All Figures

$\begin{figure} \par\includegraphics[width=7.2cm,clip]{1087f1a.eps}\vspace*{1mm} \par\includegraphics[width=7.2cm,clip]{1087f1b.eps} \end{figure}$ Figure 1: a) Redshift distribution of 79 distant clusters. b) Redshift distribution of the 45 regular clusters (open) and the 34 irregular clusters (hatched).
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{1087f2.eps} \end{figure}$ Figure 2: Comparison of the temperature distribution of our 79 distant clusters (open) to the nearby flux limited 55 sample of Edge et al. (1990) (hatched).
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In the text

$\begin{figure} \par\includegraphics[width=10.7cm,clip]{1087f3.eps} \end{figure}$ Figure 3: Centroid determination and classification of X-ray morphology. #13 A1689 and #19 A2163 are shown in the left and right panels, respectively. From top to bottom: a) the HRI image in the pixel coordinates. b) The centroids determined in circles of radii $R=n\bar{\sigma}$ ( n=2,3,..,9), where the size of the symbols are nearly proportional to n. c) The deviations the centroids relative to that determined for $3\bar{\sigma}$ in units of standard error, shown as a function of $R~[\bar{\sigma}]$ . While A1689 is classified as a regular cluster, A2163 is an irregular cluster due to the significant centroid deviation.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{1087f4a.eps}\par\includegraphics[width=8cm,clip]{1087f4b.eps} \end{figure}$ Figure 4: Radial X-ray surface brightness profile of #13 A1689 fitted with the single $\beta$ -model a) and the double $\beta$ -model b). The crosses denote the observed radial profile of the ROSAT HRI, and the step functions show the best-fit $\beta$ -models. The best-fit background levels are shown with the horizontal dashed lines. In the panel b), the contribution of the inner and the outer component of the double $\beta$ -model are also shown with the dash-dot and the dotted lines, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=5.8cm,clip]{1087f5.eps} \end{figure}$ Figure 5: $\chi ^2$ contour of the single $\beta$ -model fit for #13 A1689. The position of $\chi ^2$ minimum is denoted with a cross. The curve corresponds to the single-parameter error domain at 90% confidence.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{1087f6.eps} \end{figure}$ Figure 6: Effect of the outer cut off radius in the single $\beta$ model fitting. The results for #13 A1689 are shown as an example. The four parameters of the single $\beta$ -model, $S_0~{\rm [counts~s^{-1}~arcmin^{-2}]}$ , $\beta$ , $r_{\rm c}~{\rm [arcsec]}$ , $C~{\rm [10^{-3}counts~s^{-1}~arcmin^{-2}]}$ are shown as a function of the outer cutoff radius, $r_{\rm out}$ in unit of $\bar{\sigma}$ . The results of the constant fits to the data points at $r_{\rm out}\ge 10$ are shown with the dotted lines.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{1087f7.eps}\end{figure}$ Figure 7: Reproducibility of $r_{\rm c}$ in the single $\beta$ -model analysis. The x-axis is the assumed core radius in the simulation cluster image, and the y-axis is the core radius derived from fitting with two different models: the $\beta$ -model with and without the PSF convolution. The results for three different photon counts are shown in each panel.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{1087f8.eps} \end{figure}$ Figure 8: Effect of the inner cutoff radius in the single $\beta$ model fitting. The results for #13 A1689 are shown as an example. From top to bottom, the resultant values of $r_{\rm c}$ [arcsec], F (see Sect. 3.5 for definition) and the probability of exceeding the F value, P, are shown.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{1087f9a.eps}\vspace*{1mm} \par\includegraphics[width=7.8cm,clip]{1087f9b.eps} \end{figure}$ Figure 9: Core radii derived by the double $\beta$ -model for the nine clusters with significant double structure. The panel a) shows the relation between two core radii of the double $\beta$ -model and that of the single $\beta$ -model, where the inner and outer cores of the double $\beta$ model are shown with the filled and open circles, respectively. The panel b) shows the ratio of the two cores and the ratio of the two normalization factors in Eq. (2). The two data points whose S₂/S₁ are large ( ${\sim } 1$ ) in the panel b) correspond to the two clusters with large core radius of the single $\beta$ -model in the panel a).
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In the text

$\begin{figure} \par\includegraphics[width=13.4cm,clip]{1087f10.eps}\end{figure}$ Figure 10: X-ray temperature a), metal abundance b), the 2-10 keV flux c) and the 2-10 keV luminosity within $r<6\hbox {$^\prime $ }$ d) measured with ASCA. At z>0.1, the filled circles and the open triangles are the regular and the irregular clusters in our sample. The filled stars show the double- $\beta$ clusters in our sample. The ASCA sensitivity curve is indicated with the dotted line in the panel of kT, assuming the luminosity-temperature relation. The temperatures of clusters with z<0.1 were taken from Mohr et al. (1999). The asterisks and the filled squares denote the single- $\beta$ and double- $\beta$ clusters in the nearby sample, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=10.3cm,clip]{1087f11.eps}\end{figure}$ Figure 11: Histograms of the spectral parameters determined with ASCA for 79 clusters (open). The panels a) and b) show the best-fit temperature and the metal abundance determined from the Raymond-Smith model fitting, respectively. The X-ray flux and the luminosity within $r<6\hbox {$^\prime $ }$ in the 2-10 keV band estimated with the GIS are shown in c) and d), respectively. The results of the double- $\beta$ clusters are superposed (hatched).
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In the text

$\begin{figure} \par\includegraphics[width=13cm,clip]{1087f12.eps}\end{figure}$ Figure 12: Results from $\beta$ -model analysis. In the panels a)- e), $\beta$ , $r_{\rm c}$ , $n_{\rm e}$ , $\rho _0$ , and $t_{\rm cool}$ are shown. At z>0.1, the filled circles and the open triangles are the regular and the irregular clusters in our sample. The filled stars and the open stars show the inner core and the outer core of the double- $\beta$ clusters in our sample. At z<0.1 the filled squares and the open squares denote the inner and outer components of the double- $\beta$ clusters in the nearby Mohr et al. (1999) sample. The asterisks denote the nearby single- $\beta$ clusters. In the panel of $r_{\rm c}$ , the selection effects due to the sensitivity and the spatial resolution of the ROSAT HRI are indicated with the dotted line and the dashed line, respectively. In the panel of $t_{\rm cool}$ we show a curve on which $t_{\rm cool}$ is equal to the age of the Universe at the cluster redshift.
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In the text

$\begin{figure} \par\includegraphics[width=13.4cm,clip]{1087f13.eps}\end{figure}$ Figure 13: r₅₀₀, M₅₀₀, $M_{\rm gas}$ , and $f_{\rm gas}$ derived from $\beta$ -model analysis are shown in the panels a)- d). See Sect. 5.2.1 for definitions of the parameters. The meanings of the symbols are the same as in Fig. 12.
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In the text

$\begin{figure} \par\includegraphics[width=10.3cm,clip]{1087f14.eps}\end{figure}$ Figure 14: Histograms of the X-ray parameters determined with the single $\beta$ -model for 79 clusters (open). The results of the double $\beta$ -model fittings for the nine double- $\beta$ clusters are superposed in the panels, where the hatched and filled regions show contributions of the inner core and the outer core, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=10.3cm,clip]{1087f15.eps}\end{figure}$ Figure 15: Histograms of r₅₀₀, M₅₀₀, $M_{\rm gas}$ and $f_{\rm gas}$ derived with the single $\beta$ model for 79 clusters (open). The results of double $\beta$ clusters are superposed (hatched).
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In the text

$\begin{figure} \par\includegraphics[width=13.3cm,clip]{1087f16.eps}\end{figure}$ Figure 16: Relations of the metallicity a), the bolometric luminosity b), $\beta$ c), the central electron density d), the overdensity radius e), the cluster mass f), the gas mass g), and the gas-mass fraction h) with the X-ray temperature. In all the panels the results of the single $\beta$ -model fitting are plotted and the single- $\beta$ regular clusters, the single- $\beta$ irregular clusters, and the double- $\beta$ clusters are denoted with the filled circles, the open triangles, and the filled stars respectively. The horizontal and the vertical error bars are $1\sigma$ . In the panels b) and e)- h), the best-fit power-laws for the entire sample are shown with the solid lines. In the panels a), c) and h) the sample means are indicated with the dashed lines. In panel b), the best-fit L-T relations for two subgroups with $r_{\rm c}<0.1$ Mpc and $r_{\rm c}>0.1$ Mpc are indicated with the dashed and dash-dotted lines, respectively (see Sect. 6.1 for details). In panels e)- g), the slopes expected from the self-similar model are shown with the dotted lines.
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{1087f17.eps}\end{figure}$ Figure 17: Venn diagrams which illustrate the relation between the X-ray morphology and the optical morphology a) and the X-ray core radius and the optical morphology b) (see Sect. 6.2 for definition of the cD cluster). In the panel b), we divided the sample into three subgroups: small core ( $r_{\rm c}<0.1$ Mpc) single- $\beta$ , double- $\beta$ , and large core ( $r_{\rm c}>0.1$ Mpc) single- $\beta$ clusters.
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In the text

$\begin{figure} \par\includegraphics[width=12.95cm,clip]{1087f18.eps}\end{figure}$ Figure 18: Relations of $\beta$ a), the central electron density b), the cooling timescale c), the temperature d), and the overdensity radius e) with the core radius. The meanings of the symbols are the same as Fig. 16. The error bars are $1\sigma$ . In the panels a)- e) the best-fit power-laws obtained for 77 distant clusters are shown with the solid lines. In the panels b) and c), the best-fit power-laws for $\log r_{\rm c}\leq -1$ and $\log r_{\rm c} >-1$ are also shown with the dashed lines and the dot-dash lines, respectively. In the panel e) four dotted lines correspond to four different constant values of $r_{500}/r_{\rm c}$ .
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In the text

$\begin{figure} \par\includegraphics[width=15cm]{1087fc1a.eps} \end{figure}$ Figure C.1: ROSAT HRI images, radial surface brightness profiles and the ASCA SIS and GIS spectra of 79 clusters. In the left panels, the HRI images are smoothed by a Gaussian filter with $\sigma =7\hbox {$^{\prime \prime }$ }.5{-}15\hbox {$^{\prime \prime }$ }$ and the contours whose levels correspond to n (=3, 5, 9, 15, 31, 63) times the $1\sigma$ background level are overlaid. The backgrounds are not subtracted. The X-ray centroids that were determined and used to derive the radial profiles in the image analysis are marked with crosses. In the central panels, the crosses denote the observed radial profile of the HRI, and the step functions show the best-fit $\beta$ models. The best-fit background levels are shown with the dashed lines. For the nine double $\beta$ clusters, the results of the double $\beta$ model fitting are shown instead of the single $\beta$ model and the inner and the outer components are also shown with the dash-dot and the dotted lines, respectively. The X-ray significance radius, $r_{\rm x}$ , and the overdensity radius, r₅₀₀ are shown with the vertical dashed, and dotted lines respectively. In the right panels, the ASCA spectra fitted with the Raymond-Smith model are shown, where the crosses and crosses with circles denote the spectra obtained with the GIS and the SIS respectively, and the stepped lines show the best-fit models convolved with the telescope and detector responses. The fitting residuals are also shown in the panels.

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1b.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1c.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1d.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1e.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1f.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1g.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1h.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1i.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1j.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1k.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1l.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1m.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1n.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1o.eps}\end{figure}$

In the text

$\begin{figure}\par\includegraphics[width=15cm]{1087fc1p.eps}\end{figure}$

In the text

$\begin{figure} \par\includegraphics[width=7.2cm,clip]{1087f1a.eps}\vspace*{1mm} \par\includegraphics[width=7.2cm,clip]{1087f1b.eps} \end{figure}$	Figure 1: a) Redshift distribution of 79 distant clusters. b) Redshift distribution of the 45 regular clusters (open) and the 34 irregular clusters (hatched).
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$\begin{figure} \par\includegraphics[width=6.9cm,clip]{1087f2.eps} \end{figure}$	Figure 2: Comparison of the temperature distribution of our 79 distant clusters (open) to the nearby flux limited 55 sample of Edge et al. (1990) (hatched).
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$\begin{figure} \par\includegraphics[width=5.8cm,clip]{1087f5.eps} \end{figure}$	Figure 5: $\chi ^2$ contour of the single $\beta$ -model fit for #13 A1689. The position of $\chi ^2$ minimum is denoted with a cross. The curve corresponds to the single-parameter error domain at 90% confidence.
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$\begin{figure} \par\includegraphics[width=7.7cm,clip]{1087f7.eps}\end{figure}$	Figure 7: Reproducibility of $r_{\rm c}$ in the single $\beta$ -model analysis. The x-axis is the assumed core radius in the simulation cluster image, and the y-axis is the core radius derived from fitting with two different models: the $\beta$ -model with and without the PSF convolution. The results for three different photon counts are shown in each panel.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{1087f8.eps} \end{figure}$	Figure 8: Effect of the inner cutoff radius in the single $\beta$ model fitting. The results for #13 A1689 are shown as an example. From top to bottom, the resultant values of $r_{\rm c}$ [arcsec], F (see Sect. 3.5 for definition) and the probability of exceeding the F value, P, are shown.
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$\begin{figure} \par\includegraphics[width=13.4cm,clip]{1087f13.eps}\end{figure}$	Figure 13: r₅₀₀, M₅₀₀, $M_{\rm gas}$ , and $f_{\rm gas}$ derived from $\beta$ -model analysis are shown in the panels a)- d). See Sect. 5.2.1 for definitions of the parameters. The meanings of the symbols are the same as in Fig. 12.
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$\begin{figure} \par\includegraphics[width=10.3cm,clip]{1087f14.eps}\end{figure}$	Figure 14: Histograms of the X-ray parameters determined with the single $\beta$ -model for 79 clusters (open). The results of the double $\beta$ -model fittings for the nine double- $\beta$ clusters are superposed in the panels, where the hatched and filled regions show contributions of the inner core and the outer core, respectively.
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$\begin{figure} \par\includegraphics[width=10.3cm,clip]{1087f15.eps}\end{figure}$	Figure 15: Histograms of r₅₀₀, M₅₀₀, $M_{\rm gas}$ and $f_{\rm gas}$ derived with the single $\beta$ model for 79 clusters (open). The results of double $\beta$ clusters are superposed (hatched).
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