next up previous contents
Next: Summary of the parameters Up: Basic Equations Previous: Optical Depth

Broadening Function

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.

\begin{displaymath}\psi (v) = \frac {1}{\sqrt \pi} \ \frac {1}{b} \ \exp \ \left[- \ \left(
\frac{v - v_{o}}{b} \right) \right]^{2}
\end{displaymath}


\begin{displaymath}b = \ \sqrt \frac {2kT}{m}
\end{displaymath}

vo = velocity of the cloud relative to the observer.

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

In the case of low column density ($\tau <$ 1)

$\tau$ can be approximated to:

$\tau$ = NS $\phi_{\lambda}$

\begin{displaymath}\phi_{\lambda} = \ \frac {\lambda_{lk}}{\sqrt\pi b} e^{{-(w/b)}^{2}}
\end{displaymath} (8.2)


\begin{displaymath}\frac {w}{c} = \frac {\nu - \nu_{lk}}{\nu_{lk}}
\end{displaymath}


\begin{displaymath}S = \frac {\pi e^{2}}{m_{e}c} \ f_{lk}
\end{displaymath}

This simplified expression (2) is denoted as a ``Maxwellian'' profile in the program.

Finally if the line of sight crosses N clouds then, the resulting optical depth is:

\begin{displaymath}\tau = \sum_{i = 1}^{N} \ \tau_{i}
\end{displaymath}

In cases where the source has a (cosmological) velocity.

Let z be the redshift of the source.

An absorption is measured in the spectrum at $\lambda$ = $\lambda_{a}$corresponding to a rest wavelength $\lambda_{o}$.

This yields for the redshift of the cloud:


\begin{displaymath}Za = \frac {\lambda_{a}}{\lambda_{o}} \ -1
\end{displaymath}

The velocity of the cloud relative to the source is:

\begin{displaymath}v_{rel} = \ c \ \frac {R^{2} -1}{R^{2} +1}
\end{displaymath}


\begin{displaymath}R = \frac {1 + Z}{1 + Z_{a}}
\end{displaymath}

In practice the program computes the absorption profile in the cloud reference frame
(vo = 0) and shifts the result into the observer's rest frame $\left[ \lambda \rightarrow \frac {\lambda}{1 + Za} \right]$


next up previous contents
Next: Summary of the parameters Up: Basic Equations Previous: Optical Depth
Petra Nass
1999-06-15