Sophie

Sophie

distrib > Mageia > 5 > i586 > by-pkgid > 37ce2601040f8edc2329d4714238376a > files > 3870

eso-midas-doc-13SEPpl1.2-3.mga5.i586.rpm

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Broadening Function</TITLE>
<META NAME="description" CONTENT="Broadening Function">
<META NAME="keywords" CONTENT="vol2">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="vol2.css">
<LINK REL="previous" HREF="node181.html">
<LINK REL="up" HREF="node180.html">
<LINK REL="next" HREF="node183.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html3703"
 HREF="node183.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="icons.gif/next_motif.gif"></A> 
<A NAME="tex2html3700"
 HREF="node180.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="icons.gif/up_motif.gif"></A> 
<A NAME="tex2html3696"
 HREF="node181.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="icons.gif/previous_motif.gif"></A> 
<A NAME="tex2html3702"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="icons.gif/contents_motif.gif"></A>  
<BR>
<B> Next:</B> <A NAME="tex2html3704"
 HREF="node183.html">Summary of the parameters</A>
<B> Up:</B> <A NAME="tex2html3701"
 HREF="node180.html">Basic Equations</A>
<B> Previous:</B> <A NAME="tex2html3697"
 HREF="node181.html">Optical Depth</A>
<BR>
<BR>
<!--End of Navigation Panel-->

<H3><A NAME="SECTION001321200000000000000">
Broadening Function</A>
</H3>
Broadening is due both to the natural width of the transition and to the velocity
spread of the absorbing atoms along the line of sight.
<UL>
<LI>In the ideal case of atoms at rest.
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\phi_{\lambda}(v = o) = \frac {1}{\pi}\ \  \frac {\delta_{\lambda}(\lambda)}
{[\delta k (\lambda)]^{2} + (\lambda - \lambda_{lk})^{2}}
\end{displaymath} -->


<IMG
 WIDTH="350" HEIGHT="52"
 SRC="img349.gif"
 ALT="\begin{displaymath}\phi_{\lambda}(v = o) = \frac {1}{\pi}\ \ \frac {\delta_{\lam...
...da)}
{[\delta k (\lambda)]^{2} + (\lambda - \lambda_{lk})^{2}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P> 

<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\delta k(\lambda) = \frac {\lambda^{2}}{4 \pi c} \ \sum_{E_{r}<E_{k}} a_{kr}
\end{displaymath} -->


<IMG
 WIDTH="216" HEIGHT="72"
 SRC="img350.gif"
 ALT="\begin{displaymath}\delta k(\lambda) = \frac {\lambda^{2}}{4 \pi c} \ \sum_{E_{r}<E_{k}} a_{kr}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\hspace{8mm}      = \frac {\lambda^{2}}{4 \pi c} \ \  SAKL
\end{displaymath} -->


<IMG
 WIDTH="140" HEIGHT="48"
 SRC="img351.gif"
 ALT="\begin{displaymath}\hspace{8mm} = \frac {\lambda^{2}}{4 \pi c} \ \ SAKL
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<LI>Let <IMG
 WIDTH="26" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img352.gif"
 ALT="$\psi(v)dv$">
be the normalized distribution of atoms with 
velocity between <I>v</I> and <BR> <I>v</I> + d<I>v</I>, then:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\phi_{\lambda} = \frac {1}{\pi}\  \int_{-\infty}^{+\infty} \ \frac {\delta_{k}(\lambda)}
{[\delta_{k}(\lambda)]^{2} + [\lambda - \lambda_{lk}(1 + \frac {v}{c})]^{2}}
\ \ \psi(v)dv
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="484" HEIGHT="54"
 SRC="img353.gif"
 ALT="\begin{displaymath}\phi_{\lambda} = \frac {1}{\pi}\ \int_{-\infty}^{+\infty} \ \...
... [\lambda - \lambda_{lk}(1 + \frac {v}{c})]^{2}}
\ \ \psi(v)dv
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(8.1)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P> 

<P></P>
<LI>In the program <U>the velocity is assumed to be gaussian</U>, thus:
</UL> 
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\psi (v) = \frac {1}{\sqrt \pi} \ \frac {1}{b} \ \exp \ \left[- \ \left(
\frac{v - v_{o}}{b} \right) \right]^{2}

\end{displaymath} -->


<IMG
 WIDTH="334" HEIGHT="55"
 SRC="img355.gif"
 ALT="\begin{displaymath}\psi (v) = \frac {1}{\sqrt \pi} \ \frac {1}{b} \ \exp \ \left[- \ \left(
\frac{v - v_{o}}{b} \right) \right]^{2}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
b = \ \sqrt \frac {2kT}{m}
\end{displaymath} -->


<IMG
 WIDTH="109" HEIGHT="70"
 SRC="img356.gif"
 ALT="\begin{displaymath}b = \ \sqrt \frac {2kT}{m}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P> <I>v</I><SUB><I>o</I></SUB> = velocity of the cloud relative to the observer.
<BR>

<P>
This full expression (1) is denoted as a ``Maxwell + damping wing'' 
or ``Voigtian'' profile in the program. 

<P>
In the case of low column density (<IMG
 WIDTH="66" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="img357.gif"
 ALT="$\tau <$">
1)

<P>
 <IMG
 WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
 SRC="img358.gif"
 ALT="$\tau$">
can be approximated to:

<P>
 <IMG
 WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
 SRC="img359.gif"
 ALT="$\tau$">
= NS 
<!-- MATH: $\phi_{\lambda}$ -->
<IMG
 WIDTH="54" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="img360.gif"
 ALT="$\phi_{\lambda}$">
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\phi_{\lambda} = \ \frac {\lambda_{lk}}{\sqrt\pi b} e^{{-(w/b)}^{2}}
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
 WIDTH="176" HEIGHT="53"
 SRC="img361.gif"
 ALT="\begin{displaymath}\phi_{\lambda} = \ \frac {\lambda_{lk}}{\sqrt\pi b} e^{{-(w/b)}^{2}}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(8.2)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P> 

<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\frac {w}{c} = \frac {\nu - \nu_{lk}}{\nu_{lk}}
\end{displaymath} -->


<IMG
 WIDTH="116" HEIGHT="55"
 SRC="img362.gif"
 ALT="\begin{displaymath}\frac {w}{c} = \frac {\nu - \nu_{lk}}{\nu_{lk}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
S = \frac {\pi e^{2}}{m_{e}c} \ f_{lk}
\end{displaymath} -->


<IMG
 WIDTH="118" HEIGHT="52"
 SRC="img363.gif"
 ALT="\begin{displaymath}S = \frac {\pi e^{2}}{m_{e}c} \ f_{lk}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P></P>
This simplified expression (2) is denoted as a ``Maxwellian'' profile in the
program. <BR>

<P></P>
<EM>Finally if the line of sight crosses N clouds then, the 
resulting optical depth is:</EM>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
\tau = \sum_{i = 1}^{N} \ \tau_{i}
\end{displaymath} -->


<IMG
 WIDTH="95" HEIGHT="74"
 SRC="img364.gif"
 ALT="\begin{displaymath}\tau = \sum_{i = 1}^{N} \ \tau_{i}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
<EM>In cases where the source has a (cosmological) velocity.</EM>  

<P>
Let z be the redshift of the source.

<P>
An absorption is measured in the spectrum at <IMG
 WIDTH="42" HEIGHT="56" ALIGN="BOTTOM" BORDER="0"
 SRC="img365.gif"
 ALT="$\lambda$">
= 
<!-- MATH: $\lambda_{a}$ -->
<IMG
 WIDTH="52" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="img366.gif"
 ALT="$\lambda_{a}$">corresponding to a rest wavelength 
<!-- MATH: $\lambda_{o}$ -->
<IMG
 WIDTH="52" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="img367.gif"
 ALT="$\lambda_{o}$">.

<P>
This yields for the redshift of the cloud:

<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
Za = \frac {\lambda_{a}}{\lambda_{o}} \ -1
\end{displaymath} -->


<IMG
 WIDTH="127" HEIGHT="57"
 SRC="img368.gif"
 ALT="\begin{displaymath}Za = \frac {\lambda_{a}}{\lambda_{o}} \ -1
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The velocity of the cloud relative to the source is:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
v_{rel} = \ c \ \frac {R^{2} -1}{R^{2} +1}
\end{displaymath} -->


<IMG
 WIDTH="151" HEIGHT="51"
 SRC="img369.gif"
 ALT="\begin{displaymath}v_{rel} = \ c \ \frac {R^{2} -1}{R^{2} +1}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
R = \frac {1 + Z}{1 + Z_{a}}
\end{displaymath} -->


<IMG
 WIDTH="110" HEIGHT="57"
 SRC="img370.gif"
 ALT="\begin{displaymath}R = \frac {1 + Z}{1 + Z_{a}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
In practice the program computes the absorption profile in the cloud reference
frame
<BR>
 (<I>v</I><SUB><I>o</I></SUB> = 0) and shifts the result into the observer's rest frame

<!-- MATH: $\left[ \lambda \rightarrow \frac {\lambda}{1 + Za} \right]$ -->
<IMG
 WIDTH="75" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="img371.gif"
 ALT="$\left[ \lambda \rightarrow \frac {\lambda}{1 + Za} \right]$">

<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html3703"
 HREF="node183.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="icons.gif/next_motif.gif"></A> 
<A NAME="tex2html3700"
 HREF="node180.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="icons.gif/up_motif.gif"></A> 
<A NAME="tex2html3696"
 HREF="node181.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="icons.gif/previous_motif.gif"></A> 
<A NAME="tex2html3702"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="icons.gif/contents_motif.gif"></A>  
<BR>
<B> Next:</B> <A NAME="tex2html3704"
 HREF="node183.html">Summary of the parameters</A>
<B> Up:</B> <A NAME="tex2html3701"
 HREF="node180.html">Basic Equations</A>
<B> Previous:</B> <A NAME="tex2html3697"
 HREF="node181.html">Optical Depth</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>

Perl :: Catalyst :: PostgreSQL :: RPM Valid HTML 4.01 Transitional