5: Schematic of rays, grating to detector




where tex2html_wrap_inline6141 is the camera angle with respect to the collimator optical axis. Note that tex2html_wrap_inline6143 are measured with respect to the optical axis of the camera.

With the mirror in place, incident and reflected angles are equal in magnitude but opposite in sense:


With a grating in place:


where m is the order, usually first, n is the index of refraction of air, and tex2html_wrap_inline6153 is the spacing of rulings on the grating.

Note that the third mapping is usually needed in reverse, in order to estimate a tex2html_wrap_inline5715 for a given pixel location:



2.2.4 ICS tex2html_wrap_inline5620 CCDCSi:

This mapping involves rotation and translation constants for each of the eight CCDs in the mosaic. CCDCSi will generally be expressed in pixels.



where tex2html_wrap_inline6175 is the rotation and tex2html_wrap_inline6177 are the translations for each CCD with respect to the ICS. Note that even the fiducial CCD can have a rotation and offset with respect to the ICS, which by definition has axes parallel to those of the MFCS.

2.2.5 MF/SMCS tex2html_wrap_inline5620 TVCS:

[Not yet described.]

2.2.6 Constants:

Engineering drawings currently give the following values:


The following grating parameters will also be of interest when discussing vignetting:


2.3 Empirical Mappings


[We need to make some direct mappings between non-adjacent coordinate systems, as in astrometry off direct images, and the definitions from the theoretical mappings above do not describe these well. Furthermore, these mappings must be made empirically, although the analytic mappings may be used in many cases to provide estimates of their form and scales. These are the mappings that will be discussed:

ICS tex2html_wrap_inline5883 CCDCDi. Although discussed above, we must measure the actual tex2html_wrap_inline6215 empircally.

ICS tex2html_wrap_inline5620 FKCS. This is astrometry, going directly from the image to the fundamental celestial coordinates, via astrometric references in the field.

ICS tex2html_wrap_inline5620 RSCS. Same as above, but two or less astrometric references so only refracted coordinates are possible.

RSCS tex2html_wrap_inline5620 SMCS. The analytical mapping should work well, but can likely be improved through a differential mapping -- this should be discussed here. ]

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