<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN">
<html xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML">
<link rel="stylesheet" type="text/css" href="../../../CSS_FULL/edps_full.css">
<body><div id="contenu_olm">

<!-- DOI: 10.1051/0004-6361/200913008 -->

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

<p></p><p></p><h2 class="sec"><a name="SECTION00080000000000000000"></a>
Appendix A: Classical analysis of the magnetic braking: cloud embedded into an external medium
</h2>

<p>In the context of interstellar clouds, the classical analysis of magnetic braking  (e.g.; Mouschovias <a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#mouscho1991">1991</a>; Shu et&nbsp;al. <a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#shuetal87">1987</a>), considers a rigid and dense axisymmetric cloud. We define 
<!-- MATH: $\rho_{\rm c}$ -->
<IMG SRC="img30.png" ALT="$\rho_{\rm c}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="15">
to be its density, <i>R</i> its radius and  <i>Z</i> its height.
The cloud is surrounded by a diffuse inter cloud medium of density 
<!-- MATH: $\rho_{\rm ic}$ -->
<IMG SRC="img59.png" ALT="$\rho_{\rm ic}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="19">.
The typical timescale for magnetic braking 
<!-- MATH: $\tau_{\rm br}$ -->
<IMG SRC="img60.png" ALT="$\tau_{\rm br}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="18">,
corresponds to the time necessary for torsional Alfv&#xe9;n waves
induced by the twisting of the magnetic field line, to propagate over a distance <i>l</i>,
such that the mass of gas swept by the waves is comparable to the mass
of the cloud itself. At this point, a significant fraction of the cloud
angular momentum has been transferred to the intercloud medium.
</p><p>Two cases can be considered. First, when the magnetic field and
the rotation axis are aligned, the waves propagate along the magnetic
field at the Alfv&#xe9;n speed <IMG SRC="img61.png" ALT="$V_{\rm a}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="17">,
and 
<!-- MATH: $l \rho_{\rm ic} \simeq Z \rho_{\rm c}$ -->
<IMG SRC="img62.png" ALT="$l \rho_{\rm ic} \simeq Z \rho_{\rm c}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="59">,
leading to
<br>
</p><DIV ALIGN="CENTER"><A NAME="brake_align"></A>
<!-- MATH: \begin{eqnarray}
\tau_{\rm br} \simeq \frac{Z}{V_a} ~ \frac{\rho_{c}}{\rho_{ic}}\cdot
\end{eqnarray} -->

<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td style="text-align: center;" nowrap="nowrap"><IMG SRC="img63.png" ALT="$\displaystyle \tau_{\rm br} \simeq \frac{Z}{V_a} ~ \frac{\rho_{c}}{\rho_{ic}}\cdot$" ALIGN="MIDDLE" BORDER="0" HEIGHT="49" WIDTH="85"></td>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.1)</td></tr>
</tbody></TABLE></DIV><br clear="all"><p></p>
The second case corresponds to the magnetic field and the rotation axis
being perpendicular to each other, and the waves propagating in the
equatorial plane of the cloud. The intercloud medium, which at time<!-- MATH: $t=\tau_{\rm br}$ -->
 <IMG SRC="img64.png" ALT="$t=\tau_{\rm br}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="39">
is reached by the torsional Alfv&#xe9;n waves, is located in a cylinder of radius&nbsp;<i>l</i> and height&nbsp;<i>Z</i>. In this case,

<!-- MATH: $((\tau_{\rm br}  V_{\rm a})^2 - R^2) \rho_{\rm ic} \simeq R^2  \rho_{\rm c}$ -->
<IMG SRC="img65.png" ALT="$((\tau_{\rm br} V_{\rm a})^2 - R^2) \rho_{\rm ic} \simeq R^2 \rho_{\rm c}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="144">,
which gives
<br>
<DIV ALIGN="CENTER"><A NAME="brake_perp"></A>
<!-- MATH: \begin{eqnarray}
\tau_{\rm br} \simeq \frac{R}{V_{\rm a}}  \sqrt{\frac{\rho_{\rm c}}{\rho_{\rm ic}} + 1}.
\end{eqnarray} -->

<TABLE ALIGN="CENTER" cellpadding="0" WIDTH="100%">
<tbody><tr VALIGN="MIDDLE"><td style="text-align: center;" nowrap="nowrap"><IMG SRC="img66.png" ALT="$\displaystyle \tau_{\rm br} \simeq \frac{R}{V_{\rm a}} \sqrt{\frac{\rho_{\rm c}}{\rho_{\rm ic}} + 1}.$" ALIGN="MIDDLE" BORDER="0" HEIGHT="54" WIDTH="123"></td>
<td>&nbsp;</td>
<td>&nbsp;</td>
<td ALIGN="RIGHT" WIDTH="10">
(A.2)</td></tr>
</tbody></TABLE></DIV><br clear="all"><p></p>
Since in typical astrophysical circumstances the intercloud medium has
a density that is low with respect to the cloud density, Eqs.&nbsp;(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_align">A.1</a>) and&nbsp;(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_perp">A.2</a>) show that the braking is usually more efficient when the magnetic field is perpendicular to the rotation  axis than when 
it is parallel. However,    this conclusion is obviously correct only 
when 
<!-- MATH: $R \simeq Z$ -->
<IMG SRC="img67.png" ALT="$R \simeq Z$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="37">,
i.e., if the cloud aspect ratio  is  not too different from 1.  In particular, from Eqs.&nbsp;(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_align">A.1</a>)-(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_perp">A.2</a>) we see
that if 
<!-- MATH: $Z/R \ll 1/\sqrt{\rho_{\rm c}/\rho_{\rm ic}}$ -->
<IMG SRC="img68.png" ALT="$Z/R \ll 1/\sqrt{\rho_{\rm c}/\rho_{\rm ic}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="33" WIDTH="105">,
the braking time is shorter in the aligned case than for the perpendicular configuration.

<p>Previous studies have demonstrated that magnetized clouds are
usually very flat because of the magnetic compression exerted by the radial 
component of the magnetic field (see e.g.; Li &amp; Shu <a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#Li96">1996</a>,
HF08).
These magnetized sheets, which are called pseudo-disks, are
perpendicular to the average magnetic field. In the same way,
centrifugally supported disks are also very flat objects that are
perpendicular to the rotation axis. For these extreme configurations,
the magnetic braking time can obviously be longer
when the magnetic field and the rotation axis are perpendicular than
when they are parallel (depending on the respective values of <i>R</i>, <i>Z</i> and 
<!-- MATH: $\rho_{\rm c}/\rho_{\rm ic}$ -->
<IMG SRC="img69.png" ALT="$\rho_{\rm c}/\rho_{\rm ic}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="37">).
Note that strictly speaking, if the magnetic field and the rotation
axis are not aligned with each other, the resulting structure is fully
tridimensional rather than axisymmetric.
</p><p>It is interesting to compare Eqs.&nbsp;(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_align">A.1</a>)-(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brake_perp">A.2</a>) with
Eq.&nbsp;(<a href="/articles/aa/full_html/2009/41/aa13008-09/aa13008-09.html#brak_esti">2</a>). Even though the latter is identical to the first  for 

<!-- MATH: $\rho_{\rm c} = \rho_{\rm ic}$ -->
<IMG SRC="img70.png" ALT="$\rho_{\rm c} = \rho_{\rm ic}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="46">,
there are two major differences. First, even when  
<!-- MATH: $R \simeq Z$ -->
<IMG SRC="img67.png" ALT="$R \simeq Z$" align="bottom" BORDER="0" HEIGHT="14" WIDTH="37">,
the magnetic braking is not significantly more efficient in the
perpendicular configuration than in the aligned one. The relative
efficiency of the magnetic braking in the two configurations is,
instead, directly proportional to the cloud aspect ratio. Second, as
the cloud is compressed along the field lines, the quantity<!-- MATH: $Z / \sqrt{\rho_{\rm c}}$ -->
 <IMG SRC="img71.png" ALT="$Z / \sqrt{\rho_{\rm c}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="28" WIDTH="42">
is simply proportional to <i>Z</i><sup>1/2</sup>, implying that the magnetic braking time in the aligned configuration decreases. Again this is unlike the case of a rigid cloud
embedded in a diffuse intercloud medium. 

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