1. INTRODUCTIONA discussion of the sensitivity of the Channel Multiplier Array (CMA) on board the EXOSAT satellite has been given elsewhere (e.g. De Korte et al 1981, Space Science Rev. volume 30, EXOSAT Observers Guide part 2 and part III sect. 8). This was however based on pre-flight calibrations and did not take into account all the peculiarities of the in-orbit performance.
The availability of a large data base containing information from several hundreds of observations, carried out in the first two years of operations, together with the necessary sophisticated software, allows an accurate measurement of the sensitivity of the CMA. The purpose of this paper is to describe the sensitivity of the CMA to point sources over the entire field of view (FOV). Because the CMA at the focus of the second EXOSAT telescope has been operational for only a relatively small fraction of the mission, and no significant differences are expected in its performance compared to CMA 1, only data obtained with CMA1 is analysed here.
Formulae used to determine the limiting sensitivity of the instrument, are derived in sect. 2. Sections 3, 4 and 5 discuss the spatial distribution of the background, the off-axis point spread function (PSF) and the efficiency of the telescope optics as a function of the position in the FOV, conclusions are presented in section 6.
2. MATHEMATICAL FORMALISMThe X-Ray flux of a cosmic source can be calculated using,the following equation:
where A(E) is the effective area of the system comprising the telescope, the CMA and the filter, dN/dE is the source photon spectrum, c/r is the observed count rate, is the absorption cross section, NH the equivalent hydrogen column density in the line of sight and K.ss is a correction factor which depends on the sum signal distribution of the source photons (see EXOSAT Observers Guide part III for details). For X-ray sources which do not emit strongly in the UV, K.ss is generally very close to 1. E1 and E2 define a minimum and maximum energy outside which the effective area of the system is negligibly small. These values are generally assumed to be E1=0.05 and E2=2.0 Kev. The minimum detectable flux (fm), i.e. the limiting sensitivity of the instrument, is essentially determined by the minimum detectable count rate c/r(m) and the effective area of the system. c/r(m) depends on the exposure time (t), on the Point Spread Function (PSF), and on the mean background level (b). The exact calculation of the CMA effective area is described in detail in the EXOSAT Observers Guide part III sect. 8.1 and need not be repeated here.
It can be shown that
A(E) = A(x=0,y=0,E(x,y)) * h(x,y)
where x, y is the source position in pixels, E is the energy, and h(x,y) is a function of x and y only. As a first approximation the positional dependence of E can be neglected since this is not likely to introduce large inaccuracies. In this case
A(E,x,y) = A(E,x=0,y=0) * h(x,y)
and equation (1) can now be rewritten as
which is equivalent to
where g(x,y) is a function defined by
4a and 4b are likely to produce only very small errors and will therefore be neglected. Eq (3) shows that the minimum observable flux at the position x,y is equal to the minimum observable flux in the centre of FOV multiplied by a function of x,y only. Let PSF(i) be the fraction of the source net counts contained in a square box denoted by the index i and centered on the source centroid. The index i is defined such that box 0 is a square of 1 pixel in size, box i is a box whose size is 2i+1 pixels. (Note that throughout this paper, PSF(i) refers to the integral or cumulative Point Spread Function). A typical cumulative PSF in the central region of the FOV is shown in fig 1.**
**The curve shown in fig 1 represents the typical Point Spread Function as observed with the Lexan or the Al/par filters. There is evidence that the shape of the Boron PSF is a strong function of the source energy spectrum (see EXOSAT EXPRESS no. 10 page 45) and therefore, reference is always made to the Lexan or Al/Par PSF.
The source count rate is generally estimated as follows
where NET(i) is the source net counts in box i and t is the exposure time corrected for telemetry and sampling dead time.
NET(i) = TOTAL(i) - b*(2*i + 1)2 * t
TOTAL(i) is the total number of counts found in box i and b is the average background level in units of counts/sq.pixel/sec.
Fig 2 shows the distribution of the background intensity in the central region of the FOV, as determined in the automatic analysis, for several hundreds of CMA observations. Since the LE automatic analysis excludes all periods when the background is anomalously high the distribution shown in fig 2 is representative of the CMA background under normal conditions (i.e. in the absence of what is generally, referred to as 'solar activity'). The distribution is rather narrow and centered on b=9.0-6 counts/sec/sqpixel.
This background value is assumed throughout this paper.
The statistical error associated with c/r is
and the signal-to-noise ratio s/n is
since the expected value of NET(i) is
Fig 3 shows the signal to noise ratio plotted against box number for different values of the count rate. In this example the exposure time has been assumed to be 1 EXOSAT unit (1.04 seconds) but the graph can easily be resealed to any value of t using eq (5). For low values of the count rates the signal to noise ratio is sufficiently high only for a rather narrow range of box sizes. Obviously, for sources near the limit of detectability it is extremely important to choose the box size that gives the best possible signal to noise ratio.
This result is strictly valid only if the PSF is perfectly known. In the central region of the FOV this is true unless the source is characterized by a very hard spectrum and/or high photoelectric cut-off (NH>1.022). As an example, the cumulative PSFs of several bright sources, covering a wide range of spectra and photoelectric absorptions, and detected near the centre of the CMA FOV, are plotted in figure 4. The curves are all very similar and support the hypothesis that the CMA PSF is largely independent of the nature of the source. Note, however that for NH > 1.022 the PSF could be wider than those shown in figure 4. Since the PSFs plotted in fig 4 have been obtained from data collected using both Lexan and Al/par filters the plots indicate that, within statistical limits, the PSFs in each filter are essentially identical.
Solving eq (5) with respect to c/r gives
The minimum detectable count rate can be defined as that value of the count rate corresponding to a minimum acceptable signal-to-noise ratio; this minimum value is often taken as 5. Eq (6) has been plotted in figure 5 as a function of the exposure time t for the case of a signal to noise ratio equal to 5 (solid line). For small values of the exposure time, c/r is almost a linear function of t, corresponding to the case of the photon limited regime. In the limit of very large exposure times and/or large background values the minimum detectable count rate becomes proportional to the square root of t . This corresponds to the case of the present generation non-imaging X-ray detectors for which the background is much higher than the minimum detectable source strength.
For X-ray images, however, a good estimate of the background can be obtained in regions close to the position of the source. Here the minimum detectable count rate can be defined as that value which would produce a number of counts such that the associated probability that the event is due to a random fluctuation of the background is sufficiently small. This condition is generally satisfied if the net counts in the box used is equal to or greater than five times the square root of the expected background.
where I is the index that minimizes equation 7. The intensity of the normally expected CMA background is such that for short exposure times the expected number of background counts in the box used becomes so small that eq 7 is no longer valid and the condition that at least 10 source counts must be collected is generally substituted. Eq (7) is plotted as a dashed line in fig 5 and shows that the minimum count rate defined in this way can be more than a factor of five lower than the value obtained with eq 6 for a signal to noise ratio equal to 5.
In order to compare the minimum detectable count rate defined by eq (6) and eq (7) to the actual data, the measured count rate of more than three hundred sources detected within 200 pixels from the centre of the CMA FOV has been plotted in figure 4. The sample has been chosen so that all the sources have been detected at least twice, either in two different filters or in the same filter but on two separate occasions and should therefore be virtually free from spurious sources. It is clear from figure 5 that the use of an over- conservative definition for the minimum detectable count rate, such as that given by eq (6) for s/n=5, leads to the loss of a significant fraction of detectable sources. Hence the minimum detectable count rate definition given by eq (7) with the restriction that the net counts must be > 10 will be adopted.
3. THE-SPATIAL DISTRIBUTION OF THE BACKGROUNDAs described in the EXOSAT Observers Guide part III sect 8.1, the CMA background is not uniformly distributed over the FOV. The background- intensity shows a local maximum at the detector centre. It decreases at intermediate distances with roughly circular symmetry and finally increases again at the very edge of the FOV. Quantitatively, the effect is of the order of 10 percent in the central square degree.
A very deep image has been analysed (exposure time ~160.000 seconds) to illustrate in detail the spatial distribution of the CMA background. This image is free from extended sources and all point sources have been removed from it and substituted with a randomized mean background, estimated in regions near the sources. The filter used was 3000 Å Lexan. A comparison of this image with exposures obtained at different epochs indicates that the structure of the background does not significantly change with time. In addition, comparison with an image obtained summing exposures performed during periods of high 'solar activity' (i.e. in the presence of anomalously high particle background) shows that the spatial distribution of the CMA background depends on the overall background intensity -- on average, the spatial distribution of the background during periods of high 'solar activity' differs from the distribution normally expected by an amount of the order of 5-10%.
In order to show the spatial distribution of the CMA background on the scale of a few arc minutes the image has been rebinned such that the size of each pixel is about 2 arc minutes. Figure 6 shows the background iso- intensity contours of the resulting image. The contours have been resealed such that the central region of the image corresponds to an intensity of 100. This image can be used to calculate the CMA background in any part of the FOV, given its value in the centre.
4. THE OFF-AXIS POINT SPREAD FUNCTIONData obtained during an in-flight raster scan performed on July 5 1983 has been analysed to study the off-axis CMA PSF. This raster scan consists of 43 observations of the X-ray source CYG X-2 carried out such that the source is detected at different locations with respect to the optical axis of the instrument. The 43 pointings cover the whole field of view and the exposure times were chosen to give at least 8-900 counts during each observation.
For each pointing the CMA Point Spread Function has been determined as follows
1) estimate the mean background in a nearby region using an area sufficiently large to guarantee a small statistical error and rescale the value thus obtained to the position of the source taking into account the background spatial distribution described in the previous paragraph.
2) calculate the source centroid X,Y
3) estimate the net counts in square boxes centered on the source centroid, subtracting from the total number of counts in the box the expected background in the same area calculated as described in p,oint 1) and corrected for spatial non-uniformities
PSF(i) = (TOTAL(i) - b#(2i+1)2#F(i,X,Y)#t) / TOTAL
b is the average background in units of counts/sec/sqpixel, t is the exposure time and F(i,x,y) is a function which corrects for the spatial distribution of the background, and TO,AL is the total number of detected photons from the source.
The above procedure has been carried out for all pointings of the raster scan. Resulting PSFs have been grouped according to the distance from the optical axis (which, for the case of LE1 is located at X=136 Y=61 in linearized coordinates) and are shown in figure 7. Note that within a distance of roughly 200 pixels from the image centre, the PSFs remain constant in shape and essentially identical to the one shown in fig. 1. At larger distances from the centre the PSF broadens by an amount dependent on the distance. The dependence on the orientation of the source position with respect to the instrument axis is small and is probably negligible. Broadening of the PSF with distance from the optical axis is also shown in fig 8 where the index number (I(50%)) corresponding to the box containing 50% of the source photons is plotted against distance from the centre. This parameter is essentially constant for distances smaller than ~200 pixels and then increases approximately linearly with distance.
5. THE EFFICIENCY OF THE OPTICSThe dependence of the CMA effective area on position in the FOV mainly results from the variability of the efficiency of the telescope optics across the image. The exact dependence, as measured during pre-flight calibrations is given in the EXOSAT Observers Guide part III sect 8.1, fig A. A raster scan of the CRAB Nebula has been analysed to check whether the ground calibration is directly applicable to the in-orbit data. Use of the CRAB Nebula raster scan has two disadvantages.
First, the CRAB is not a point source and therefore the estimate of its count rate must be made in large boxes, therefore increasing the overall errors. This is especially true at the edge of the FOV where the Point Spread Function is very wide and where the effect of the obscuration due to the ME flap is largest (ref page 67 and the EXOSAT Observers Guide part 3 sect. 8.1)
Secondly, the CRAB raster scan did not cover the entire field of view.
For each observation, the CRAB count rate has been estimated and corrected for the efficiency of the optics at the position of the source to give the best estimate of the source count rate in the centre of the FOV. The results are summarized in Table 1, showing in Columns 1 and 2 the source position in pixels, in columns 3 and ~ the measured count rate and the relative error, in columns 5 and 6 the corrected count rate and error, and in column 7 the distance from the image centre in arc minutes.
All the values listed in column 5 are close to the mean value of 12.26 c/sec. Pointings with the source closer then approximately 20 arc minutes to the centre give a corrected count rate distribution with a standard deviation of 0.35 cts/s (~2.8% of the mean). The standard deviation increases to 0.97 c/s when only distances greater than 20 arcmin are considered and the mean value becomes 12.02. Given that minor corrections such as the dependence of the detector quantum efficiency on the position in the FOV and the corrections for the sum signal distribution have been neglected, the above result is very satisfactory.
6. CONCLUSIONIn the previous sections, the formulae necessary to calculate the CMA minimum detectable count rate have been derived and examples given to illustrate the sensitivity of the instrument in the central part of FOV. The estimation of the minimum detectable count rate can now be extended to any position in the FOV by making use of eq. (7) i.e.
b, I, and PSF(I) all depend on the position (x,y). Near the centre of the FOV, PSF(I)~0.5 and for a generic position (x,y), PSF(I) should be calculated using the point spread functions shown in fig 7. For each PSF derived from the CYG X-2 raster scan the value of the index i (Im(x,y)) which minimizes eq (7) (i.e. the value of i for which c/r(m) is minimum) has been calculated.
The background b(x,y) is estimated assuming circular symmetry around x=136 y=61 and taking into account the spatial distribution described in sect 3.
Eq 7 can therefore be rewritten as
The minimum detectable count rate defined by eq (8) is plotted in figure 9 as a function of distance from the detector centre for the case of t=10000 seconds. Before this count rate can be converted into flux it must be corrected for the efficiency of the telescope optics at the position x,y. This correction is applied using the calibration values given in the LE current calibration file (CCF). Eq (2) now becomes
where K.opt (=1/h(x,y), see eq. 2) is the correction due to efficiency of the telescope optics. Consequently eq (3) becomes
g(x,y), originally defined in eq (3), describes the functional dependence of the CMA sensitivity on the position in the FOV and is plotted in figure 10 assuming circular symmetry for b(x,y) and K.opt.
It has been stressed elsewhere that the conversion of the CMA count rate to flux is a strong function of the source energy spectrum and of the total amount of equivalent hydrogen column density in the line of sight. The derived limiting sensitivity can be very different for sources characterized by different spectra and/or photoelectric absorptions. The reader is advised to refer the EXOSAT Observers Guide part III chapter 8 where the conversion factors necessary to convert count rate in the centre of the FOV into flux in erg/cm2/sec for a wide range of spectral parameters are given. These conversion factors can easily be scaled to any position in the CMA image using eq 9 or figure 10.
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