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<!-- DOI: 10.1051/0004-6361/200811233 -->

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

<p>

<p>

<h2 class="sec"><a name="SECTION00080000000000000000"></a>
Appendix A: Selection of the best-fit  and the accuracy of the parameter values
</h2>

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig9">&#160;</A><!-- end Label--><A NAME="1375"></A><A NAME="figure1154"
 HREF="img82.png"><IMG
 WIDTH="200" HEIGHT="89" SRC="Timg82.png"
 ALT="\begin{figure}
\rotatebox{270} {\resizebox{8cm}{!}{
\includegraphics{1233fia1.eps}
}}
\end{figure}"></A><!-- HTML Figure number: 9 --></td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
Residuals (observations minus synthetic maps) of the
   <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1&nbsp;line brightness from our best model fitting for the
   GoHam disk. The spatial scale and contours are the same as in 
   the observations and predictions, Figs.&nbsp;<a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#fig1">1</a>, <a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#fig6">6</a>.  </p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=9&DOI=10.1051/0004-6361/200811233" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>

<h3 class="sec2"><a name="SECTION00081000000000000000"></a>
A.1 Criteria for acceptable models
</h3>

<p>
The general criterion we chose to select acceptable models was the
comparison of the predicted images with the observed ones. Some authors
(e.g., Dutrey el al. 2007; Pety et&nbsp;al. 2006; Isella et&nbsp;al. 2007)
perform such a comparison in
the Fourier transformed plane of the visibilities. The selected model
parameters are then those yielding the smallest residuals, after
considering `blind' variations of the parameter values. Their method
has the advantage of being objective and that uncertainties
introduced by the cleaning process are avoided.
Other authors
(<A NAME="aaref9"></A><A NAME="tex2html43"
 HREF="#fuente06">Fuente et&nbsp;al.  2006</A>; <A NAME="aaref20"></A><A NAME="tex2html44"
 HREF="#mannings00">Mannings &amp; Sargent  2000</A>,<A NAME="aaref19"></A><A NAME="tex2html45"
 HREF="#mannings97">1997</A>) follow, however, a more intuitive
approach, comparing directly the images. 

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig10">&#160;</A><!-- end Label--><A NAME="1377"></A><A NAME="figure1166"
 HREF="img83.png"><IMG
 WIDTH="200" HEIGHT="89" SRC="Timg83.png"
 ALT="\begin{figure}
\rotatebox{270}{\resizebox{8cm}{!}{
\includegraphics{1233fia2.eps}
}}
\end{figure}"></A><!-- HTML Figure number: 10 --></td>
<td class="img-txt"><span class="bold">Figure A.2:</span><p>
Predictions of the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1&nbsp;line brightness from our
   best model fitting for the GoHam disk. The spatial scale and 
   contours are the same as in the observations, Fig.&nbsp;<a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#fig2">2</a>.
</p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=10&DOI=10.1051/0004-6361/200811233" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig11">&#160;</A><!-- end Label--><A NAME="1379"></A><A NAME="figure1175"
 HREF="img84.png"><IMG
 WIDTH="200" HEIGHT="89" SRC="Timg84.png"
 ALT="\begin{figure}
\rotatebox{270}{\resizebox{8cm}{!}{
\includegraphics{1233fia3.eps}
}}
\end{figure}"></A><!-- HTML Figure number: 11 --></td>
<td class="img-txt"><span class="bold">Figure A.3:</span><p>
Residuals (observations minus synthetic maps) of the
   <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1&nbsp;line brightness from our best model fitting for the
   GoHam disk. The spatial scale and contours are the same as in 
the observations and predictions, Figs.&nbsp;<a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#fig2">2</a>, A.2.  </p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=11&DOI=10.1051/0004-6361/200811233" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>
In our case, the high number of free parameters prevents a blind
analysis of the resulting residuals for any combination of parameter
values, which would imply an exceedingly high amount of
calculations. Even the definition of ``free parameter'' in this complex
structure is difficult, and it is unclear at what extent we can vary
all the disk properties. Even if theoretical constraints on our model
parameters are taken into account (see Sect.&nbsp;3), we continue to have
more than&nbsp;15 (more or less free) model parameters: 5 defining the
geometry, 2 for the dynamics, 5
for the temperature, 3
for the density, and finally the <sup>12</sup>CO/<sup>13</sup>CO/C<sup>17</sup>O
abundance ratio. It is difficult to decrease such a high number of
parameters because of the different components identified in
the maps: the hotter and less dense fringes separated from the equator,
an outer less dense region, the cold central part of the disk, and the
central hotter region.

<p>
On the other hand, our procedure allows an intuitive analysis of the
relevance of the different parameters. We can infer that some
parameters are not so relevant in fitting some observational
properties, and that certain observational features are related to only
one or two model parameters. For instance, the total size of the disk
is selected to match the total extent of the image, which depends
weakly on, e.g., the rotational velocity.

<p>
It has been argued that in sources comparable in angular size to the
telescope resolution, the comparison performed in the image plane is
inaccurate. This is untrue in our case, since GoHam occupies an area
about 30&nbsp;square arcseconds, almost a factor of 20 larger than the beam
of our <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 data. The cleaning process always introduces
noise, but, besides numerical noise, it is mostly due to uncertainties
in the calibration of the complex visibilities, i.e., the true beam
shape not being exactly equal to the theoretical beam function (for the
<I>uv</I>&nbsp;coverage of the observation). Subtraction of the convolved `clean
components' and the subsequent convolution to obtain the final image
introduces then spurious features, mainly when the <I>uv</I>&nbsp;sampling is
poor. However, such unavoidable uncertainties in the amplitude and
phase calibration also appear if the fitting is performed in the 
<I>uv</I>&nbsp;plane, and the limitation in the dynamic range of the observations due
to them applies to both the sky plane and the Fourier transformed
plane. On the other hand, we also note that Fourier transformation of
the predicted brightness (which must be calculated for a finite grid of
points in space coordinates from the standard radiative transfer
equation) also introduces numerical noise.

<p>
As we see below in actual cases, it is practically impossible to define
a single best-fit model in our case, because of the complex data set
and model. Instead, we adopt criteria to consider whether a set of
parameter values is acceptable and, when they are not satisfied,
provide limits to these values. This process will only be detailed for
the most representative parameters: radius and width of the disk,
central mass (i.e., rotational velocity law), typical densities and
temperatures. There is no significant differences among the acceptable
models, both in their predictions and in values of the relevant disk
properties; one of them was chosen as our best-fit model.

<p>
To select the acceptable models, we basically use the comparison
of the predictions with the observations of the <sup>12</sup>CO and <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 lines, in which <I>S</I>/<I>N</I> and the spatial and spectral
resolutions are particularly high. We checked that the predictions
for the <I>J</I>&nbsp;=&nbsp;3-2 lines are also satisfactory. The criteria used to select
acceptable models are:

<p>

<B>c1)</B> The differential image, i.e., the difference between observed
and synthetic
velocity channels, must have, in the regions of each channel where
emission is present, an <I>rms</I> noise not exceeding <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">1.5&nbsp;times
that present in adjacent regions with no emission. Those regions, of
noise not higher than 1.5 times that found in adjacent ones, are only
slightly noticeable in the differential maps.

<p>

<B>c2)</B> In the differential <I>J</I>&nbsp;=&nbsp;2-1 image, the residuals must be
smaller than 2&nbsp;times the spurious contours (due to noise) seen in
adjacent regions (<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">0.6&nbsp;Jy/beam, two contours, in our <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 images), and no residual <IMG
 WIDTH="13" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img85.png"
 ALT="$\ga$">0.3&nbsp;Jy/beam must appear systematically,
i.e., in the same spatial offsets for several velocity channels. We
note that one can identify noise features more intense than 0.3&nbsp;Jy/beam
in the observed <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 maps, so residuals not exceeding the above
limits are again not clearly different than the observation noise.

<p>
These conditions are relaxed for velocities around 4.5-5.5&nbsp;km&nbsp;s<sup>-1</sup> LSR,
which present a strong emission clump with no counterpart in the
equivalent blue-shifted emission.
This excess cannot be due to opacity or excitation effects, since it is
more prominent in the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 maps than in the <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 ones and much more than for <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;3-2. The excess is also clear in
the C<sup>17</sup>O <I>J</I>&nbsp;=&nbsp;3-2 maps. The fact that this excess is so remarkable in
the less optically thick <sup>13</sup>CO emission strongly suggests that it is
mainly due to the presence of a gas condensation, rotating at about 
2.5&nbsp;km&nbsp;s<sup>-1</sup> at a distance of about 
<!-- MATH: $5 \times  10^{15}$ -->
<IMG
 WIDTH="50" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$5 \times 10^{15}$">&nbsp;cm from the star. Our
model shows axial symmetry and therefore cannot explain this excess; we
chose to fit mainly the emission from the rest of the nebula. A
tentative model fitting of this feature is presented in&nbsp;A.3, and the
consequences and possible origins of the presence of this condensation
are discussed in Sect.&nbsp;4.1.

<p>
In Fig. A.1, we present the residuals of our fitting of the <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1&nbsp;maps. Predictions and residuals for <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 are shown in
Figs.&nbsp;A.2 and&nbsp;A.3. Except for the velocities around 5&nbsp;km&nbsp;s<sup>-1</sup>, the
typical rms noise outside the emitting regions is about 
0.1-0.11&nbsp;Jy/beam (slightly less for <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1), and does not exceed 
1.5-1.7&nbsp;Jy/beam in the regions with emission. We see that the largest
residuals in the differential images do not reach two contours (noise
reaching one level is also seen out of the emitting regions).

<p>
<A NAME="tab2"></A><p class="inset-old"><a href="/articles/aa/full_html/2009/24/aa11233-08/tableA.1.html"><span class="bold">Table A.1:</span></a>&#160;&#160;
Relative range of acceptable values, around those given in
  Table&nbsp;<a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#tab1">1</a>, of the main parameters
  defining the model disk in
  Gomez's Hamburger.</p>
<p>
Of course, between 4.5 and 5.5&nbsp;km&nbsp;s<sup>-1</sup>, the situation is worse, reflecting
the asymmetry in the disk density mentioned above. We can notice some
other minor features in the images that are not accounted for by our
model. For instance, the <sup>13</sup>CO&nbsp;line emission at 3.7&nbsp;km&nbsp;s<sup>-1</sup> is less
extended than expected, which is not the case at 4.1&nbsp;km&nbsp;s<sup>-1</sup>. We also note
in our <sup>13</sup>CO maps a weak extent towards the north at about 1.3&nbsp;km&nbsp;s<sup>-1</sup>,
but practically at the noise level. In <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1, we see a similar
protrusion, but not exactly at the same position. We finally note in
some observed panels that the emission from regions close to the star
(see, e.g., the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 emission at <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">0.5&nbsp;km&nbsp;s<sup>-1</sup> and the
<sup>12</sup>CO emission at 3.7&nbsp;km&nbsp;s<sup>-1</sup>) is more extended than the predictions. The
width of our disk is given mainly by the equation of hydrostatics
(Sect.&nbsp;3), under the standard theory of massive flaring disks; it is
possible that these usual theoretical requirements are not fully
satisfied in the innermost regions of the disks, although further
analysis of this phenomenon obviously requires higher-quality
observations.

<p>

<h3 class="sec2"><a name="SECTION00082000000000000000"></a>
A.2 Uncertainty in the fitted parameters
</h3>

<p>
We estimated the uncertainties in derived values of the model
parameters by varying the values of each one (while the others remain
unchanged) and checking the values for which the above conditions, <B>c1</B> and <B>c2</B>, were clearly not satisfied. This procedure was
completed for the main, most representative parameters, for instance,
the disk radius&nbsp;
<!-- MATH: $R_{\rm out}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.png"
 ALT="$R_{\rm out}$">,
the characteristic density, etc. The
results are summarized in Table&nbsp;<a href="/articles/aa/full_html/2009/24/aa11233-08/aa11233-08.html#tab1">1</a>. We also present below some cases in
which the uncertainty in the derived parameters requires some
discussion.

<p>
We did not attempt to consider fully the parameter uncertainties when
two or more parameters are allowed to vary. For example, the density
range is slightly larger than that given in the table if we allow the
temperature also to vary, since both variations are in some way 
compensated. In general, however, the ranges do not differ very much
from our standard uncertainty brackets. The density is
mostly fixed by the emission of <sup>13</sup>CO and that of <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 from outer regions, while the temperature law is
mainly given by the <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 and <I>J</I>&nbsp;=&nbsp;3-2&nbsp;maps.

<p>
We note the uncertain determination of the density in the outer disk
(
<!-- MATH: $r \sim R_{\rm out}$ -->
<IMG
 WIDTH="48" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img91.png"
 ALT="$r \sim R_{\rm out}$">), which is only given by the <sup>12</sup>CO emission,
since <sup>13</sup>CO is not detected in this region (see Table&nbsp;A.1). We also
recall that the assumption of level population thermalization may 
be invalid for these diffuse regions, which may imply that the density in
them is higher than the values given here, perhaps closer to

<!-- MATH: $3 \times 10^{3}$ -->
<IMG
 WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img92.png"
 ALT="$3 \times 10^{3}$">&nbsp;cm<sup>-3</sup>. 

<p>
Important problems with the thermalization assumption are unexpected
for the relatively high densities of the remaining nebula and the
analyzed transitions (Sect.&nbsp;3). The <I>J</I>=6-5 transition is significantly
more sensitive to these effects, due to the relatively high Einstein
coefficients of high-<I>J</I> transitions. We also expect underpopulation of
the <I>J</I>=6 and <I>J</I>=5&nbsp;levels in the high-<I>z</I> low-density regions, with

<!-- MATH: $n \sim  10 ^{4}$ -->
<IMG
 WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img93.png"
 ALT="$n \sim 10 ^{4}$">&nbsp;cm<sup>-3</sup>, leading to 6-5 brightness temperatures
of under 10&nbsp;K. This line would mainly come from this high temperature
surface, and we think that this phenomenon is responsible for the
non-detection of <sup>12</sup>CO <I>J</I>=6-5 in our observations. Any further
discussion is impossible because of the lack of accurate information
about the <I>J</I>=6-5 emission. This relative underpopulation of high-<I>J</I>levels in the layers at high absolute values of&nbsp;<I>z</I> may lead to a
relative overpopulation of the <I>J</I>=2&nbsp;level, and therefore to more
emission than expected in <I>J</I>&nbsp;=&nbsp;2-1. This effect could lead to slightly
smaller jumps of the temperature, perhaps closer to factor&nbsp;2.

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig12">&#160;</A><!-- end Label--><A NAME="1383"></A><A NAME="figure1215"
 HREF="img94.png"><IMG
 WIDTH="200" HEIGHT="89" SRC="Timg94.png"
 ALT="\begin{figure}
\rotatebox{270}{\resizebox{8cm}{!}{
\includegraphics{1233fia4.eps}
}}
\end{figure}"></A><!-- HTML Figure number: 12 --></td>
<td class="img-txt"><span class="bold">Figure A.4:</span><p>
Predictions of the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1&nbsp;line brightness from our
   best model fitting for the GoHam disk, including the increase in
   temperature and density in a southern clump, as described in
   Appendix&nbsp;A.3. The spatial scale and contours are the same as in the
   observations, Fig.&nbsp;2, and our standard model predictions, Fig.&nbsp;A.2.
  </p></td>
</tr><tr><td colspan="2"><a href="http://dexter.edpsciences.org/applet.php?pdf_id=12&DOI=10.1051/0004-6361/200811233" target="DEXTER">Open with DEXTER</a></td></tr>

</table></div>
<p>
Some parameter pairs are quite dependent each on other. This is
particularly true for the density and CO total abundance,
since we assume LTE and then the line opacity depends on the product
of the density and the relative abundance. Both parameters can
therefore vary freely, provided that their product remains constant. The
indetermination is solved assuming a relative <sup>12</sup>CO abundance of
10<sup>-4</sup>. This is also the case for parameters that we have not
considered separately in this uncertainty analysis, such as the
temperature at a given point, <I>T</I>(<I>R</I><SUB>0</SUB>), and the slope of the
temperature law, <IMG
 WIDTH="18" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img70.png"
 ALT="$\alpha_T$">;
instead, as mentioned, we discuss
the uncertainty in the characteristic temperature. Finally, we note the
case of the rotation velocity and the conditions in the hot and dense
center of the disk. Strong emission from these regions could mimic the
emission of a faster rotating disk in the extreme velocity
channels. So, the rotation velocity is mainly determined from the
emission extent in the channels at moderate velocities, the emission at
the extreme channels depending on both the Keplerian velocity and the
emissivity of the central regions. In general, the uncertainty in the
parameters defining the inner, denser region (
<!-- MATH: $r <  2  \times R_{\rm in}$ -->
<IMG
 WIDTH="65" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img95.png"
 ALT="$r < 2 \times R_{\rm in}$">)
is high, because this clump is not resolved and the
number of independent observational constraints is small (as also
concluded in Paper&nbsp;I). We can say that an inner region of higher
density and temperature is necessary to attain our strong requirements
on the fitting quality and that it must be smaller than about 
10<sup>16</sup>&nbsp;cm, but we cannot provide details of this region.

<p>
The outer hot region, for high values of <I>z</I>, is hardly resolved in our
<sup>12</sup>CO&nbsp;maps. Therefore, we could also reproduce our data assuming that
this region is significantly thinner and brighter (in general, hotter)
than in our standard model. However, those disk models are less
probable than our standard one, since the jump in temperature we found
is already quite high (see Sect.&nbsp;3). Moreover, for very high
temperatures in the high-<I>z</I> rim, we should also increase the typical
density significantly (to be able to reproduce <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 data,
which, for the assumed dependence of density on&nbsp;<I>z</I>, is only moderately
opaque in the high-<I>z</I> regions). This would then imply too low values
of <I>X</I>(<sup>13</sup>CO), when attempting to reproduce the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 data,
leading to improbably high values of <I>X</I>(<sup>12</sup>CO)/<I>X</I>(<sup>13</sup>CO), of
significantly over&nbsp;100. In any case, we note that the properties of
this high-<I>z</I> bright rim depend on the relative calibration of the
<I>J</I>&nbsp;=&nbsp;2-1 and <I>J</I>&nbsp;=&nbsp;3-2&nbsp;lines; allowing&nbsp;15% variations in the relative
calibrations, we can fit the data with temperature jumps ranging
between&nbsp;2 and&nbsp;5 (and hot-layer widths that do not differ significantly
from our standard value).

<p>

<h3 class="sec2"><a name="SECTION00083000000000000000"></a>
A.3 Model (tentative) fitting of the southern brightness maximum
</h3>

<p>
We mentioned that there is a relative maximum in the southern part
of the disk that has no counterpart in the north. This maximum is seen
in all our maps, but is more prominent in the <sup>13</sup>CO emission, which is
mostly optically thin, as well as in the C<sup>17</sup>O line. Therefore, this
brightness maximum must be due mainly to an increase in the density of
some southern regions, although an increase in temperature is also
necessary to explain the excess observed in <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1.

<p>
In our standard model analysis, we mostly tried to fit the emission
from the rest of the nebula (Sects.&nbsp;3, A1, A2). Any attempt to fit the
emission excess from this southern clump is very uncertain, because of
the lack of previous experience in trying to study condensations of
this kind, including the lack of theoretical modeling, and because of
the poor information contained in our observations, which scarcely
resolve its extent.

<p>
Nevertheless, we note that this emission excess can be detected over a
remarkable range of velocities, between about&nbsp;4.1 and 5.7&nbsp;km&nbsp;s<sup>-1</sup> LSR. This means that the emitting condensation cannot be very small,
the projected velocity dispersion being caused by gas emission from
different distances from the star or from regions showing a significant
variation in the velocity projection along the line of sight. In both
cases, we expect typical sizes <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">
<!-- MATH: $5 \times 10^{15}$ -->
<IMG
 WIDTH="50" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$5 \times 10^{15}$">&nbsp;cm 
(<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">1
<!-- MATH: $\hbox{$^{\prime\prime}$ }$ -->
<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img14.png"
 ALT="$\hbox{$^{\prime\prime}$ }$">
for the adopted distance). We can also assume that the
condensation occurs in the equatorial disk regions in which <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 emission originates, because the maximum is so prominent in this
line. We have assumed that the emission comes from a region defined by

<!-- MATH: $R_{\rm in} < r < 10^{16}$ -->
<IMG
 WIDTH="83" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img96.png"
 ALT="$R_{\rm in} < r < 10^{16}$">&nbsp;cm, and <I>z</I><SUB><I>s</I></SUB>/<I>r</I> &lt; 0.6, where
<I>z</I><SUB><I>s</I></SUB> is the distance between a given point and the plane containing
the star and perpendicular to the equator that gives the extreme
projections for the rotation velocity. We assume that the temperature
and density of this region vary with respect to the standard laws
assumed for the rest of the nebula.

<p>
To explain the (moderate) emission excess found in <sup>12</sup>CO <I>J</I>&nbsp;=&nbsp;2-1, we
estimate that we must assume an increase in temperature in the
southern condensation of a factor <IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.png"
 ALT="$\sim$">1.5, with respect to our
standard laws for nearby regions. Finally, we estimate the
excess density in the condensation by comparing the model
results with the intensity of <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1. Since the <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 emission from these inner regions is not fully optically thin and the
beam dilution is not negligible, a
significant density increase, of a factor&nbsp;10, is necessary. A total
mass increase of about 
<!-- MATH: $6 \times 10^{-3}$ -->
<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$6 \times 10^{-3}$">&nbsp;
<!-- MATH: $M_{{\odot}}$ -->
<IMG
 WIDTH="23" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.png"
 ALT="$M_{{\odot}}$">
is deduced. 

<p>
The resulting brightness distribution is shown in Fig.&nbsp;A.4.  We can
see that the brightness excess of <sup>13</sup>CO <I>J</I>&nbsp;=&nbsp;2-1 in this southern clump
is reasonably well reproduced, but the location of the predicted
maximum is slightly closer to the star than the observed one. This
cannot be avoided assuming a larger distance from the star for the
clump, because then the velocity of the feature would be less
positive. The velocity field in this region must then also be 
disturbed.  See discussion on the interpretation of this emission
excess in Sect.&nbsp;4.1.

<p>
<br>

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