2. Observations and data reduction
The X-ray observations were obtained with the ROSAT X-ray telescope (Trümper et al. 1991) in combination with the high-resolution imager (HRI, David et al. 1995). The list of the observations is given in Table 1. The standard data reduction was done with the Extended Scientific Analysis System (Zimmermann et al. 1996), as follows. To take into account the re-calibration of the pixel size (Hasinger et al. 1998), we multiply the pixel coordinates of each photon with respect to the HRI center with 0.9972. A search for sources is made by comparing counts in a box with the counts in a ring surrounding it, and by moving this detection box across the image. The sources thus detected are excised from the image and a background map is made for the remaining photons. A search for sources is then made by comparing the number of photons in a moving box with respect to the number expected on the basis of the background map. Finally, at each position in which a source was found, a maximum-likelihood technique is used to compare the observed photon distribution with the point spread function of the HRI (Cruddace et al. 1988). This produces a maximum-likelihood value ML such that the probability that the source is due to chance at one trial position is given by . We retain sources for further discussion if ML. (To make sure that all such sources are found, we enter in the maximum likelihood technique all sources that have ML10 according to the sliding box searches.)
Table 1. Log of the ROSAT HRI observations of globular clusters analysed in this paper. For each cluster observation, the observation date(s) and exposure time are given. We further give the shift in applied to bring the X-ray coordinate frame of the longest observation to the optical coordinate frame J2000.
Upper limits at the position of known sources were determined by counting the number n of actually detected photons at the position of the source (and in an area surrounding it corresponding to the uncertainty in the position); we then assign as upper limit the lowest expected number m for which the probability of measuring a number n or smaller is less than 5% according to the Poisson distribution.
The maximum-likelihood technique also provides an indication whether the source is extended. If such indication is present, we apply further analysis to test whether the source is a multiple point source.
The further analysis is also based on maximum likelihood techniques, but the analysis is limited to a small area of the detector, near its center. This allows the simplifications that the background in the analysed area is a constant (as opposed to a polynomial function of the pixel coordinates), and that the point spread function is that for the center of the image (David et al. 1995). Suppose that a model to be tested predicts photons at detector pixel i. The probability that photons are observed is then given by the Poisson probability:
The probability that the model describes the observations is given by the product of the probabilities for all i in the region considered: . For computational ease we maximize the logarithm of this quantity:
The last term in this equation doesn't depend on the assumed model, and - in terms of selecting the best model - may be considered as a constant. Thus maximizing is equivalent to minimizing L, where
If one compares two models A and B, with number of fitted parameters and and with likelihoods of and , respectively, the difference is a distribution with degrees of freedom, for a sufficient number of photons (Cash 1979; Mattox et al. 1996).
Our analysis of possibly multiple sources thus proceeds as follows. First we compute for a model with constant background and for the best model with background plus one source, and compare it with the -distribution with 3 degrees of freedom. If , the presence of one source has a significance more than three sigma. Next we compare the best model with two sources with the best model with one source, to prove the significance of a second source; the best models with three and two sources to prove the significance of a third source, etc. until no more significant sources are found.
The addition of one source adds three fitted parameters, one for its number of counts and two for its position. In the case of NGC 6397 optical counterparts have been suggested for three X-ray sources. For these we also make a fit in which the distances in right ascension and declination between these three sources is fixed to the optically determined values. The three sources in that case only add five fitted parameters, two for the position of one of them, and three for the fluxes.
To determine the error in a parameter, we start from the best fit value . We then fix the parameter at and make a new fit, allowing all other parameters to vary. The value of d for which increases by 1 is quoted as the 1-sigma error.
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000