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<!-- DOI: 10.1051/0004-6361/200912048 -->

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


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<h2 class="sec"><a name="SECTION000100000000000000000"></a><A NAME="section:NLTE"></A>

Appendix A: NLTE modeling
</h2>


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<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig:beta"></A><!-- end Label--><A NAME="1094"></A><A NAME="figure872" HREF="img96.png"><IMG SRC="Timg96.png" ALT="\begin{figure}
\par\includegraphics[width=8.5cm,clip]{12048figA1.eps}
\end{figure}" HEIGHT="186" WIDTH="84"></A><!-- HTML Figure number: 15 -->
</td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
      <b> a)</b>, <b> b)</b>: departure coefficients <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">
of the Ba&nbsp; <small>II</small> atomic levels  vs. height in a representative granular
(<i> left</i>) and intergranular (<i> right</i>) models.  Thick solid
and thick dash lines: the <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients for the ground

<!-- MATH: $6{\rm s}~ \rm{{^2}S}_{1/2}$ -->
      <IMG SRC="img94.png" ALT="$6{\rm s}~ \rm{{^2}S}_{1/2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="47">
and upper 
<!-- MATH: $6{\rm p}~ \rm{{^2}P}_{3/2}$ -->
      <IMG SRC="img95.png" ALT="$6{\rm p}~ \rm{{^2}P}_{3/2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="49">
levels
of the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line, respectively. Dash-dotted lines: the
      <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients for the upper level of the Ba&nbsp; <small>II</small> 4934&nbsp;&#xc5; resonance line. Dotted lines: the <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients for the
Ba&nbsp; <small>II</small> levels with excitation potentials  above 5&nbsp;eV. <b> c)</b>,
      <b> d)</b>: the line source function <i>S</i><sub><i>L</i></sub> of the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line
(dash) and Planck function <i>B</i> (solid) in units of the temperature
vs. height for the same models. <b> e)</b>, <b> f)</b>: the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line profiles for the representative granule and itergranule.
Solid and dotted lines: NLTE and LTE, respectively. <b> g)</b>, <b> h)</b>:
spatially averaged granular and intergranular profiles. 
</td>
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<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig:NLTE_height"></A><!-- end Label--><A NAME="1049"></A><A NAME="figure893" HREF="img97.png"><IMG SRC="Timg97.png" ALT="\begin{figure}
\par\includegraphics[width=8.5cm]{12048figA2.eps}
\end{figure}" HEIGHT="54" WIDTH="90"></A><!-- HTML Figure number: 16 -->
</td>
<td class="img-txt"><span class="bold">Figure A.2:</span><p>
The  NLTE (solid line) and LTE (open circles) heights of formation of the core of the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line along the slice of the snapshot <i>y</i><sub><i>i</i></sub>=0.6&nbsp;Mm. Dash-dotted line: continuum height
of formation at 4554&nbsp;&#xc5;. Dotted line: height of formation of the
line wing for the wavelength position 
<!-- MATH: $\Delta \lambda=-76.5$ -->
      <IMG SRC="img22.png" ALT="$\Delta \lambda =-76.5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="69">&nbsp;m&#xc5;.
The background image is the snapshot vertical velocity <i>V</i><sub><i>z</i></sub>.
Negative (upflow) velocities <i>V</i><sub><i>Z</i></sub> correspond to granules (dark),
while positive (downflow) velocities to intergranules (light).
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<p>Figure&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a> (panels a to f) shows the population
departure coefficients, the Ba&nbsp; <small>II</small> <IMG SRC="img1.png" ALT="$\lambda $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">4554&nbsp;&#xc5; line source
functions and line profiles for two spatial grid points of the
3D snapshot representing  the typical granular and intergranular
models. We use these models to illustrate the difference in the
NLTE results for granules and intergranules. The population
departure coefficients are defined as 
<!-- MATH: $\beta =n_{\rm NLTE}/n_{\rm
LTE}$ -->
<IMG SRC="img98.png" ALT="$\beta =n_{\rm NLTE}/n_{\rm
LTE}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="89">
where 
<!-- MATH: $n_{\rm NLTE}$ -->
<IMG SRC="img99.png" ALT="$ n_{\rm NLTE}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="35">
and 
<!-- MATH: $n_{\rm LTE}$ -->
<IMG SRC="img100.png" ALT="$n_{\rm LTE}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="27">
are the NLTE and
LTE atomic level populations, respectively. The Complex behaviour
of the <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients shown in Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>a, b is a
result of the interaction of several NLTE mechanisms described in
detail by <A NAME="aaref9"></A><a name="tex2html75" href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#bruls:etal:1992">Bruls et&nbsp;al. (1992)</a>; <A NAME="aaref10"></A><a name="tex2html76" href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#carlsson:etal:1992">Carlsson et&nbsp;al. (1992)</a>; <a name="tex2html77" href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#Shchukina:Trujillo:2001">Shchukina &amp; Trujillo Bueno (2001)</a>. Here we just point out that for the
barium atom the most important of them are ultraviolet line
pumping, ultraviolet overionization, resonance line scattering and
photon losses.
The resonance line scattering and photon losses manifest
themselves as a divergence of the lower 
<!-- MATH: $6{\rm s}~ \rm {{^2}S}_{1/2}$ -->
<IMG SRC="img94.png" ALT="$6{\rm s}~ \rm{{^2}S}_{1/2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="47">and upper 
<!-- MATH: $6{\rm p}~\rm {{^2}P}_{1/2}$ -->
<IMG SRC="img101.png" ALT="$6{\rm p}~\rm {{^2}P}_{1/2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="49">,

<!-- MATH: $6{\rm p}~ \rm {{^2}P}_{3/2}$ -->
<IMG SRC="img95.png" ALT="$6{\rm p}~ \rm{{^2}P}_{3/2}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="49">
levels
of the Ba&nbsp; <small>II</small> resonance lines. This divergence results from the
surface losses near the layer where the optical depth is equal to
unity. The losses propagate by scattering to far below that layer.
Interestingly, for the integranule the divergence of the
<IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients arises in the innermost layers. This happens
because the photon losses occur mainly through the line wings of
the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line. As follows from
Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:NLTE_height">A.2</a> the line wings in integranules are
formed considerably deeper than in granules. Such a difference in
the formation heights is a result of the Doppler shift of the line
opacity coefficient caused by the velocity field.  As a
consequence, in the intergranular model (see
Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>a, b) the divergence starts already in the
lower photosphere while in the granular model it happens only in
upper photosphere at heights around 400&nbsp;km.

</p>
<p>Another important conclusion that follows from
Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:NLTE_height">A.2</a> concerns the height of formation of the
Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line. The lower departure coefficient <IMG SRC="img102.png" ALT="$\beta_l$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="13">is close to unity. So the scaling of the line opacity with this
coefficient cannot lead to an appreciable difference between the
NLTE and LTE heights of formation of this line.

</p>
<p>The excess of Ba&nbsp; <small>II</small>  ions at the levels with excitation
potentials  above 5&nbsp;eV  visible in the granule model is produced
by the pumping via the  ultraviolet Ba&nbsp; <small>II</small>   lines starting at

<!-- MATH: $6s~ \rm {{^2}S}$ -->
<IMG SRC="img103.png" ALT="$6s~ \rm {{^2}S}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="34">,

<!-- MATH: $5d~ \rm {{^2}D}$ -->
<IMG SRC="img104.png" ALT="$5d~ \rm {{^2}D}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="30" WIDTH="36">,

<!-- MATH: $6p~ \rm {{^2}P}^o$ -->
<IMG SRC="img105.png" ALT="$6p~ \rm {{^2}P}^o$" ALIGN="MIDDLE" BORDER="0" HEIGHT="31" WIDTH="41">
levels.
For the intergranule model the overpopulation arises only in the
uppermost layers. Such behaviour of the <IMG SRC="img19.png" ALT="$\beta $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">-coefficients
corresponds to the temperature stratification of the models. The
overpopulation of the high excitation levels of Ba&nbsp; <small>II</small> in granules
occurs because here the excitation temperature of the ultraviolet
pumping radiation field appreciably exceeds the electron
temperature. In integranules such superthermal radiation, and
hence  the level overpopulation,  is present  only above the
temperature minimum region. In addition, in the intergranular
photospheric layers the photon losses in the ultraviolet lines are
more pronounced than in granules.

</p>
<p></p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="fig:stot"></A><!-- end Label--><A NAME="1056"></A><A NAME="figure923" HREF="img106.png"><IMG SRC="Timg106.png" ALT="\begin{figure}
\par\includegraphics[width=9cm]{12048figA3.eps}
\end{figure}" HEIGHT="122" WIDTH="87"><!-- HTML Figure number: 17 -->
</td>
<td class="img-txt"><span class="bold">Figure A.3:</span><p>
The total source function 
<!-- MATH: $S_{\rm tot}(\Delta \lambda)/B$ -->
      <IMG SRC="img23.png" ALT="$S_{\rm tot}(\Delta \lambda )/B$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="65">
measured in the Planck function units   for three wavelength
points 
<!-- MATH: $\Delta \lambda=-76.5; {-}42.5; {-}8.5$ -->
      <IMG SRC="img24.png" ALT="$\Delta \lambda =-76.5; {-}42.5; {-}8.5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="141">&nbsp;m&#xc5; situated  in the
blue wing of the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line profile. <i> Left</i>:
Maps of the 
<!-- MATH: $S_{\rm tot}(\Delta\lambda)/B$ -->
      <IMG SRC="img23.png" ALT="$S_{\rm tot}(\Delta \lambda )/B$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="65">
at the mean intensity
formation heights at these wavelengths. Horizontal line
corresponds to the slice <i>y</i><sub><i>i</i></sub>=0.6&nbsp;Mm and the filled circles to
the surface positions <i>x</i><sub><i>i</i></sub>=1.8&nbsp;Mm (granule), <i>x</i><sub><i>i</i></sub>=2.4&nbsp;Mm
(intergranule). <i> Right</i>: scatter plots of 
<!-- MATH: $S_{\rm tot}(\Delta
\lambda)/B$ -->
      <IMG SRC="img23.png" ALT="$S_{\rm tot}(\Delta \lambda )/B$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="65">
and velocities <i>V</i><sub><i>z</i></sub> for the same wavelengths. The
velocities are  taken at heights of formation of continuum
intensity. 
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<p>The 
<!-- MATH: ${\beta}_u / {\beta}_l$ -->
<IMG SRC="img107.png" ALT="${\beta}_u / {\beta}_l$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="33">
ratio of the upper and lower level
departure coefficients of  the Ba&nbsp; <small>II</small> 4554&nbsp;&#xc5; line sets the
departure  of its line source function <i>S</i><sub><i>L</i></sub> from the Planck
function&nbsp;<i>B</i>. Figure&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>&nbsp;(c, d) shows   that this
departure (reflecting the corresponding departure coefficient
divergence in the upper panels of this figure) is larger in the
intergranular than in the granular model.

</p>
<p>Figure&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:stot">A.3</a> demonstrates that such  behaviour is typical
also for the total source function 
<!-- MATH: $S_{\rm {\rm tot}}$ -->
<IMG SRC="img108.png" ALT="$S_{\rm {\rm tot}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="23">
at the
wavelengths corresponding to the inner wings (
<!-- MATH: $\Delta \lambda  <
76.5$ -->
<IMG SRC="img109.png" ALT="$\Delta \lambda <
76.5$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="61">&nbsp;m&#xc5;). On average, in intergranular regions it drops below
the Planck function while in granules the effect is less
pronounced. Moreover, in granular areas with strong upflows the
total source function can exceed the Planck function. This excess
can be understood if one takes into account that the resonance
source line function is described by the two-level approximation,
i.e. it approximately equals mean intensity&nbsp;<i>J</i>. In the regions
with small photon losses (like granules) the <i>J</i> &gt; <i>B</i>, hence,
<IMG SRC="img110.png" ALT="$S_{\rm L}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="18">
and 
<!-- MATH: $S_{\rm {\rm tot}}$ -->
<IMG SRC="img108.png" ALT="$S_{\rm {\rm tot}}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="23">
have to be greater than&nbsp;<i>B</i> as well.

</p>
<p>Figure&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>e, f show the NLTE and LTE disc-centre
line profiles for the individual  granular and integranular
models. The profiles displayed in Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>g result
from averaging of the emergent intensities corresponding only to
the granular models. Averaged intergranular profiles are shown in
Fig.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>h. These two bottom panels quantify the
statistical effect produced by the deviation from the LTE in two such
types of the atmospheric models. The main conclusions that may
be drawn from the results presented in
Figs.&nbsp;<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:beta">A.1</a>-<a href="/articles/aa/full_html/2009/42/aa12048-09/aa12048-09.html#fig:stot">A.3</a> are the
following:
</p>
<UL>

  <LI>the source function deficit, as compared to the LTE assumption,
is the main mechanism that controls the formation of  the Ba&nbsp;
    <small>II</small> <IMG SRC="img1.png" ALT="$\lambda $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">4554&nbsp;&#xc5; line. The line opacity deficit is small
and, hence, unimportant;
  </LI>
  <LI>
the divergence between 
<!-- MATH: $S_{\rm tot}$ -->
    <IMG SRC="img111.png" ALT="$S_{\rm tot}$" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="24">
and&nbsp;<i>B</i> changes the shape of
the individual profiles, particularly  the intergranular ones.
  </LI>
  <LI>On average, the deviations from the LTE lead to deepening (i.e.
strengthening) of  the spatially averaged Ba&nbsp; <small>II</small> <IMG SRC="img1.png" ALT="$\lambda $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">4554&nbsp;&#xc5;
line profiles. The NLTE effects are most pronounced around the line
core and are generally more important in the intergranular regions than
in the granular ones. The mean difference between the NLTE and LTE line
core residual intensities does not exceed&nbsp;5% for granules
and&nbsp;10% for intergranules.
</LI>
  <LI>Towards the wings, the LTE becomes a valid description for the
Ba&nbsp; <small>II</small> <IMG SRC="img1.png" ALT="$\lambda $" ALIGN="MIDDLE" BORDER="0" HEIGHT="26" WIDTH="11">4554&nbsp;&#xc5;   line profile;
  </LI>
</UL>

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