3. The expansion of Cas A
With our method we measure the expansion factor, f, which is just the ratio of the sizes of two images. The physically more interesting parameter is the expansion timescale, , which would be the age of the remnant in case no deceleration had taken place. The relation between the two is:
where s denotes the plate scale ratio (0.995 in case an EHRI image is compared to a RHRI image) and is the time difference between the two exposures. As is clear from this equation an error in f weighs more heavily when is close to 1.
Fig. 2 (right) shows the region of the image used for determining the expansion. In Table 2 we list the values for f and the expansion time scales for each time interval. In case images of two different instruments are compared we have to ensure that the different detector characteristic (such as shown in Fig. 1) do not affect the estimate of the expansion timescale significantly. We tested this using the following method.
If we know the expansion age of Cas A we can, by correcting the Einstein image for the expansion, estimate the differences of the response of the two detectors to the emission of Cas A. We do not know the exact expansion properties of Cas A, but we do have the best fit values listed in Table 2. So by correcting for the expansion we can obtain a, first order, quantitative estimate of the effect of different detector properties on the exposure of Cas A. We have visualized these differences by dividing the combined Einstein images by the 1995/96 RHRI image on a pixel by pixel basis. In order to reduce the poisson noise we smoothed the images with a gaussian with . The resulting ratio image is shown in Fig. 2 (right). Most of the structure seen in the ratio image is now allegedly arising from differences in detector responses to the spectral characteristics of the X-ray emission of Cas A and to possible efficiency variations in the detectors. The absence of ring-like structures in the ratio image indicates that we have corrected adequately for the expansion of Cas A. Note that the most salient feature in the ratio image, the high ratio in the West, can be attributed to the fact that the interstellar absorption peaks in the West of the remnant as OH and HI absorption studies indicate (Bieging & Crutcher 1986, Schwarz et al. 1997; see also Keohane et al. 1996).
To assess the potential sensitivity of our result to the differences displayed in Fig. 2, we corrected the model image with these ratio's and repeated the expansion measurements for this revised model. Although some interdependence is now present in the image comparison, this procedure is legitimate to bring out the influence of systematic effects due to relative differences in the instrumental response functions. If we determine the expansion age of Cas A using our revised model, we find expansion rates that do not differ more than 10 yr from our best fit expansion rates. From this we conclude that the systematic error caused by different instrument characteristics is . An additional systematic error is the uncertainty in the plate scale which is . Combining the two systematic errors we find that the total systematic error in the expansion age in case an EHRI image is involved is of the order of yr. This means that systematic errors dominate the statistical errors. From now on we will always include estimates of systematic errors.
Table 2 lists the individual expansion measurements. Our overall best fit value for the expansion age of Cas A is (for the epoch 1996). This value is a weighted average of the three uncorrected measurements, with the weight being the value including a 19 yr systematic error for the expansion ages determined by a comparison between a ROSAT and an Einstein HRI image. Note that, although the three measurements are not completely independent since they involve the 1995/96 image, our approach is again justified by the superior statistical quality of the 1995/96 image. The expansion age corresponds to an expansion rate of %yr-1. If we adopt a distance to Cas A of 3.4 kpc (Reed et al. 1995), the expansion age translates into a velocity of km/s for the bright ring at a radius of . The inferred velocity of the blast wave at a radius of about is km/s.
The expansion age of 501 yr is remarkable in the sense that it is significantly lower than the expansion age found in the radio (see AR95). The average expansion timescale of all radio knots was found to be yr, for the subsample of faint knots AR95 derived yr (for the epoch 1987). An expansion age comparable to the latter value was derived for the diffuse radio emission. However, AR95 reported substantial variation of the expansion timescale with azimuth. Their shortest timescales, i.e. the timescales of the Eastern and Southeastern sectors, are consistent with our overall expansion timescale. Furthermore, in X-rays the Western region is relatively faint, whereas in the radio it is the brightest region. So there may be a bias in the radio measurements towards the Western region and a bias in our measurement towards the expansion of the Eastern region.
We also searched for variations in the expansion with azimuthal angle. We found that, except for the Western region, all expansion timescales were consistent with the overall expansion timescale within the errors. The sectors for doing the measurements were chosen to be similar to the sectors used by AR95, i.e. we used 6 sectors each spanning . For the Western region (sector V in AR95) we found for and a weighted average of for the time intervals involving the EHRI. In this case a correction for the instrumental efficiencies was applied. All parameters except the expansion factor were fixed to the appropriate values listed in Table 2. The expansion timescale is much larger for a comparison between the EHRI images and the 1995/96 image if we do not correct for the differences in instrumental efficiencies, namely a weighted average for measurements involving the EHRI of and a total weighted average of . So, unfortunately, the effect of the differences in detector characteristics is largest for the Western region making the measurement unreliable (cf. Fig. 2 right). So we have an indication that the Western region is expanding more slowly than the rest of the remnant, although the evidence based on the X-ray emission alone is not conclusive. A measurement of the overall expansion timescale of Cas A excluding the Western region gives yr.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998