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A Progress Report

1. Introduction

EXOSAT, ESA's first scientific three-axis stabilised satellite, is maintained in stable pointing with an accuracy of approximately ± 2-3 arc seconds using gyros for attitude measurement and control of a propane gas thruster system. The overall attitude measurement accuracy is about 5-6 arc seconds. A comparison of the determined EXOSAT position and the accurately known optical/radio position of a number of X-ray sources has shown systematic differences of 4 to 15 arc seconds with a mean of approximately 10". This note describes work carried out to investigate and improve the accuracy of EXOSAT source position determination.

2. Coordinate Transformations

Details of coordinate transformations, use of -the attitude information and the telescope misalignment data have been given in the FOT Handbook (Sect.7.1). For completeness, the mathematics involved is presented here.

Generally, the transformation from the coordinate system (U,V,W) to a new coordinate system (X, Y, Z) involving rotation and no transition can be written as follows:

transformation matrix

where Xu, Xv, Xw are the three components of the X unit vector with respect to U, V and W (so called direction cosines) etc.

If the coordinate frames are orthogonal then the transformation matrix T is orthogonal and the inverse of T is equal to the transpose.

T-1 = TT
so that

transformation matrix

Three coordinate systems are defined as follows:

1) The star tracker reference frame (spacecraft axes): (XST, YST, ZST)

2) The LE detector reference frame (after linearisation): (XLE, YLE, ZLE).

3) The Celestial Mean Equatorial (ME) 1950 reference frame: (XME, YME, ZME)

Two transformations are required to establish the relation between the detector reference frame and the Mean Equatorial frame:

A) ME -------- ST
B) ST -------- LE

A) The auxiliary data on the FOT contain the spacecraft attitude defined as the three components of the roll, pitch and yaw axes with respect to the ME reference frame, ie:

roll = (rx, ry, rz)

pitch = (pX, pY, pZ)

yaw = (yX, yY, yZ)

The transformation ME -------- ST is then written as:

transformation matrix

where AUX1 .... AUX9 are the values in the Auxiliary data file on the FOT, record 3 bytes 8-124 (see FOT Handbook, Sect. 3.8 p. 11 ) . Note that these values are converted from the requested nominal pointing direction. The actual pointing direction is obtained from the star positions in the star tracker field of view ( refer to next section). RPY is the attitude matrix printed in the automatic analysis output.

A 5" offset is applied to the Y-axis limit cycle to overcome an operational problem of the AOCS electronics (ref. Express No.4 p.34). This affects all observations carried out from 15.9.83 at 07.00 UT (day 253, SHF time key 116924400) and pointing positions must be corrected appropriately by an additional transformation.

transformation matrix

where epsilon symbol = +5".

B) The relationship between the ST and LE frames is given by the telescope misalignment data in the LE CCF (data type BD) The transformation ST ------ LE is written as:

transformation matrix

where MIS1 .... MIS9 are given in the LE CCF (data type BD) bytes 0 - 35 (CMA) multiplied by 109.

The misalignment is the displacement of the LE frame with respect to the ST frame (see Fig. 1) and can be derived in terms of 3 consecutive rotational matrices about the roll, pitch and yaw axes:

rotational matrices

where delta alpha, delta beta and delta gamma are the anti-clockwise rotational angles (misalignment angles) as seen from the respective positive axis in the direction of the origin (fig. 1).

equation with test imbeded text

equation with test imbeded text

equation with test imbeded text

3. Results

An analysis of a number of EXOSAT X-ray sources with known optical or radio counterparts and no detectable proper motion has been carried out and the results are summarised in Table 1. Source positions were taken from the compilations of Burbridge et al. (1977), Lang (1980), Weller et al. (1980) and Clements (1981). The fourth column of Table 1 gives the day of observation and the next three columns the resulting number of counts (cts) and the position xb, yb in the L1 CMA, determined as the baricentre of the distribution of the source counts corrected for background obtained from regions outside the source location.

EXOSAT RA and Dec. positions are derived using the formula given in the FOT Handbook (Sect. 7.1 p.46). The 5" offset of the Y-axis limit cycle and refinements to the accuracy of the calculation were included. Columns delta alpha and delta delta give the deviations (arcsec) of the EXOSAT position with respect to the known optical/radio position (column 2 and 3).

Secondly, a correction has been applied for the typical limit cycle of ± 2-3 arcseconds maintained during an observation. Image 'deblurring' was carried out by the simple method of calculating the mean offset (pixels) in pitch (delta pitch) and yaw (delta yaw) from the attitude data in the HK records and changing the positions accordingly (see Fig. 1):

Xd = Xb + delta pitch
Yd = Yb - delta yaw

from which the deviations delta alpha and delta delta, of the order of 10", were calculated. In general, delta pitch and delta yaw are less than ~2".

Two X-ray sources (3C273 and 1156+295) in the sample have twice been observed about half a year apart, with opposite orientation of the spacecraft. Fig. 3 shows the computed 'deblurred' positions for 3C273 together with the orientation of the Ex, Ey plane of the L1 CMA. The source was observed using different filters yielding a number of positions which are within a region of about 4" diameter. A systematic error is clearly present, the two sets of observations each being located opposite the true source position and about 30" apart. Observations of 1156+295 show the same behaviour suggesting a telescope to star tracker misalignment error. This error has been calculated empirically by taking for each set of observations the difference in the Ex and Ey direction:

3C273 : X = 24.3", Y = -18.2"
1156+205 : X =23.", Y = -22.5"
Mean : X = 23.66", Y =-20.36"

giving the following errors in the misalignment angles (see Fig.1).

delta beta = X/2= -11.83"
delta gamma = Y/2= +10.18"

If this correction is applied to the misalignment angles, new positions and deviations delta alpha and delta delta are derived. These deviations are typically less than ~7".

In July 1985, improved star tracker calibration reference data was implemented at ESOC (p.2) yielding a better knowledge of the actual pointing direction. Generally, the calculated (requested) pointing direction is within ~1" of the actual one. As a final correction to the EXOSAT positions, this new calibration data gave slightly improved deviations delta alpha north-south and delta delta north-south. delta north-south, the corresponding distance, is now typically ~6".

There are a few exceptions to this, some of which may be caused by the linearisation of the detector read-out positions. The linearisation for LE1 is based on an in-flight raster scan (Cyg X-2), which is less accurate than the ground-based calibration for the LE2 CMA. However, close to the detector origin, where the linearisation is most accurate, delta north-south the difference between known RA, DEC and the EXOSAT position is less than 4-5".

In some cases, poor statistics (<~ 50 cts) may yield inaccurate positions, see eg. the third pointing of MKN 421, where Yb is about 1 pixel less than the two previous values.

4. Conclusions and Status of Analysis Software

For a small sample of X-ray sources accurate positions have been determined by including in the analysis the following factors:
  • improvement in the calculation accuracy
  • 5" offset applied to the Y-axis limit cycle.
  • image deblurring of the 2-3" limit cycle.
  • possible errors in telescope star tracker misalignment.
  • improved star tracker calibration data.
Systematic errors are now negligible and generally the deviation with respect to the known optical/radio position is <~ 6", well within the design goal of the system.

Further work is in progress to:

  • implement a more sophisticated deblurring method where each photon is treated individually.
  • use a correlation technique in which the source distribution is compared to the PSF based on (in-flight) calibrations.
  • analyse several sets of observations separated by half a year to determine the misalignment errors accurately, together with any time dependence.
The Observatory software has been modified as a result of this work as follows:

(A) Interactive Analysis system

  • 5" offset of the Y axis limit cycle included.
  • improved accuracy of thc calculation.
  • new set of misalignment angles.
No deblurring to account for the limit cycle is presently included since in certain cases the algorithm appears to give misleading figures - the problem is under investigation.

(B) Auto analysis software

No modifications
(1) the new star tracker calibration data is automatically used as from 30.7.85. Observations carried out prior to 30.7.85 have been analysed using the old reference data although any differences in derived pointing are generally small (~1") compared to the other effects described above.
(2) the misalignment angles have not been modified in the CCF, pending further study.
Users primarily interested in very accurate position determination are therefore requested to contact the Observatory Team or apply to use the IA analysis system.

E. Gronenschild


Burbridge, G.R., Crowne, A.H., Smith, Harding, E. Ap.J.Suppl., 33, 113 (1977).
Clements, E.D. M.N.R.A.S. (1981), 197, 829.
Lang, K.R. Astrophysical Formulae, p.195 (1980).
Weller. K.W., Johnston, K.J. M.N.R.A.S. (1980), 190, 269.


FIG 1 description below
Figure 1
Orientation of EXOSAT coordinate reference frames of the star tracker and the LE detector, showing the misalignment angles delta alpha, delta beta and delta gamma
FIG 2a description below FIG 2b description below
Figure 2
The spacecraft attitude in the mean equatorial (ME) coordinate frame(a). The pointing direction is given by the vector XLE, the detector origin by alpha and delta. The north angle theta is defined as shown (b).

FIG 3 description below
Figure 3
The observations of 3C273. Two sets of observations were carried out, separated by half a year. The first set yielded positions in the lower right- hand corner, the second set in the upper left-hand corner (numbers refer to the sequence of pointings given in Table 1). The orientation of the detector reference frame Ex, Ey is also indicated. Final source positions (index NS) after deblurring, incorporation of the new misalignment angles and taking the actual pointing positions based on a new star tracker calibration are now close to the true source position.
table 1 listing the misalignment angles
Table 1

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Last modified: Thursday, 26-Jun-2003 13:48:32 EDT