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<!-- DOI: 10.1051/0004-6361/200811263 -->

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

<p>

<p>

<h2 class="sec"><a name="SECTION000120000000000000000"></a><A NAME="appendix_dust_evol"></A>
Appendix A: Modeling dust evolution
</h2>

<p>
Interstellar dust plays a major role in the evolution of the postshock gas because its survival is required for <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;formation. For a constant dust-to-gas mass ratio, the dust dominates the cooling efficiency of the gas at high (<I>T</I>&gt; 10<sup>6</sup>&nbsp;K) temperatures (<A NAME="aaref15"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Dwek87ApJ">Dwek  1987</a>), but for such high temperatures, the timescale for dust destruction is shorter than the gas cooling time  (<A NAME="aaref63"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Smith1996ApJ">Smith et&nbsp;al.  1996</a>). This Appendix details how the evolution of the dust-to-gas mass ratio in the postshock gas is calculated. This calculation is coupled to that of the gas cooling, detailed in Appendix&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#appendix_hotgascooling">B</a>.

<p>
The dominant dust destruction process depends on the shock velocity. In this study, we consider the effect of thermal and inertial sputtering by grain-ion collisions.  This is the dominant mode of destruction in a hot (
<!-- MATH: ${\sim}10^6{-}10^8$ -->
<IMG
 WIDTH="60" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img178.png"
 ALT="${\sim}10^6{-}10^8$">&nbsp;K) plasma, resulting from fast (
<!-- MATH: $100{-}1000$ -->
100-1000&nbsp;km&nbsp;s<sup>-1</sup>) intergalactic shocks (e.g. <A NAME="aaref17"></A><A NAME="tex2html96"
 HREF="#Dwek1996ApJ">Dwek et&nbsp;al.  1996</A>; <A NAME="aaref13"></A><A NAME="tex2html97"
 HREF="#Draine1979ApJ">Draine &amp; Salpeter  1979</A>).

<p>
It is assumed that dust grains have an initial radius equal to the effective  (mean)  radius of 
<!-- MATH: $a_{\rm eff} = 0.05~\mu$ -->
<IMG
 WIDTH="74" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img179.png"
 ALT="$a_{\rm eff} = 0.05~\mu$">m. This radius is calculated from the MRN dust size distribution (Sect.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#subsec_coolfunct_plasma">B.1</a>). At each time step, the fraction of dust remaining in the gas&nbsp;
<!-- MATH: $f_{\rm dust}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img113.png"
 ALT="$f_{\rm dust}$">,
is computed in parallel to the postshock gas cooling:

<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
f_{\rm dust} = \left( \frac{a}{a_{\rm eff}} \right) ^{3}~{\rm with} \ a =  a_{\rm eff} - \int _{t_0} ^{t} \dot{a} ~ {\rm d}t,
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="238" HEIGHT="74"
 SRC="img180.png"
 ALT="\begin{displaymath}%
f_{\rm dust} = \left( \frac{a}{a_{\rm eff}} \right) ^{3}~{\rm with} \ a = a_{\rm eff} - \int _{t_0} ^{t} \dot{a} ~ {\rm d}t,
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.1)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
where 
<!-- MATH: $\dot{a} = {\rm d}a / {\rm d}t$ -->
<IMG
 WIDTH="59" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img181.png"
 ALT="$ \dot{a} = {\rm d}a / {\rm d}t$">.
For a shock velocity 
<!-- MATH: $V_{\rm s} > 300$ -->
<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img182.png"
 ALT="$V_{\rm s} > 300$">&nbsp;km&nbsp;s<sup>-1</sup>, thermal sputtering dominates over inertial sputtering, so we ignore inertial sputtering. In this case, the grain sputtering rate is nearly constant for  <IMG
 WIDTH="37" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img183.png"
 ALT="$T \ga 3$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>6</sup>&nbsp;K and strongly decreases at lower temperatures (<A NAME="aaref68"></A><A NAME="tex2html98"
 HREF="#Tielens1994ApJ">Tielens et&nbsp;al.  1994</A>; <A NAME="tex2html99"
 HREF="#Dwek1996ApJ">Dwek et&nbsp;al.  1996</A>; <A NAME="tex2html100"
 HREF="#Draine1979ApJ">Draine &amp; Salpeter  1979</A>). The rate of decrease in grain size is given&nbsp;by
<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\frac{{\rm d} a}{{\rm d} t} = \frac{m_{\rm sp}}{2 ~ \rho _0} ~ n_{\rm H} ~ \sum A_i \left\langle Y_i ~ v \right\rangle,
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="173" HEIGHT="68"
 SRC="img184.png"
 ALT="\begin{displaymath}%
\frac{{\rm d} a}{{\rm d} t} = \frac{m_{\rm sp}}{2 ~ \rho _0} ~ n_{\rm H} ~ \sum A_i \left\langle Y_i ~ v \right\rangle,
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.2)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
where 
<!-- MATH: $m_{\rm sp}$ -->
<IMG
 WIDTH="23" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img185.png"
 ALT="$m_{\rm sp}$">
and <IMG
 WIDTH="15" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img186.png"
 ALT="$\rho _0$">
are the average mass of the sputtered atoms  and the density of the grain material, <I>A</I><SUB><I>i</I></SUB>&nbsp;the abundance of impacting ion&nbsp;<I>i</I> and&nbsp;<I>Y</I><SUB><I>i</I></SUB> is the sputtering yield<A NAME="tex2html17"
 HREF="#foot2027"><sup><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="/icons/foot_motif.png"></sup></A> of ion&nbsp;<I>i</I>. The quantity&nbsp;
<!-- MATH: $\left\langle Y_i ~ v \right\rangle$ -->
<IMG
 WIDTH="34" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img187.png"
 ALT="$\left\langle Y_i ~ v \right\rangle$">
is the average of the impacting ion velocities&nbsp;<I>v</I> times the yield over the Maxwellian distribution. The right hand side of this equation is computed from the results given in <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Tielens1994ApJ">Tielens et&nbsp;al. (1994)</a>.

<p>
For 
<!-- MATH: $V_{\rm s} < 300$ -->
<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img188.png"
 ALT="$V_{\rm s} < 300$">&nbsp;km&nbsp;s<sup>-1</sup>, we integrate the equation of deceleration of the dust grains with respect to the gas
<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\frac{{\rm d} v_{\rm g}}{{\rm d} t}  = - \frac{\beta ~ \pi ~ a ^2 ~ \rho ~ v_{\rm g}^{2}}{m},
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq_dust_decel"></A><IMG
 WIDTH="125" HEIGHT="72"
 SRC="img189.png"
 ALT="\begin{displaymath}%
\frac{{\rm d} v_{\rm g}}{{\rm d} t} = - \frac{\beta ~ \pi ~ a ^2 ~ \rho ~ v_{\rm g}^{2}}{m},
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.3)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
where <IMG
 WIDTH="14" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img190.png"
 ALT="$v_{\rm g}$">
is the grain velocity with respect to the gas, <I>m</I>&nbsp;and <I>a</I> are its mass and radius, <IMG
 WIDTH="11" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img191.png"
 ALT="$\rho$">&nbsp;is the gas density, and <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img192.png"
 ALT="$\beta$">&nbsp;is the enhancement of the collisional drag in a plasma relative to that of a neutral medium. The initial grain velocity is set to&nbsp;3/4 of the shock velocity (<A NAME="aaref36"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#1994ApJ...433..797J">Jones et&nbsp;al.  1994</a>). We take <IMG
 WIDTH="34" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img193.png"
 ALT="$\beta =1$">,
which maximizes the dust destruction (see <A NAME="aaref4"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#1995ApJ...454..254B">Borkowski &amp; Dwek  1995</a>, for details). Therefore, our computation of the dust survival is conservative. When inertial sputtering is the dominant destruction process, the erosion rate is given by (<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Tielens1994ApJ">Tielens et&nbsp;al.  1994</a>)
<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\frac{{\rm d} a}{{\rm d} t} = \frac{m_{\rm sp}}{2 ~ \rho _0} ~ v_{\rm g} ~ n_{\rm H} \sum A_i  Y_i.
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="159" HEIGHT="70"
 SRC="img194.png"
 ALT="\begin{displaymath}%
\frac{{\rm d} a}{{\rm d} t} = \frac{m_{\rm sp}}{2 ~ \rho _0} ~ v_{\rm g} ~ n_{\rm H} \sum A_i Y_i.
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(A.4)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
By integrating these differential equations, the radius of the dust grains as a function of their velocity has been deduced (see <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#1995ApJ...454..254B">Borkowski &amp; Dwek  1995</a>, for a complete parametrization of 
<!-- MATH: ${\rm d}a/{\rm d}t$ -->
<IMG
 WIDTH="35" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img195.png"
 ALT="${\rm d}a/{\rm d}t$">
in the case of inertial sputtering).

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_DustMassFracRemain_InitTemp">&#160;</A><!-- end Label--><A NAME="2528"></A><A NAME="figure2055"
 HREF="img197.png"><IMG
 WIDTH="89" HEIGHT="71" SRC="Timg197.png"
 ALT="\begin{figure}
\par\includegraphics[angle=90,width=8cm,clip]{11263fA1.eps}
\end{figure}"></A><!-- HTML Figure number: 7 --></td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
Dust mass fraction as a function of the initial (postshock) temperature, after isobaric cooling to 10<sup>4</sup>&nbsp;K. The pressure is fixed to 
<!-- MATH: $P_{\rm ps} / k_{\rm B} = 2.3$ -->
<IMG
 WIDTH="75" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$P_{\rm ps} / k_{\rm B} = 2.3$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">

<!-- MATH: $10^{5}~[\rm cm^{-3}~K]$ -->
<IMG
 WIDTH="78" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img196.png"
 ALT="$10^{5}~[\rm cm^{-3}~K]$">.
The dotted line shows the result of the model which includes both inertial and thermal sputtering. As a comparison, the dashed line only shows the effect of thermal sputtering.</p></td>
</tr>
</table></div>
<p>
Figure&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_DustMassFracRemain_InitTemp">A.1</a> shows the dust mass fraction remaining in the gas after isobaric cooling from the postshock temperature 
<!-- MATH: $T_{\rm ps}$ -->
<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img101.png"
 ALT="$T_{\rm ps}$">
to 10<sup>4</sup>&nbsp;K as a function of the shock velocity. The dotted line includes both thermal and inertial sputtering. For comparison, the dashed line only includes dust destruction by thermal sputtering. Our calculation for thermal sputtering with a single grain size (
<!-- MATH: $a_{\rm eff} = 0.05~\mu$ -->
<IMG
 WIDTH="74" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img179.png"
 ALT="$a_{\rm eff} = 0.05~\mu$">m) is in very good agreement with the study by <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Smith1996ApJ">Smith et&nbsp;al. (1996)</a> who used  a grain size distribution. For 
<!-- MATH: $T_{\rm ps} > 2$ -->
<IMG
 WIDTH="44" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img198.png"
 ALT="$T_{\rm ps} > 2$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>6</sup>&nbsp;K, i.e.&nbsp;
<!-- MATH: $V_{\rm s} > 400$ -->
<IMG
 WIDTH="54" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img199.png"
 ALT="$V_{\rm s} > 400$">&nbsp;km&nbsp;s<sup>-1</sup>, most of the dust is destroyed before the gas has cooled  to 10<sup>4</sup>&nbsp;K. The gas that is heated to less than 
<!-- MATH: ${\sim}10^{6}$ -->
<IMG
 WIDTH="32" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img119.png"
 ALT="${\sim}10^6$">&nbsp;K keeps a major part of its dust content and may therefore form&nbsp;<IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">.

<p>

<h2 class="sec"><a name="SECTION000130000000000000000"></a><A NAME="appendix_hotgascooling"></A>
Appendix B: Modeling the cooling of hot gas
</h2>

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_cool_function">&#160;</A><!-- end Label--><A NAME="2529"></A><A NAME="figure2080"
 HREF="img201.png"><IMG
 WIDTH="177" HEIGHT="119" SRC="Timg201.png"
 ALT="\begin{figure}
\par\includegraphics[angle=90,width=15.7cm,clip]{11263fB1.eps}
\end{figure}"></A><!-- HTML Figure number: 8 --></td>
<td class="img-txt"><span class="bold">Figure B.1:</span><p>
Time-dependent cooling efficiency [erg&nbsp;cm<sup>3</sup>&nbsp;s<sup>-1</sup>] as a function of temperature during the gas cooling, for different initial conditions. The blue dashed line, 
<!-- MATH: $\Lambda _{\rm dust}^{\rm init} (T)$ -->
<IMG
 WIDTH="49" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.png"
 ALT="$\Lambda _{\rm dust}^{\rm init} (T)$">,
represents the <EM>initial</EM> cooling function of the gas including the dust contribution, computed for a MRN&nbsp;interstellar dust size distribution (0.01 to <IMG
 WIDTH="38" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img200.png"
 ALT="$0.25~\mu$">m dust particles). The black dashed lines are the dust cooling functions for different shock velocities  (i.e.&nbsp;different initial temperatures) which take into account the destruction by sputtering during the cooling. The cooling function due to atomic processes, 
<!-- MATH: $\Lambda _{\rm gas} (T)$ -->
<IMG
 WIDTH="45" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img20.png"
 ALT="$\Lambda _{\rm gas} (T)$">,
is displayed for an isobaric and non-equilibrium (time-dependent) cooling, and for solar metallicities (<I>Z</I>=1). The dominant cooling elements at various temperatures are indicated on the curves. The red lines are the  total cooling functions, 
<!-- MATH: $\Lambda _{\rm total} = \Lambda _{\rm dust} + \Lambda _{\rm gas}$ -->
<IMG
 WIDTH="115" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img21.png"
 ALT="$\Lambda _{\rm total} = \Lambda _{\rm dust} + \Lambda _{\rm gas}$">,
for different initial temperatures or shock velocities. The shock velocities are indicated in km&nbsp;s<sup>-1</sup>. Their starting points are indicated by the&nbsp;arrows.</p></td>
</tr>
</table></div>
<p>
This Appendix details how we model the cooling of the hot gas. The cooling functions used in the calculations are presented in Sect.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#subsec_coolfunct_plasma">B.1</a>. The results are discussed in Sect.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#subsec_evollowdensgas">B.2</a>. 

<p>
The time dependence of both the gas temperature and the dust-to-gas mass ratio has been calculated. We start from gas at an initial  postshock temperature&nbsp;
<!-- MATH: $T_{\rm ps}$ -->
<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img101.png"
 ALT="$T_{\rm ps}$">
with a galactic dust-to-gas ratio, a solar metallicity and assume equilibrium ionization. From a range of postshock temperatures up to 10<sup>4</sup>&nbsp;K, the isobaric gas cooling is calculated by integrating the energy balance equation which gives the rate of decrease of the gas temperature:
<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\frac{5}{2}  ~ k_{\rm B} ~ \frac{{\rm d} T}{{\rm d} t} = - \mu ~ n_{\rm e} \left(f_{\rm dust} ~ \Lambda _{\rm dust} + \Lambda _{\rm gas}\right),
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="Eq_ratetemp"></A><IMG
 WIDTH="232" HEIGHT="65"
 SRC="img202.png"
 ALT="\begin{displaymath}%
\frac{5}{2} ~ k_{\rm B} ~ \frac{{\rm d} T}{{\rm d} t} = - \...
..._{\rm dust} ~ \Lambda _{\rm dust} + \Lambda _{\rm gas}\right),
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(B.1)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
where <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\mu$">
is the mean particle weight (
<!-- MATH: $\mu = 0.6~\rm a.m.u$ -->
<IMG
 WIDTH="80" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img203.png"
 ALT="$\mu = 0.6~\rm a.m.u$">
for a fully ionized gas), <IMG
 WIDTH="15" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img204.png"
 ALT="$n_{\rm e}$">&nbsp;is the electron density, <IMG
 WIDTH="17" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img51.png"
 ALT="$k_{\rm B}$">&nbsp;the Boltzman constant, 
<!-- MATH: $f_{\rm dust}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img113.png"
 ALT="$f_{\rm dust}$">&nbsp;the dust-to-gas mass ratio, 
<!-- MATH: $\Lambda _{\rm dust}$ -->
<IMG
 WIDTH="31" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img205.png"
 ALT="$\Lambda _{\rm dust}$">
and 
<!-- MATH: $\Lambda _{\rm gas}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img206.png"
 ALT="$\Lambda _{\rm gas}$">
are respectively the dust and gas cooling efficiencies per unit mass of dust and gas, respectively.

<p>
The time-dependent total cooling function is the sum of the dust cooling (weighted by the remaining fraction of dust mass, see Appendix&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#appendix_dust_evol">A</a>) and the gas cooling contributions. 
We neglect the effect of the magnetic field on the compression of the gas. This assumption is in agreement with observations and numerical simulations, provided that the gas is not gravitationally bound (see Sect.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#subsec_gal_shocks">3.1</a> for details and references). The thermal gas pressure&nbsp;
<!-- MATH: $P_{\rm th}$ -->
<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img207.png"
 ALT="$P_{\rm th}$">
is constant in the calculations.

<p>

<h3 class="sec2"><a name="SECTION000131000000000000000"></a>
<A NAME="subsec_coolfunct_plasma"></A>
B.1 Physical processes and cooling functions
</h3>

<p>

<h4 class="sec3"><a name="SECTION000131100000000000000"></a>
B.1.1 Cooling by atomic processes in the gas phase
</h4>

<p>
The calculations of the non-equilibrium (time-dependent) ionization states and isobaric cooling rates of a hot, dust-free gas for the cooling efficiency <EM>by atomic processes</EM> has been taken from <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Gnat2007ApJS">Gnat &amp; Sternberg (2007)</a>. The electronic cooling efficiency, 
<!-- MATH: $\Lambda _{\rm gas}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img206.png"
 ALT="$\Lambda _{\rm gas}$">,
includes the removal of electron kinetic energy via recombination with ions, collisional ionization, collisional excitation followed by line emission, and thermal bremsstrahlung. 
<!-- MATH: $\Lambda _{\rm gas}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img206.png"
 ALT="$\Lambda _{\rm gas}$">
is shown in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_cool_function">B.1</a> (black solid line<A NAME="tex2html19"
 HREF="#foot2127"><sup><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]" SRC="/icons/foot_motif.png"></sup></A>)  and reproduces the standard ``cosmic cooling curve'' presented in many papers in the literature  (e.g. <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#sutherland1993ApJS">Sutherland &amp; Dopita  1993</a>). Note that most of the distinct features that appear for the ionization equilibrium case are smeared out in the nonequilibrium cooling functions and that the nonequilibrium abundances reduce the cooling efficiencies by factors of&nbsp;2 to&nbsp;4 (<A NAME="tex2html101"
 HREF="#sutherland1993ApJS">Sutherland &amp; Dopita  1993</A>; <A NAME="tex2html102"
 HREF="#Gnat2007ApJS">Gnat &amp; Sternberg  2007</A>). The chemical species that dominate the gas cooling are indicated in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_cool_function">B.1</a> near the 
<!-- MATH: $\Lambda _{\rm gas}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img206.png"
 ALT="$\Lambda _{\rm gas}$">&nbsp;cooling curve. Over most of the temperature range, the radiative energy losses are dominated by electron-impact excitation of resonant line transitions in metal ions. At&nbsp;
<!-- MATH: $T \simeq 2$ -->
<IMG
 WIDTH="37" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img209.png"
 ALT="$T \simeq 2$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>4</sup>&nbsp;K, the cooling is mainly due to the collisional excitation of hydrogen Ly<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.png"
 ALT="$\alpha $">&nbsp;lines. For 
<!-- MATH: $T \ga 10^7$ -->
<IMG
 WIDTH="48" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img210.png"
 ALT="$T \ga 10^7$">&nbsp;K, the bremsstrahlung (free-free) radiation becomes dominant.

<p>

<h4 class="sec3"><a name="SECTION000131200000000000000"></a>
B.1.2 Cooling by dust grains
</h4>

<p>
Following the method of <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Dwek87ApJ">Dwek (1987)</a>, the cooling function of the gas <EM>by the dust</EM> via electron-grain collisions has been calculated. We adopt an MRN (<A NAME="aaref45"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#1977ApJ...217..425M">Mathis et&nbsp;al.  1977</a>) size distribution of dust particles, between 0.01 to <IMG
 WIDTH="38" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img200.png"
 ALT="$0.25~\mu$">m in grain radius. All the details can be found in <A NAME="tex2html103"
 HREF="#Dwek87ApJ">Dwek (1987</A>,<A NAME="aaref14"></A><A NAME="tex2html104"
 HREF="#Dwek86ApJ">1986)</A>, and we briefly recall here the principles of this calculation. For temperatures lower than 10<sup>8</sup>&nbsp;K, electron&nbsp;- dust grain collisions cool the gas with a cooling efficiency, 
<!-- MATH: $\Lambda _{\rm dust} (T)$ -->
<IMG
 WIDTH="50" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img211.png"
 ALT="$\Lambda _{\rm dust} (T)$">

<!-- MATH: $[\rm erg~s^{-1}~cm^{3}]$ -->
<IMG
 WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img212.png"
 ALT="$[\rm erg~s^{-1}~cm^{3}]$">,
given&nbsp;by:
<br><p></p>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
%
\Lambda ^{\rm init}_{\rm dust} (T) = \frac{\mu ~ m_{\rm H} ~ Z_{\rm d}}{\langle m_{\rm d} \rangle} \left( \frac{32}{\pi ~ m_{\rm e}} \right)^{1/2} \! \pi (k_{\rm B}  T)^{3/2} \! \int \! a^{2}  h(a, T)  f(a) ~ {\rm d}a\;
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="366" HEIGHT="74"
 SRC="img213.png"
 ALT="\begin{displaymath}%
\Lambda ^{\rm init}_{\rm dust} (T) = \frac{\mu ~ m_{\rm H} ...
...(k_{\rm B} T)^{3/2} \! \int \! a^{2} h(a, T) f(a) ~ {\rm d}a\;
\end{displaymath}"></td>
<TD WIDTH=10 ALIGN="RIGHT">
(B.2)</td></tr>
</TABLE>
</DIV><BR CLEAR="ALL"><p></p>
where <IMG
 WIDTH="16" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img214.png"
 ALT="$Z_{\rm d}$">
is the dust-to-gas mass ratio, <IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img215.png"
 ALT="$m_{\rm H}$">&nbsp;the mass of an hydrogen atom, <IMG
 WIDTH="18" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img216.png"
 ALT="$m_{\rm e}$">&nbsp;the mass of electron, <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\mu$">&nbsp;the mean atomic weight of the gas (in amu), <I>f</I>(<I>a</I>)&nbsp;the grain size distribution function (normalized to&nbsp;1) in the dust size interval, 
<!-- MATH: $\langle m_{\rm d} \rangle$ -->
<IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img217.png"
 ALT="$\langle m_{\rm d} \rangle$">&nbsp;the size-averaged mass of the dust, and <I>h</I>(<I>a</I>, <I>T</I>)&nbsp;the effective grain heating efficiency (see e.g. <A NAME="aaref16"></A><A NAME="tex2html105"
 HREF="#Dwek1981">Dwek &amp; Werner  1981</A>; <A NAME="tex2html106"
 HREF="#Dwek87ApJ">Dwek  1987</A>). This <EM>initial</EM> cooling efficiency is shown with the <EM>blue</EM> dashed line of Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_cool_function">B.1</a> for an MRN dust size distribution and the solar neighbourhood dust-to-gas mass ratio of 7.5&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">&nbsp;10<sup>-3</sup>.

<p>
The comparison in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_cool_function">B.1</a> of the <EM>initial</EM> cooling efficiency by the dust, 
<!-- MATH: $\Lambda _{\rm dust}^{^{\rm init}} (T)$ -->
<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img218.png"
 ALT="$\Lambda _{\rm dust}^{^{\rm init}} (T)$">,
and by atomic processes, 
<!-- MATH: $\Lambda _{\rm gas}$ -->
<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img206.png"
 ALT="$\Lambda _{\rm gas}$">,
shows that, for a dust-to-gas mass ratio of 
<!-- MATH: $Z_{\rm d} = 7.5$ -->
<IMG
 WIDTH="51" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img219.png"
 ALT="$Z_{\rm d} = 7.5$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>-3</sup>, the dust is initially the dominant coolant of the hot postshock gas for 
<!-- MATH: $T \ga 10^6$ -->
<IMG
 WIDTH="48" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img220.png"
 ALT="$T \ga 10^6$">&nbsp;K.  During the gas cooling, we calculate how much the grains are eroded and deduce the mass fraction of dust that remains in the cooling gas as a function of the temperature.  The total cooling function (<EM>red</EM> curves in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_cool_function">B.1</a>) is deduced by summing the gas and dust cooling rates.   We show different cooling efficiencies for different initial temperatures, corresponding to different shock velocities.

<p>

<h3 class="sec2"><a name="SECTION000132000000000000000"></a><A NAME="subsec_evollowdensgas"></A>
B.2 Results: evolution of the hot gas temperature  and dust survival
</h3>

<p>
The temperature profiles are shown in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_temp_profiles">B.2</a> for different shock velocities (100 to 1000&nbsp;km&nbsp;s<sup>-1</sup>), which corresponds to a range of preshock densities of <IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img221.png"
 ALT="${\sim}0.2$">
to 2&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>-3</sup>&nbsp;cm<sup>-3</sup> for a postshock pressure of 2&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">&nbsp;10<sup>5</sup>&nbsp;K&nbsp;cm&nbsp;<sup>-3</sup>.

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_temp_profiles">&#160;</A><!-- end Label--><A NAME="2536"></A><A NAME="figure2181"
 HREF="img222.png"><IMG
 WIDTH="87" HEIGHT="60" SRC="Timg222.png"
 ALT="\begin{figure}
\par\includegraphics[angle=90,width=7.8cm,clip]{11263fB2.eps}
\end{figure}"></A><!-- HTML Figure number: 9 --></td>
<td class="img-txt"><span class="bold">Figure B.2:</span><p>
Evolution of the gas temperature with time at constant pressure, from the postshock temperature to 10<sup>4</sup>&nbsp;K for different shock velocities (indicated on the curves in km&nbsp;s<sup>-1</sup>). The dashed line is for 
<!-- MATH: $V_{\rm s} = 600$ -->
<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.png"
 ALT="$V_{\rm s} = 600$">&nbsp;km&nbsp;s<sup>-1</sup>, which corresponds to the SQ&nbsp;hot (5&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>6</sup>&nbsp;K) plasma. The red vertical thick  line at 5&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>6</sup>&nbsp;yr indicates the collision&nbsp;age.</p></td>
</tr>
</table></div>
<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_dust_temp">&#160;</A><!-- end Label--><A NAME="2603"></A><A NAME="figure2189"
 HREF="img224.png"><IMG
 WIDTH="87" HEIGHT="57" SRC="Timg224.png"
 ALT="\begin{figure}
\par\includegraphics[angle=90,width=7.8cm,clip]{11263fB3.eps}
\end{figure}"></A><!-- HTML Figure number: 10 --></td>
<td class="img-txt"><span class="bold">Figure B.3:</span><p>
Dust mass fraction remaining as a function of the gas temperature, from the postshock temperature to 10<sup>4</sup>&nbsp;K for different shock velocities&nbsp;<IMG
 WIDTH="17" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img23.png"
 ALT="$V_{\rm s}$">
[km&nbsp;s<sup>-1</sup>] labelled on the curves. The gas cooling is isobaric, and the thermal gas pressure is set to the average pressure of the SQ&nbsp;hot gas 
<!-- MATH: $\displaystyle P_{\rm ps}/k_{\rm B} \simeq 2.3$ -->
<IMG
 WIDTH="75" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.png"
 ALT="$\displaystyle P_{\rm ps}/k_{\rm B} \simeq 2.3$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">

<!-- MATH: $10^{5}~{\rm cm^{-3}~K}$ -->
<IMG
 WIDTH="69" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img223.png"
 ALT="$10^{5}~{\rm cm^{-3}~K}$">.
The dashed line is for 
<!-- MATH: $V_{\rm s} = 600$ -->
<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.png"
 ALT="$V_{\rm s} = 600$">&nbsp;km&nbsp;s<sup>-1</sup>. The red dashed lines are the cooling isochrones at&nbsp;10<sup>5</sup>, 5&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>5</sup> and&nbsp;10<sup>6</sup>&nbsp;yr.</p></td>
</tr>
</table></div>
<p>
Figure&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_dust_temp">B.3</a> shows the remaining fraction of dust mass as a function of the gas temperature. In this plot, the thermal gas pressure is constant and equals the measured average thermal gas pressure of the Stephan's Quintet hot gas  
<!-- MATH: $\displaystyle P_{\rm ps}/k_{\rm B} \simeq 2$ -->
<IMG
 WIDTH="65" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img225.png"
 ALT="$\displaystyle P_{\rm ps}/k_{\rm B} \simeq 2$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">

<!-- MATH: $10^{5}~[\rm cm^{-3}~K]$ -->
<IMG
 WIDTH="78" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img196.png"
 ALT="$10^{5}~[\rm cm^{-3}~K]$">.
This plot illustrates a dichotomy in the evolution of the dust to gas mass ratio. On the top right side, a significant fraction of the dust survives in the gas, whereas on the left side, almost all the dust is destroyed on timescales shorter than the collision age (5&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>6</sup>&nbsp;yr). In the intercloud gas shocked at high velocities (
<!-- MATH: $V_{\rm s} > 400$ -->
<IMG
 WIDTH="54" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img199.png"
 ALT="$V_{\rm s} > 400$">&nbsp;km&nbsp;s<sup>-1</sup>), the dust mass fraction drops rapidly. In clouds where the gas is shocked at 
<!-- MATH: $V_{\rm s} < 300$ -->
<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img188.png"
 ALT="$V_{\rm s} < 300$">&nbsp;km&nbsp;s<sup>-1</sup>, the gas retains a large fraction of its dust content (&gt;20%).
For the gas that is heated to temperatures 
<!-- MATH: $T > 10^{6}$ -->
<I>T</I> &gt; 10<sup>6</sup>&nbsp;K, the dust lifetime is shorter than the gas cooling time and most of the dust is destroyed. At&nbsp;lower temperatures (
<!-- MATH: $T < 10^{6}$ -->
<I>T</I> &lt; 10<sup>6</sup>&nbsp;K), the dust cooling rate is lower than the gas cooling rate. Then, the dust never contributes significantly to the cooling of the postshock gas. This last result is in agreement with the earlier study by <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Smith1996ApJ">Smith et&nbsp;al. (1996)</a>.

<p>

<h2 class="sec"><a name="SECTION000140000000000000000"></a>
<A NAME="appendix_H2"></A>
Appendix C: Modeling H<SUB>2</SUB> formation
</h2>

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_CoolFunct_isobar_vs_T">&#160;</A><!-- end Label--><A NAME="2539"></A><A NAME="figure2217"
 HREF="img226.png"><IMG
 WIDTH="87" HEIGHT="62" SRC="Timg226.png"
 ALT="\begin{figure}
\par\includegraphics[width=7.8cm,clip]{11263fC1.eps}
\end{figure}"></A><!-- HTML Figure number: 11 --></td>
<td class="img-txt"><span class="bold">Figure C.1:</span><p>
Cooling efficiency [erg&nbsp;s<sup>-1</sup>&nbsp;cm<sup>-3</sup> by <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;S(0) to S(5)&nbsp;rotational lines (thin red line) and all <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;lines (thick red line) for the gas cooling at constant thermal pressure. Initially, the gas is ionized (see text for details), the density is 
<!-- MATH: $n_{\rm H} = 10$ -->
<IMG
 WIDTH="48" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img26.png"
 ALT="$n_{\rm H} = 10$">&nbsp;cm<sup>-3</sup> and temperature is <I>T</I> =10<sup>4</sup>&nbsp;K. For comparison, the local cooling rate of all the coolants is shown (green dashed line).</p></td>
</tr>
</table></div>
<p>

<h3 class="sec2"><a name="SECTION000141000000000000000"></a>
<A NAME="subsec_lowT_chemistry"></A>
C.1 Cooling function for  <I>T</I> &lt; 10<sup>4</sup>&nbsp;K
</h3>

<p>
This section describes the cooling of clouds below 10<sup>4</sup>&nbsp;K and H<SUB>2</SUB>&nbsp;formation. We use the chemistry, and the atomic and molecular cooling functions described in <A NAME="aaref23"></A><a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#2003MNRAS.341...70F">Flower et&nbsp;al. (2003)</a> and references therein. The principal coolants are&nbsp;O, <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">,

<!-- MATH: $\rm H_{2}O$ -->
<IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img227.png"
 ALT="$\rm H_{2}O$">
and OH (see Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_timescales_1E4K_vs_nH_SQ">3</a> in <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#2003MNRAS.341...70F">Flower et&nbsp;al.  2003</a>). The time evolution of the gas temperature and composition is computed at a fixed thermal gas pressure equal to that of the intercloud gas (2&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>5</sup>&nbsp;K&nbsp;cm<sup>-3</sup>). The metallicity and the gas-to-dust mass ratio are assumed to be the solar neighbourhood values. The initial ionization state of the gas is the out-of-equilibrium value resulting from cooling from higher temperatures (<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Gnat2007ApJS">Gnat &amp; Sternberg  2007</a>). We assume a standard value for the cosmic ray ionization rate of <IMG
 WIDTH="33" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img140.png"
 ALT="$\zeta = 5$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>-17</sup>&nbsp;s<sup>-1</sup> and the UV&nbsp;radiation field is not considered here (see Sect.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#subsec_mass_nrj_budget">2.2</a>). The initial temperature is 10<sup>4</sup>&nbsp;K and density 
<!-- MATH: $n_{\rm H} = 10$ -->
<IMG
 WIDTH="48" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img26.png"
 ALT="$n_{\rm H} = 10$">&nbsp;cm<sup>-3</sup>. Hydrogen is initially highly ionized. During the postshock gas cooling, hydrogen recombination occurs, the neutral gas cools, <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;starts to form and further cools and condenses the&nbsp;gas.

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="Fig_species_time">&#160;</A><!-- end Label--><A NAME="2540"></A><A NAME="figure2240"
 HREF="img228.png"><IMG
 WIDTH="88" HEIGHT="62" SRC="Timg228.png"
 ALT="\begin{figure}
\par\includegraphics[width=8cm,clip]{11263fC2.eps}
\end{figure}"></A><!-- HTML Figure number: 12 --></td>
<td class="img-txt"><span class="bold">Figure C.2:</span><p>
Isobaric cooling of the postshock gas and <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;formation. The temperature (blue line, labeled on the right side) and abundance profiles relative to&nbsp;<IMG
 WIDTH="18" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img27.png"
 ALT="$n_{\rm H}$">
are shown.  The initial conditions and model are the same as in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_CoolFunct_isobar_vs_T">C.1</a>. The final density of the molecular gas is 
<!-- MATH: $n(\rm H_2) = 2$ -->
<IMG
 WIDTH="58" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.png"
 ALT="$n(\rm H_2) = 2$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">
10<sup>4</sup>&nbsp;cm<sup>-3</sup>, for a temperature of&nbsp;<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.png"
 ALT="${\sim }10$">&nbsp;K.</p></td>
</tr>
</table></div>
<p>
Figure&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_CoolFunct_isobar_vs_T">C.1</a> presents the contribution of <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;line emission to the total cooling function as a function of the gas temperature. It&nbsp;shows the cooling functions of <EM>(i)</EM> all the <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;lines (thick red line); <EM>(ii)</EM>&nbsp;the <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;rotational lines S(0) to&nbsp;S(5) detected by <EM>Spitzer</EM>(thin red line);  <EM>(iii)</EM>&nbsp;the total cooling efficiency in which all elements are included (green dashed line). The cooling efficiencies of some other major coolants (O&nbsp;in blue, 
<!-- MATH: $\rm H_{2}O$ -->
<IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img227.png"
 ALT="$\rm H_{2}O$">
in purple, and OH in orange) are also indicated. H<SUB>2</SUB>&nbsp;excitation at low temperatures (<IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.png"
 ALT="${\sim }10$">&nbsp;K) is dominated by the contribution associated with <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;formation.

<p>

<h3 class="sec2"><a name="SECTION000142000000000000000"></a><A NAME="subsec_cloudcoolingH2form"></A>
C.2 H<SUB>2</SUB> formation
</h3>

<p>
The chemical abundance profiles in the postshock gas are illustrated in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_species_time">C.2</a> as a function of time, for the same model as in Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_CoolFunct_isobar_vs_T">C.1</a>
From the time when the gas attains 10<sup>4</sup>&nbsp;K, it takes <IMG
 WIDTH="20" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img229.png"
 ALT="${\sim}3$">&nbsp;<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.png"
 ALT="$\times $">&nbsp;10<sup>5</sup>&nbsp;yrs to form&nbsp;<IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">.
This time scales inversely with the dust-to-gas mass ratio. The <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;formation gives rise to a shoulder in the temperature profile around 200&nbsp;K. At this point, the energy released by  <IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$\rm H_2$">&nbsp;formation is roughly balanced by the cooling due to atomic oxygen (see Fig.&nbsp;<a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#Fig_timescales_1E4K_vs_nH_SQ">3</a> in <a href="/articles/aa/full_html/2009/29/aa11263-08/aa11263-08.html#2003MNRAS.341...70F">Flower et&nbsp;al.  2003</a>).

<p>

<p>
<br>

</div></body></html>