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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.

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.

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

= 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 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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