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This Legacy journal article was published in Volume 1, May 1992, and has not been
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Positional Accuracy in the EXOSAT CMA (LE1) Database
Julian Osborne
University of Leicester
1. Introduction
In measuring the misalignment angles of the L1 CMA with respect to the star
tracker (EXOSAT Express,17, 3), a radius was given for the 90%
confidence position error region associated with on-axis sources optimally
extracted. In this article the position accuracy of sources in the L1 database
is described. The X-ray positions that make up the basic data of this article
were extracted at random from the CMA (LE1) database by Hervé Le Meur (a
summer student at the EXOSAT observatory). Optical positions were obtained from
the CDS database SIMBAD. The selected X-ray sources were required to conform to
the following conditions:
- 'zero' statistical uncertainty about reality of X-ray detection.
- EXOSAT field containing no potentially extended objects
- no positions derived from polypropalene filter observations
- reasonable optical candidate (typically an M or B star)
- precise optical position available (i.e. <= 1'' position error)
Optical identifications were made based on the nearest reasonable candidate.
Proper motion corrections were applied where appropriate. 193 X-ray--optical
position pairs were selected in this way. In addition to these, 10 sources with
high x and low y were identified to fill a hole in the position
distribution in the field of view.
2. The Star Tracker Mode
The EXOSAT Attitude and Orbit Control System could be programmed to maintain
the spacecraft attitude on the basis of two stars in the star tracker
field-of-view, or on the basis of one star in the star tracker f.o.v. plus the
position of the sun determined by the coarse and fine sun sensors. It was
anticipated that the two-star mode would yield the most precise attitudes,
whereas the one-star mode would use less control gas.
The following table gives the mean position errors for all on-axis sources as a
function of star tracker mode. There is weak evidence (~2  ) that 'two star' positions are better than 'one
star'
positions. This difference is not visible outside of the central 10 arcminutes of the field of view.
Table 1
EXOSAT Position Errors versus Star Tracker Mode
Star Tracker
Mode
|
Off-axis Angle
Range
|
Mean Radial
Error
|
Number
|
| 1 star |
0' - 10' |
11.53''
1.55'' |
13 |
| 1 star |
10' - 20' |
9.71''
1.63'' |
12 |
| 2 star |
0' - 10' |
8.32''
0.62'' |
51 |
| 2 star |
10' - 20' |
9.34''
0.93'' |
34 |
3. The Time of the Observation
Mean radial position errors for all the positions within 10 arcminutes of the
optical axis are plotted as a function of time in figure 1. The decrease in
accuracy seen is also visible in similar plots using only 'one star' and 'two
star' positions.
4. The Location in the Field of View
Positional accuracy is expected to decrease away from the optical axis due to
the increasing size of the point spread function and the lower quality
linearization at high off-axis angles. Figure 2 shows mean radial position
errors as a function of off-axis angle for the four quadrants of the detector
bounded by lines parallel to the x and y axes and passing through
the optical axis. The quadrants are shown in their correct relative location.
It is clear from these plots that position error is not simply a function of
off-axis distance, and that this statement is true not only far from the axis,
but also relatively close to it.
The position error behavior is made clearer in figure 3, which shows sketched
contours of constant position error superimposed on the detector plane (some
'smoothing' has been done). The optical axis is marked with a cross, and the
positions of all the detected sources used in this study are shown by stars.
The contour levels are 20'' to 120'' in steps of 20''. Clearly the position of
those contours enclosing only one or two sources is uncertain. However, the
reality of the bad position patch centered on (450, -350) is well established
as it contains 7 observations of 6 different objects made over 2 years.
This part of the detector plane apparently suffers greater linearization errors
than elsewhere. This is demonstrated by the fact that the error vector angles
all lie within a 90deg. cone and also by their similar absolute error values.
These are listed in table 2, where 
is the clockwise angle with respect to the instrument (and image) x
axis of the optical position from the X-ray position, and d is the
distance between the two positions.
Table 2
EXOSAT Position Errors in the Bad Position Patch
x
|
y
|
Exosat position ( ,
 )
|
Optical position
(,
)
|
d (arcseconds)
|
(degrees)
|
| 330 |
-390 |
11 33 56.8 |
-61 09 50 |
11 33 56.6 |
-61 11 24 |
94.0 |
42.3 |
| 372 |
-360 |
13 24 01.7 |
-62 22 10 |
13 24 04.5 |
-62 23 23 |
75.6 |
52.4 |
| 486 |
-324 |
18 49 34.6 |
-30 47 59 |
18 49 29.1 |
-30 47 44 |
72.4 |
67.7 |
| 453 |
-350 |
18 49 35.7 |
-30 48 21 |
18 49 29.1 |
-30 47 44 |
92.7 |
55.5 |
| 565 |
-311 |
05 34 00.0 |
+26 52 53 |
05 34 01.3 |
+26 53 42 |
52.0 |
-21.0 |
| 501 |
-294 |
07 40 13.7 |
+50 32 01 |
07 40 17.4 |
+50 33 15 |
82.0 |
25.9 |
| 553 |
-300 |
17 12 02.0 |
38 10 26 |
17 11 57.0 |
-38 09 25 |
84.8 |
48.4 |
Improved positions (2, 2) can be derived for sources with nominal
positions (1,
1) lying within the '80 arcsecond' contour
of the bad
position patch by correcting for the mean position error from table 2
(i.e. = 38.7deg. ,`d = 79''):
 = 360deg.- -  o
sin 2 = sin 1 cosd + cos 1 sind cos
for 0deg. < < 180deg.:
2 = 1 + cos -1 (cos`d - sin 2 sin 1 cos 2 cos 1)
for < 0deg. or > 180deg.:
2 = 1 - cos -1 (cos - sin 2 sin 1 cos 2 cos 1)
Where o is the CMA north angle
(see the FOT handbook, section
7.1, page 46). It is given in the compressed image header and can be displayed
using EXOSAT database software.
The mean radial error for the sources listed in table 2 reduces from `d
= 79'' to 29'' when the above correction is applied.
There is a region of reduced sensitivity (~ 75% of nominal) close to the bad
position patch which is approximately bounded by the coordinates (399, -426),
(446, -413), (446, -443). This is not responsible for distorting the positions
derived for the sources in table 2, although such a distortion could occur for
sources overlying this region.
Figure 4 shows the mean radial position error for all sources as a function of
off-axis angle. Sources in the 'bad position patch' have had their positions
corrected. Also shown are the radii which enclose 90% of the radial position
errors (measured without reference to any specific distribution function), and
the CMA point spread function radius which encloses 90% of the counts of a
point source (derived from EXOSAT Express, 18, 45; 'radius' is a box
half-side). It is clear that position errors are much smaller than the size of
the point spread function.
The mean and 90% error radii plotted in figure 4 are listed in table 3.
Contrary to expectation, the data appear to show evidence for a small rise in
mean position error close to the optical axis. At 2.2 , it is not
clear that this is a real effect, and in any case the area of the detector
involved is very small. For (|x|<20, |y|<20), the position at which most
targets were observed, the 90% confidence radius is 12.0'', slightly larger
than the 8'' determined earlier without recourse to the automatic analysis and
database software (EXOSAT Express, 17, 3).
Table 3
EXOSAT Radial Position Errors (Corrected Data)
Off-Axis Angle Mean Error 90 % Error Radius Number
(arcminutes) (arcseconds) (arcseconds)
0.00 - 6.25 13.7 23.5* 7
6.25 - 12.50 8.9 15.3 75
12.50 - 18.75 8.8 15.4 27
18.75 - 25.00 14.5 26.1 16
25.00 - 31.25 24.1 38.7 31
31.25 - 37.50 27.5 44.0 27
37.50 - 43.75 37.5 67.7 12
43.75 - 50.00 67.3 115.5* 8
* estimated from mean error.
Figure captions.
figure 1 The
mean radial positional error within 10 arcminutes of
the optical axis as a function of observation date, together with the standard
error on that value. Although this plot includes data from both star tracker
modes, the trend to increasing error is still seen in the data associated with
both star tracker modes, 1 and 2, plotted separately. The number of positions
in the bins (increasing in time) is 19 , 18, 12, 15.
figure 2 The
positional accuracy as a function of distance from the
optical axis. The data are separated into four detector quadrants, bounded by
lines of constant x and constant y that pass through the optical axis. The
plots are shown in their correct relative locations. The crosses show the mean
radial error ( 1 s.e.) of EXOSAT positions
with respect to the counterpart optical positions.
figure 3
Contours of constant positional accuracy are overlaid on
the detector plane. The contour levels are 20'' to 120'' in steps of 20''. The
positions of all the sources used in this study are shown by stars, the optical
axis is marked by a cross. A small degree of smoothing (by eye) was applied to
the data before contouring.
figure 4 The
positional accuracy of all sources as a function of
distance from the optical axis. Sources in the 'bad position patch' have had
their positions corrected. The crosses show the mean radial error ( 1 s.e.)
of EXOSAT positions with respect to the counterpart optical positions. The
solid line is the contour which encloses 90% of the position errors in each bin
(estimated from mean for the first and last bin). The dashed line shows the
radius of the point spread function which encloses 90% of the counts.
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