Upper limits

  As an object for further discussion it is helpful to have a copy of the upper limits map produced by PSS for the demonstration image. A copy can be created using the command,

   > PSS ast_demo:pss_demo 1.01 mode=upmap map=demo_map NOEXPERT \

Two features of the upper limit map are worth noting immediately,

• The two simulated sources stand out strongly. It is apparent that these sources (FWHM about 3 arcminutes) contaminate the surrounding data for quite some distance
• The remainder of the image is clumpy on a scale similar to that of the psf.

The upper limit flux map is clumpy simply because of local variations in the background. Test positions in an image sited in areas deficient in counts with respect to the local background constrain a necessarily positive source flux more tightly than positions in areas with excess counts. The justification for this statement can be found in Appendix .

The restricted scope of the above analysis must be emphasised. The upper limit flux map is the flux of a point source whose surface brightness is exactly represented by the psf system, at a confidence level 68%. This raises the question of what is required by a particular upper limits analysis -- the answers to two questions should be clear in the mind of anyone using PSS in UPLIM mode,

• Is the test source position known well? The definition of `well' depends on the data, but basically relates uncertainties in a test position to the scale of clumpiness in the upper limit flux surface, which in turn depends on the characteristic size of the psf. There are two possible sources of uncertainty. The first is in the (assumed) X-ray data, where uncertainties are usually limited by residual errors in spacecraft attitude reconstruction. These errors may be manifested as either systematic shifts across a field or simply as a random positional blurring of the events constituting an image. The sizes of these errors are of order 10 and 5 arcseconds for the WFC and PSPC respectively -- about one third of the smallest pixel resolution for both detectors. The second source of error is that in the test position. This is generally smaller than the first source of error for wavebands other than X-ray or -ray, and can be generally regarded as zero for stellar data. Combine estimates of your two sources of error in quadrature and compare with the FWHM of the psf being used. If the former is smaller than the latter by a factor of then using UPLIM mode is acceptable. This is simply because possible errors in the registration of the test position on the image are much smaller than the scales over which the upper limit surface is varying significantly. If the condition is not satisified then UPMAP mode should be used to map the upper limit surface over a small area around the test position, and the upper limit taken to the maximum value in that area.
• Is the test source psf well known? In the case where the test is being performed on an assumed point source this question reduces to one of whether the psf system as accurately modelling the psf. This topic is addressed in Appendix A.1. Where upper limits are required for sources of unknown surface brightness distribution, the only technique avaliable using PSS is to construct various likely surface brightness models (see Section 6.8) and use whatever mode has been selected on the basis of the answer to previous question for these different models.

Practical examples may serve to clarify the above. Suppose we had a poorly constrained source position corresponding to image position in Figure 3, with an error circle of radius 5 arcmin. By our first criterion we should use UPMAP mode, as our uncertainity is comparable to if not greater than the clumpiness in the upper limit surface. The largest upper limit flux within the circle about 26 counts. A supposed stellar object at the same position with an uncertainty of say 1 arcsecond allows use of UPLIM mode and would give a result of only 2.5 counts.