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Astron. Astrophys. 339, 201-207 (1998)

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2. Data and method

2.1. The data

Our measurement of the expansion of Cas A is based on archival data of the ROSAT High Resolution Imager (RHRI) and the Einstein HRI (EHRI). Table 1 gives an overview. A detailed description of the RHRI and some information on the EHRI can be found in David et al. (1997). For the EHRI one can consult Giaccconi et al. (1979). Both instruments are microchannel plate detectors and are similar in design; they were in fact built by the same hardware group.


Table 1. The observations used for this research.

The instruments have a spatial resolution of [FORMULA] FWHM, but the EHRI point spread function has broader wings. Note that the actual resolution of both instruments may be slightly worse due to time dependent residual errors in the aspect solutions, which are of the order [FORMULA]. The point spread function does not vary within off-axis angles of [FORMULA], which is twice the radius of Cas A. Consequently, off-axis effects are not important for our analysis. The energy responses of the instruments are different as can be seen in Fig. 1. The interstellar column density towards Cas A is large, so most of the photons with energies below 1 keV are absorbed. This implies that most detected photons have energies between 1 and 2 keV, but the EHRI image has an additional contribution from photons with energies up to [FORMULA]4 keV. As for differences in plate scale, a study of M31 sources showed that 1 RHRI pixel = [FORMULA] EHRI pixel, 1 RHRI pixel being [FORMULA] (David et al. 1997).

[FIGURE] Fig. 1. The ROSAT HRI (solid line) and the Einstein HRI (broken line) effective areas.

The data we used were in the form of photon lists which have already been subjected to a basic reduction process. From these photon lists we made images which were converted to the coordinate system of the 1995/96 RHRI image. After conversion the images were rebinned by a factor of 4 to a nominal pixel size of [FORMULA], so we oversample the resolution with a factor 2 to 3, compliant with Nyquist's criterion.

We corrected the long exposure of 1995/96 for the fact that it actually consists of many single exposures. The problem is that the attitude reconstruction of ROSAT is about [FORMULA]. By matching the images of the individual exposures we improved the quality of the combined image. However, because we only used exposures longer than 5 ks, our final image consists of 95 ks instead of 180 ks of data. The rms spread in the attitude corrections was about [FORMULA]. The procedure used for matching images was similar to the procedure for measuring expansion, which is described below.

2.2. The method

The expansion of Cas A in X-rays can be qualitatively seen easily by blinking the EHRI and the RHRI 1995/96 images. In fact, due to the differences in plate scale, the actual expansion is even larger than viewed in this way. Our method of measuring the expansion is straightforward: we scale the latest image in such a way that no expansion between the two images is discernible anymore. Thereby we neglect small scale changes such as those occurring in individual knots.

Instead of judging by eye whether two images match after rescaling one image, we used the maximum-likelihood method for a poissonian distribution (Cash 1979) to see if the two images match. The main advantage of the maximum-likelihood method over a [FORMULA] fitting is that the latter assumes that the number of counts per pixel has a gaussian distribution, whereas the maximum-likelihood method can be used for arbitrary distributions. The maximum-likelihood method for a poissonian distribution involves the maximization of the so called C statistic. The difference in the C statistic (also called the log likelihood ratio) for different parameter values has a [FORMULA] distribution (Cash 1979).

In all cases we use the RHRI 1995/96 image as the model image. We can neglect the statistical errors introduced by the model image, since it is statistically far superior to any of the images used for comparison. For example, the second best image from a statistical point of view is the Einstein image from 1979 which has a total number of photons a factor 7 less than the model image. So the average statistical error per pixel is entirely dominated by the other images. Moreover, as we shall indicate below, the systematic errors dominate the statistical errors.

In addition to fitting an expansion factor, we also fitted the attitude correction, which includes an image rotation, and an additional uniform background level. The level of this uniform background can be determined empirically by introducing it as a free parameter, provided all pixel values of the model image remain positive as required for the likelihood analysis. As can be seen in Table 2 this condition was met in all three cases. All corrections were only applied to the model image (the RHRI 1995/96 image); this image was rescaled, shifted and rotated using a method described in Parker (1994): each new pixel value is based on a linear interpolation of the 4 nearest pixels. The model image was normalized in such a way that it contains the same number of counts as the images used for comparison. In order to reduce poisson noise from the model image we smoothed the image with a gaussian filter with [FORMULA]. This hardly influences the resolution of the model image. The best fit expansion rate was barely influenced by the smoothing procedure.


Table 2. Results of our overall expansion measurements. f is the expansion between two images not accounting for differences in plate scale, [FORMULA] is the expansion timescale (epoch 1996), [FORMULA] and [FORMULA] denote the attitude correction in pixels [FORMULA] needed to match two images (positive values moves the model image towards the Southeast), [FORMULA] denotes the rotational correction applied and the last column gives an additional background rate that was added to the model image before normalizing the image to the image to which is was compared. Only statistical errors are given. The systematic errors are of the order of 20 yr for the measurements involving the Einstein HRI (i.e. [FORMULA] yr). See the text for a discussion of the estimated systematic errors.

In principle it is also possible to optimize for the expansion center. However, there is a correlation between the expansion center and the attitude correction, so they cannot be fitted independently. For this reason the expansion center was fixed to the expansion center found by Reed et al. (1995) to be the best fit to the position/velocity distribution of optical knots ([FORMULA], [FORMULA] ). We experimented with other expansion centers compiled by Reed et al. (1995) and found that the expansion factors were very little affected by our choice of the actual expansion center (the expansion rate changed by less than [FORMULA]). The expansion was evaluated within a given region of the image; the region could be either a circle or a polygon.

The optimization of the likelihood was done by scanning the relevant parameter space in increasingly smaller steps. By applying this method in an iterative way we tried to circumvent potential local maxima.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998