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<!-- DOI: 10.1051/0004-6361/200913156 -->

<h2 class="sec">Online Material</h2>

<h2 class="sec"><a name="SECTION000100000000000000000"></a>
Appendix A: Relation between pseudo-entropy  and entropy ratios</h2>

<p>We have used the entropy profiles for the 31&nbsp;clusters in our
sample for which we have performed radial analysis to test if the
pseudo-entropy ratios (defined in Paper&nbsp;I) are good indicators of
the behavior of the true three dimensional entropy (
<!-- MATH: $S\equiv T_{\rm X}/n_{\rm e}^{2/3}$ -->
<IMG SRC="img156.png" ALT="$S\equiv T_{\rm X}/n_{\rm e}^{2/3}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="33" WIDTH="73">). We&nbsp;have therefore calculated the entropy ratio&nbsp;
<!-- MATH: $S_{\rm IN}/S_{\rm OUT}$ -->
<IMG SRC="img157.png" ALT="$S_{\rm IN}/S_{\rm OUT}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="60">,
starting from the three-dimensional entropy profiles obtained with the
de-projected temperature and density profiles. We show in Fig.&nbsp;<a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#test_entro">A.1</a>
that there is a strong correlation between the ``true'' entropy ratio
and the ``pseudo'' entropy ratios. We fitted the points in Fig.&nbsp;<a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#test_entro">A.1</a> with a linear model taking into account the errors on both coordinates (with&nbsp;the IDL Astrolib procedure FITEXY.PRO<A NAME="tex2html18" HREF="#foot1790"><sup><IMG ALT="[*]" SRC="/icons/foot_motif.png" align="bottom" BORDER="1"></sup></A>), and we found
<br></p><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\frac{S_{\rm IN}}{S_{\rm OUT}}=-0.051\pm0.004+(0.792 \pm 0.011)*\frac{s_{\rm IN}}{s_{\rm OUT}}\cdot
\end{equation} -->
<TABLE ALIGN="CENTER" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td ALIGN="CENTER" nowrap="nowrap"><A NAME="eq_a1"></A><IMG SRC="img158.png" ALT="\begin{displaymath}%
\frac{S_{\rm IN}}{S_{\rm OUT}}=-0.051\pm0.004+(0.792 \pm 0.011)*\frac{s_{\rm IN}}{s_{\rm OUT}}\cdot
\end{displaymath}" HEIGHT="69" WIDTH="315">
</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.1)
</td>
</tr>
</table></div><br clear="all"><p></p>
As expected, the entropy ratios are smaller than pseudo-entropy ratios, since projection effects smooth out the gradients.

<p>In order to quantify the scatter around the best fit relation, we have
applied to the residuals a maximum likelihood algorithm that postulates
a parent distribution described by a mean and an intrinsic dispersion (<A NAME="aaref44"></A><a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#mac88">Maccacaro et&nbsp;al.  1988</a>). To&nbsp;do this we have calculated an&nbsp;``effective error'' on the <i>y</i>&nbsp;coordinate, which takes into account the errors on both coordinates, defined&nbsp;as: 
<!-- MATH: $\sigma_{\rm eff}^2=\sigma_y^2+b^2\sigma_x^2$ -->
<IMG SRC="img159.png" ALT="$\sigma_{\rm eff}^2=\sigma_y^2+b^2\sigma_x^2$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="99">,
where <i>b</i>&nbsp;is the best-fit value for the slope of the linear relation between the two data sets (in&nbsp;this case <i>b</i>=0.792, Eq.&nbsp;(<a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#eq_a1">A.1</a>)). The mean value of the residual distribution is consistent with zero&nbsp;(0.021), with an intrinsic dispersion 
<!-- MATH: $\sigma=0.043$ -->
<IMG SRC="img160.png" ALT="$\sigma=0.043$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="60">.

</p><p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="test_entro"></A><!-- end Label--><A NAME="1795"></A><A NAME="figure1485" HREF="img161.png"><IMG SRC="Timg161.png" ALT="\begin{figure}
\par\includegraphics[width=8cm,clip]{13156a1.eps}
\end{figure}" HEIGHT="63" WIDTH="89"></A><!-- HTML Figure number: 8 -->
</td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
Entropy ratio versus pseudo-entropy ratio.
</td>
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<p></p><h2 class="sec"><a name="SECTION000110000000000000000"></a>Appendix B: Figures of individual clusters</h2>

<p>In this Appendix, we provide details for the low-entropy metal rich
regions in the clusters with CC&nbsp;remnants. For&nbsp;each cluster,
we show the EPIC X-ray flux image in the 0.4-2&nbsp;keV energy range
(see <a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#rossetti06">Rossetti et&nbsp;al. 2007</a>, for details), the pseudo-entropy ratio image and the optical image taken from the&nbsp;DSS.

</p><p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.1"></A><!-- end Label--><A NAME="1796"></A><A NAME="figure1493" HREF="img162.png"><IMG SRC="Timg162.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig1.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 9 -->
</td>
<td class="img-txt"><span class="bold">Figure B.1:</span><p>
Abell 1650. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X''&nbsp;mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=9&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.2"></A><!-- end Label--><A NAME="1797"></A><A NAME="figure1503" HREF="img163.png"><IMG SRC="Timg163.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig2.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 10 -->
</td>
<td class="img-txt"><span class="bold">Figure B.2:</span><p>
MKW3s. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=10&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<table>
<tr><td><!-- init Label --><A NAME="figB.3"></A><!-- end Label--><A NAME="1798"></A><A NAME="figure1513" HREF="img164.png"><IMG SRC="Timg164.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig3.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 11 -->
</td>
<td class="img-txt"><span class="bold">Figure B.3:</span><p>
A4038. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=11&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.4"></A><!-- end Label--><A NAME="1799"></A><A NAME="figure1523" HREF="img165.png"><IMG SRC="Timg165.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig4.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 12 -->
</td>
<td class="img-txt"><span class="bold">Figure B.4:</span><p>
A1644. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=12&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.5"></A><!-- end Label--><A NAME="1800"></A><A NAME="figure1533" HREF="img166.png"><IMG SRC="Timg166.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig5.eps}
\end{figure}" HEIGHT="168" WIDTH="199"></A><!-- HTML Figure number: 13 -->
</td>
<td class="img-txt"><span class="bold">Figure B.5:</span><p>
Abell 1689. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=13&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.6"></A><!-- end Label--><A NAME="1801"></A><A NAME="figure1543" HREF="img167.png"><IMG SRC="Timg167.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig6.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 14 -->
</td>
<td class="img-txt"><span class="bold">Figure B.6:</span><p>
Abell 3558. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>:
pseudo-entropy ratio map (zoom), with X-ray contours overlaid. White
regions have no data because of point sources subtraction (<a href="/articles/aa/full_html/2010/02/aa13156-09/aa13156-09.html#rossetti06">Rossetti et&nbsp;al.  2007</a>). <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and&nbsp;``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=14&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.7"></A><!-- end Label--><A NAME="1802"></A><A NAME="figure1554" HREF="img168.png"><IMG SRC="Timg168.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig7.eps}
\end{figure}" HEIGHT="169" WIDTH="195"></A><!-- HTML Figure number: 15 -->
</td>
<td class="img-txt"><span class="bold">Figure B.7:</span><p>
Abell 576. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and ``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=15&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.8"></A><!-- end Label--><A NAME="1803"></A><A NAME="figure1564" HREF="img169.png"><IMG SRC="Timg169.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig8.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 16 -->
</td>
<td class="img-txt"><span class="bold">Figure B.8:</span><p>
Abell 754. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the upper right panel is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The blue circle and&nbsp;X show the position of the
BCG, while the green circle and&nbsp;X show the elliptical galaxy
2MASSX&nbsp;J09091923-0941591. The scale refers to the pseudo-entropy
ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=16&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<table>
<tr><td><!-- init Label --><A NAME="figB.9"></A><!-- end Label--><A NAME="1804"></A><A NAME="figure1573" HREF="img170.png"><IMG SRC="Timg170.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig9.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 17 -->
</td>
<td class="img-txt"><span class="bold">Figure B.9:</span><p>
Abell 3562. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and&nbsp;``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=17&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="figB.10"></A><!-- end Label--><A NAME="1805"></A><A NAME="figure1583" HREF="img171.png"><IMG SRC="Timg171.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig10.eps}
\end{figure}" HEIGHT="169" WIDTH="192"></A><!-- HTML Figure number: 18 -->
</td>
<td class="img-txt"><span class="bold">Figure B.10:</span><p>
Abell 3571. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and&nbsp;``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=18&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<table>
<tr><td><!-- init Label --><A NAME="figB.11"></A><!-- end Label--><A NAME="1806"></A><A NAME="figure1593" HREF="img172.png"><IMG SRC="Timg172.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig11.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 19 -->
</td>
<td class="img-txt"><span class="bold">Figure B.11:</span><p>
Abell 3667. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo entropy ratio
is&nbsp;&lt;0.8. The yellow circle (<i> lower left</i>) and&nbsp;``X'' mark the position of the BCG. The scale refers to the pseudo entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=19&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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<table>
<tr><td><!-- init Label --><A NAME="figB.12"></A><!-- end Label--><A NAME="1807"></A><A NAME="figure1603" HREF="img173.png"><IMG SRC="Timg173.png" ALT="\begin{figure}
\par\includegraphics[width=18cm,clip]{fig12.eps}
\end{figure}" HEIGHT="169" WIDTH="199"></A><!-- HTML Figure number: 20 -->
</td>
<td class="img-txt"><span class="bold">Figure B.12:</span><p>
Abell 2256. <i> Upper left panel</i>: X-ray flux image. <i> Upper right panel</i>: pseudo-entropy ratio map (zoom), with X-ray contours overlaid. <i> Lower left panel</i>:
optical image (zoom), with X-ray contours overlaid. The red box in the
first panel marks the region zoomed in in the other two panels. The red
polygon in the other two panels is the contour of the ``CC-remnant'',
i.e.&nbsp;it marks the bins where the pseudo-entropy ratio
is&nbsp;&lt;0.8. The green circle (<i> lower left</i>) and&nbsp;``X'' mark the position of the BCG. The scale refers to the pseudo-entropy ratio map.
</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=20&DOI=10.1051/0004-6361/200913156" target="DEXTER">Open with DEXTER</a></td></tr>

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