<!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>