HRC-I Rotation Angle

Capella Data
HR 1099 — 1999-Sep-02
XRCF "Spatial-Linearity" Tests

The HRC-I orientation with respect to the spacecraft has a nominal "45 degree" rotation relative to the spacecraft axis, as shown in figure 1. This angle has not been well calibrated. The recent series of Capella observations that were performed for deriving improved HRC-I degapping corrections provide an ideal data set for determining the angle between the HRC-I crossed-grid charge detector (CGCD) axes and the Science Instrument Module (SIM) translation axis (spacecraft Z-axis). In this set of observations Capella was observed "on-axis" with a different SIM-Z offset used for each observation.

HRC focal-plane layout
Figure 1:The HRC focal-plane layout

Capella Data

The first twenty observations of Capella in the calibration sequence were performed during the schedule weeks: DEC0505, DEC1905 (and its re-plan - DEC2205), and JAN0906. Figure 2 displays the observed spot images of Capella for the different SIM translation offsets from the nominal observing position (labeled in the upper-left corner of the image). The circle in each image is a 1 arcsec radius centered on Capella's expected position given the aspect solution. Observations with larger SIM translation offsets are farther from the expected position, consistent with a relative rotation between the CGCD and the spacecraft Z-axis.

Capella spot images
Figure 2: Images of the spot from Capella for the 20 observations (ObsIDs 654-6559 inclusive) performed during 2005 December and 2006 January. The SIM was at a different translation offset for each of the observations; the offset in mm is indicated in the upper left-hand corner of each spot image. The circle in each image has a radius of 1 arcsec and is centered on the nominal position of Capella given its J2000 coordinates and the observed proper motion (RA = 05:16:41.4026, Dec = +45:59:50.190).

The trend for spot to get progressively farther from the nominal location as the SIM translation gets larger can be explained by the failure of standard processing to correctly account for the rotation angle between the HRC-I crossed-grid charge detector and the SIM translation axis.

The rotation angle can be determined from where the nominal on-axis position is for each SIM translation offset given the observed event positions in CHIP coordinates. Unfortunately, the mean observed CHIP coordinates are not good for this purpose 1) because of the spacecraft dither and 2) because Capella was not observed directly on-axis. However both of these "offsets" can be removed using the known position of Capella and the aspect solution.

  1. Get the nominal RA and Dec for the observation from the aspect solution and determine the RA and Dec offset of Capella from nominal
  2. Add the dither on the sky from the aspect solution to produce a time history of the pointing on the sky
  3. Using the nominal roll from the aspect solution rotate the time history of the sky pointing to the spacecraft axes and invert to get the projected path in DET coordinates
  4. Add the optical bench "bending" offsets from the aspect solution to the projected path of the dither in DET coordinates
  5. Rotate by the nominal "45 degrees" to find the time history of the offset from nominal in CHIP coordinates
Errors on the derived CHIP positions can be calculated from the errors in the aspect solution. These errors will not account for systematic errors due to limitations in degapping or systematic shifts in the expected fiducial light positions due to inadequate knowledge of the retro-reflector/collimator. The un-accounted for errors are expected to be on the order of a couple of HRC pixels. [Note: in an earlier version of this memo the errors on the derived positions were given by the standard deviations from the mean, a clear overestimate of the uncertainty in the derived CHIP position.] Figure 3 shows the resulting "nominal on-axis" CHIP positions for the twenty observations. The range of SIM translation offsets covered was -21 mm to +57 mm. The best-fit line is over-plotted on the data; the fit slope of 0.99583 ± 0.00006 corresponds to an angle between the HRC-I U-axis (CHIPX) and the SIM translation axis (spacecraft Z-axis) of 44.880 ± 0.002 degrees.

Nominal on-axis CHIPY vs CHIPX
Figure 3: Nominal on-axis CHIPY vs CHIPX for the 20 SIM translation offsets, over-plotted with the best-fit line.

A similar result can be obtained using the independent fits of each of the CHIP axes to the SIM translation offset. Figure 4 shows the deviations of the CHIP coordinates from the best-fit line as a function SIM translation offsets. The best-fit lines are

CHIPX = (7530.23 ± 0.14) + (-110.043 ± 0.005) × SIM-Z offset
CHIPY = (7743.15 ± 0.14) + (-109.583 ± 0.005) × SIM-Z offset

The ratio of the slopes corresponds to a rotation angle of 44.880 ± 0.002. Combining the slopes also provides a measure of the HRC-I pixel size. The value of 6.4392 ± 0.0002 µm is somewhat larger than the value from subassembly measurements (6.4294 µm) and this discrepancy should be investigated further. It may be that the discrepancy can be attributed to the SIM translation calibration (i.e. the reported translation step size is ~0.15% too large). The fits also provide values for the nominal on-axis CHIP position for the nominal SIM translation position (7530.23, 7743.15).

Nominal on-axis CHIP coordinates vs SIM-Z offset
Figure 4: Nominal on-axis CHIPX (upper) and CHIPY (lower) deviations from the best-fit line vs SIM translation offset

Comparison to other measurements

HR 1099 — 1999-Sep-02

Ping Zhao used the 1999-Sep-02 calibration observations of HR 1099 to derive a rotation angle of 44.704 ± 0.003 degrees and a pixel size of 6.4390 ± 0.0008 µm. While the derived pixel sizes agree, the rotation angles do not and this disagreement must be understood. I have re-analyzed the HR 1099 observations (ObsIDs 1211-1218 and 1261) using just the source spot locations (as in Ping's analysis) and using the method outlined above, after reprocessing the level 1 event lists to incorporate the latest HRC calibration information.

Spot Locations

The CHIP coordinates for the spot locations don't require correction for dither as the observations were performed with a zero dither amplitude. Figure 5 shows the result for fitting the spot chip locations as a function of SIM translation offset. For errors on the CHIP coordinates I have used the standard deviation from the mean divided by the square-root of the number of events. The large scatter in the spot locations at a given SIM translation offset can be partially attributed to the fact that these observations were quadrant-shutter focus tests and the HRC-I was slightly off the best-focus position. The derived HRC-I rotation angle is 44.706 in agreement with Ping's result.

Spot CHIP coordinates vs SIM-Z offset for all HR 1099 observations
Figure 5: Spot CHIPX (upper) and CHIPY (lower) deviations from the best-fit line as a function of SIM translation offset.

Derived On-axis Locations

Figure 6 shows the results for fitting the data from all nine of the observations where the source spot locations are used to derive the on-axis location as I did for the Capella data; position errors are derived from the aspect solution. From these fits I obtain a HRC-I rotation angle of 44.837 degrees. However, one of the CHIPX points at -54 mm (ObsID 1216) is much different from the others and distorts the resulting fit.

Nominal on-axis CHIP coordinates vs SIM-Z offset for all HR 1099 observations
Figure 6: Nominal on-axis CHIPX (upper) and CHIPY (lower) deviations from the best-fit line vs SIM translation offset for the 1999-Sep-02 observations of HR 1099. All nine observations are included.

Figure 7 shows the result of re-fitting the data while ignoring the data from ObsID 1216. The fit is improved, although the chi-squared values are still large (CHIPX fit: 141.6, CHIPY fit: 41.5). The fit functions are:

CHIPX = (7532.63 ± 0.26) + (-110.021 ± 0.007) × SIM- Z offset
CHIPY = (7740.27 ± 0.26) + (-109.506 ± 0.007) × SIM- Z offset

These fits imply an HRC-I rotation angle of 44.865 ± 0.003, which is in closer agreement with the Capella results than with Ping's.

Nominal on-axis CHIP coordinates vs SIM-Z offset for the HR 1099 observations except ObsID 1216
Figure 7: Nominal on-axis CHIPX (upper) and CHIPY (lower) deviations from the best-fit line vs SIM translation offset for the 1999-Sep-02 observations of HR 1099. Data from ObsID 1216 are ignored in this fit.

The difference in the analysis between the two methods is in accounting for the fact that the source is not observed on-axis and that the spacecraft roll, optical-bench "bending", and source off-axis position changes from observation to observation. Using the "raw" source spot positions does not account for these changes that we expect to have an impact on where the source appears in the detector.

XRCF "Spatial-Linearity" Tests

During instrument calibration at the XRCF, we performed spatial linearity tests in which the HRC-I was moved behind the HRMA through translations of the FAM. While this is not using the SIM for motion, it can provide a comparison of the HRC-I alignment with-respect-to its mounting and the mounting of the HRC instrument. In an earlier memo (HRC-I Spatial Linearity Test G-IHI-SL-1.001 and the FAM Step Size) I found that the HRC-I was mis-aligned to the FAM travel direction by 10.6 arcmin which would be a rotation angle of 44.82.

I have re-examined the data from that XRCF test, G-IHI-SL-1.001, as well as data from a second spatial-linearity test, G-IHI-SL-1.002, to derive alignments for both detector axes. The level 1 data is incomplete in the archive and had been processed with earlier versions of the CXCDS; so, I re-ran hrc_process_events on the level 0 event files. Correction for FAM motion was not applied in the processing. The FAM motion files were used to determine time intervals when the FAM was settled at one of the measurement points and for each interval the mean CHIP location of the source image was determined. The FAM pointings in the test sampled a large number of locations in the central ~10 mm by ~10 mm region of the HRC-I as well as points along each diagonal. In order to avoid a bias from over-weighting by the points in the center they were ignored in the analysis, as were points that has FAM X-axis (focus) positions off of the nominal value. The locations of the selected points from the two tests are shown in figure 8.

Positions of selected spots from XRCF TestID G-IHI-SL-1.001
Positions of selected spots from XRCF TestID G-IHI-SL-1.002
Figure 8: Positions of the FAM and X-Ray spot images in CHIP coordinates for the selected pointings from TestIDs upper G-IHI-SL-1.001 and lower G-IHI-SL-1.002. Blue and red squares encircle the first and last points for orientation reference.

The alignment of each HRC-I axis relative to the FAM axes was performed independently. The FAM axes were assumed to be orthogonal and to have the same translation scale. The relations

CHIPX = scaleCHIPX × (cos(ThetaCHIPX)FAMY - sin(ThetaCHIPX)FAMZ)
CHIPY = -scaleCHIPY × (sin(ThetaCHIPY)FAMY + cos(ThetaCHIPY)FAMZ)
were fit to find the best scale and rotation angle for the HRC-I axes relative to the FAM. (Note: unlike the analysis above the angle here is relative to the Y-axis.) Figure 9 shows the confidence regions for the fits as well as the deviations of the CHIP locations from the best-fit values.

Confidence regions for fit to G-IHI-SL-1.001 data Confidence regions for fit to G-IHI-SL-1.002 data
Deviations of CHIP positions from the best fit to G-IHI-SL-1.001 data Deviations of CHIP positions from the best fit to G-IHI-SL-1.002 data
Results of fits of each of the HRC-I CHIP axes to the FAM axes for TestIDs left G-IHI-SL-1.001 and right G-IHI-SL-1.002.

The best-fit alignment angles and scales are in reasonable agreement between the two tests if we allow for the possibility that the sizes of the confidence intervals are underestimated. The rotation angles for the G-IHI-SL-1.001 are in good agreement with the earlier analysis. The average of the angles for two axes is 45.174 degrees, which once the details of how the WCS-Y location moves with the FAM Y-axis motion of the previous analysis is accounted for becomes 44.826 degrees. Similarly the average of the scales for the two axes is in good agreement with the previous analysis. Once again while the rotational alignment measurements cannot be directly applied to the mounting relative to the SIM translation axis, the rough agreement between the XRCF data and the on-orbit data is likely to indicate the relative alignment of the HRC-I axes to the HRC mounting points.

One additional result from these fits is that the two HRC-I axes appear to be ~0.03 degrees (~1.8 arcmin) off of orthogonal. One way to think of this is that the grooves used to wrap the wires of the CGCD are shifted along one edge of the detector by ~70 µm. The impact of this non-orthogonality is a position shift of ~7 pixels from one side of the HRC-I active area to the other; a small enough amount that it can be ignored.

Conclusions

The Capella observations, because of the large number of SIM translation offsets spanning an appreciable range of the HRC-I detector, provide an excellent dataset for measuring the relative alignment between the HRC-I CGCD readout and the spacecraft axes. After accounting for the non-ideal aspects of the individual observations (e.g the observed off-axis position of the source), I find:
  1. The relative angle between the HRC-I U-axis (CHIPX) and the SIM translation direction (spacecraft Z-axis) is 135.120 ± 0.002 degrees.
    (Note that I have expressed the value as greater than 90 degrees to better reflect the orientation shown in Figure 1.)
  2. The location of the nominal on-axis point for the current default SIM translation position on the HRC-I is
    CHIPX = 7532.63 ± 0.26
    CHIPY = 7740.27 ± 0.26
  3. The reported SIM translation step size is ~0.15% too large if the subassembly calibration of the HRC-I pixel size is correct.
  4. The HRC-I detector axes are not precisely orthogonal; however, the difference is small (1.8 arcmin).

Last modified: Wed Mar 23 13:07:55 EDT 2011

Dr. Michael Juda
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Mail Stop 70
Cambridge, MA 02138, USA
Ph.: (617) 495-7062
Fax: (617) 495-7356
E-mail: mjuda@cfa.harvard.edu