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<!-- DOI: 10.1051/0004-6361/200913597 -->

<h2 class="sec">Online Material</h2>
 
<p></p><h2 class="sec"><a name="SECTION00080000000000000000"></a>
Appendix A: Note on the influence of both the&nbsp;Riemann solver and the numerical resolution</h2>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="mu20_res"></A><!-- end Label--><A NAME="711"></A><A NAME="figure561" HREF="img50.png"><IMG SRC="Timg50.png" ALT="\begin{figure}
\par\includegraphics[width=7.4cm,height=8cm,clip]{13597fg10.eps}
\end{figure}" HEIGHT="90" WIDTH="83"></A><!-- HTML Figure number: 6 -->
</td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
Case <IMG SRC="img1.png" ALT="$\mu =20$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="42">:
density maps in the <i>xy</i>-plane at time 
<!-- MATH: $t\sim1.16~t_{\rm ff}$ -->
<IMG SRC="img49.png" ALT="$t\sim1.16~t_{\rm ff}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="63">
for 4 barotropic EOS calculations with HLLD, 
<!-- MATH: $N_{\rm J}=6$ -->
<IMG SRC="img5.png" ALT="$N_{\rm J}=6$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="41">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
LF, 
<!-- MATH: $N_{\rm J}=20$ -->
<IMG SRC="img7.png" ALT="$N_{\rm J}=20$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="49">
and 
<!-- MATH: $N_{\rm exp}=4$ -->
<IMG SRC="img8.png" ALT="$N_{\rm exp}=4$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
LF, 
<!-- MATH: $N_{\rm J}=10$ -->
<IMG SRC="img9.png" ALT="$N_{\rm J}=10$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="49">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
LF, 
<!-- MATH: $N_{\rm J}=6$ -->
<IMG SRC="img5.png" ALT="$N_{\rm J}=6$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="41">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">.

</td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=6&DOI=10.1051/0004-6361/200913597" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p></p><div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="a037_fld_baro"></A><!-- end Label--><A NAME="712"></A><A NAME="figure575" HREF="img52.png"><IMG SRC="Timg52.png" ALT="\begin{figure}
\par\includegraphics[width=12.5cm,height=7cm,clip]{13597fg11.eps}
\end{figure}" HEIGHT="78" WIDTH="141"></A><!-- HTML Figure number: 7 -->
</td>
<td class="img-txt"><span class="bold">Figure A.2:</span><p>
Density and temperature maps in the equatorial plane at time <i>t</i>=38.3&nbsp;kyr (
<!-- MATH: $1.15~t_{\rm ff}$ -->
<IMG SRC="img51.png" ALT="$1.15~t_{\rm ff}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="42">), for the 3 unmagnetized cases: FLD approximation and 
<!-- MATH: $\gamma=7/5$ -->
<IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">
(<i> left</i>), the FLD approximation and 
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">
(<i> middle</i>), and with a barotropic EOS using 
<!-- MATH: $\rho_{\rm ad}=2.3\times10^{-13}$ -->
<IMG SRC="img13.png" ALT="$\rho _{\rm ad}=2.3\times 10^{-13}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="101">&nbsp;g&nbsp;cm<sup>-3</sup> and 
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">
(<i> right</i>). Velocity vectors are overplotted on the density maps
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</table></div>We present a short convergence study to investigate the effect of both the Riemann solver and the numerical resolution 
on the fragmentation process. We show in Sect.&nbsp;<a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#mu20">3.1</a>
that the use of the LF Riemann solver leads to inaccurate results. We
perform calculations using the barotropic EOS and the HLLD or LF
solvers at various numerical resolutions. The latter is determined
by&nbsp;2&nbsp;parameters: <IMG SRC="img17.png" ALT="$N_{\rm J}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="18">,
the number of points per Jeans length, and 
<!-- MATH: $N_{\rm exp}$ -->
<IMG SRC="img53.png" ALT="$N_{\rm exp}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="28">,
which indicates the extent of the mesh around a refined cell (if a cell is flagged for refinement, then the 
<!-- MATH: $N_{\rm exp}$ -->
<IMG SRC="img53.png" ALT="$N_{\rm exp}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="28">
cells around the flagged cell, in each direction, will also be refined). A large 
<!-- MATH: $N_{\rm exp}$ -->
<IMG SRC="img53.png" ALT="$N_{\rm exp}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="28">
provides a smoother transition between the levels of the AMR grid. All calculations presented in Sect.&nbsp;<a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#mu20">3.1</a> have been run with 
<!-- MATH: $N_{\rm exp}=4$ -->
<IMG SRC="img8.png" ALT="$N_{\rm exp}=4$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">.
We report results of 4 calculations, performed for: HLLD, 
<!-- MATH: $N_{\rm J}=6$ -->
<IMG SRC="img5.png" ALT="$N_{\rm J}=6$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="41">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
LF, 
<!-- MATH: $N_{\rm J}=20$ -->
<IMG SRC="img7.png" ALT="$N_{\rm J}=20$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="49">
and 
<!-- MATH: $N_{\rm exp}=4$ -->
<IMG SRC="img8.png" ALT="$N_{\rm exp}=4$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
LF, 
<!-- MATH: $N_{\rm J}=10$ -->
<IMG SRC="img9.png" ALT="$N_{\rm J}=10$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="49">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">;
and LF, 
<!-- MATH: $N_{\rm J}=6$ -->
<IMG SRC="img5.png" ALT="$N_{\rm J}=6$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="41">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">.

<p>Figure&nbsp;<a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#mu20_res">A.1</a> portrays the density maps in the equatorial plane for the 4&nbsp;calculations, at time 
<!-- MATH: $t\sim1.16~t_{\rm ff}$ -->
<IMG SRC="img49.png" ALT="$t\sim1.16~t_{\rm ff}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="63">.
The calculations performed with the LF solver, 
<!-- MATH: $N_{\rm J}=6$ -->
<IMG SRC="img5.png" ALT="$N_{\rm J}=6$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="41">
and 
<!-- MATH: $N_{\rm exp}=1$ -->
<IMG SRC="img6.png" ALT="$N_{\rm exp}=1$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="51">
clearly diverge from the others, since it fragments. The lack
ofresolution clearly induces inaccurate fragmentation. The three other
calculations are qualitatively similar (no fragmentation) and agree
with <a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#Hennebelle_Teyssier_2008">Hennebelle &amp; Teyssier (2008)</a>
results (obtained using a Roe-type Riemann solver). This indicates that
the numerical resolution should be enhanced with the LF solver to avoid
spurious effects caused by the diffusivity of the solver. We note that
these differences are more important in the case of FLD calculations,
since depending on the strength of the magnetic braking, radiative
transfer can have two opposite effects. If magnetic braking is
insignificant (with poor resolution and a diffuse Riemann solver), the
material at the centrifugal barrier cools and then fragments. On the
other hand, when the magnetic braking is efficient (with HLLD or a high
resolution), the infall velocity and consequently, the accretion
luminosity, become higher, which prevents fragmentation occurring (as
material loses angular momentum and heats up). </p><p></p><h2 class="sec"><a name="SECTION00090000000000000000"></a>Appendix B: Case <IMG SRC="img2.png" ALT="$\mu $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="10">
= 1000</h2>

<p>For this quasi-hydro case, the initial dense core is highly
gravitationally unstable. We performed three types of calculations with
the LF Riemann solver: one with the barotropic EOS and<!-- MATH: $\gamma=5/3$ -->
 <IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">,
and two with the FLD and different  adiabatic exponents (
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">
and 
<!-- MATH: $\gamma=7/5$ -->
<IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">). The value 
<!-- MATH: $\gamma=7/5$ -->
<IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">
is more appropriate for <i>T</i>&gt; 100&nbsp;K, when the rotational degrees of freedom of H<sub>2</sub> are excited
(e.g., <a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#Machida_et_al_2008">Machida et&nbsp;al.  2008</a>). Using 
<!-- MATH: $\gamma=7/5$ -->
<IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">
yields cooler adiabatic cores than with 
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">
at the same density. Higher densities and temperatures in the core are thus reached more rapidly. 

</p><p>Figure&nbsp;<a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#a037_fld_baro">A.2</a> portrays density and temperature maps in the equatorial plane for the three aforementioned calculations at a time <i>t</i>=1.15 
<!-- MATH: $t_{\rm ff}$ -->
<IMG SRC="img54.png" ALT="$t_{\rm ff}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="14">.
Temperature maps range from&nbsp;11.4&nbsp;K to&nbsp;20&nbsp;K. As pointed out in <a href="/articles/aa/full_html/2010/02/aa13597-09/aa13597-09.html#Commercon_2008">Commer&#xe7;on et&nbsp;al. (2008)</a>,
the horizon of predictability is very short when the initial thermal
support is low. Calculations do not show a convergence to the same
fragmentation pattern, since we do not integrate the same system
equations. The FLD calculations yield different fragmentation modes for
the two values of <IMG SRC="img19.png" ALT="$\gamma$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">'s: for 
<!-- MATH: $\gamma=7/5$ -->
<IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">,
we obtain a central object and two satellites (4 at later times), whereas for 
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">,
we obtain a ring fragmentation pattern with orbiting fragments linked
by a bar. The barotropic EOS calculation does not produce a central
object.
From the temperature map of the FLD case with<!-- MATH: $\gamma=7/5$ -->
 <IMG SRC="img11.png" ALT="$\gamma =7/5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="48">,
we see that the central region is quite hot. Fragments are compressed more and radiate more energy than for 
<!-- MATH: $\gamma=5/3$ -->
<IMG SRC="img12.png" ALT="$\gamma =5/3$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="47">.
Fragments also form more rapidly than the time required for the
radiative feedback to become significant (at beginning of the second
collapse).
</p><p></p><p></p>
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