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.

• In the ideal case of atoms at rest.
  
  $$\phi_{\lambda}(v = o) = \frac {1}{\pi}\ \ \frac {\delta_{\lam... ...da)} {[\delta k (\lambda)]^{2} + (\lambda - \lambda_{lk})^{2}}$$
  
  
  
  
  $$\delta k(\lambda) = \frac {\lambda^{2}}{4 \pi c} \ \sum_{E_{r}<E_{k}} a_{kr}$$
  
  
  
  
  $$\hspace{8mm} = \frac {\lambda^{2}}{4 \pi c} \ \ SAKL$$
  
  

• Let $\psi(v)dv$ be the normalized distribution of atoms with velocity between v and
  v + dv, then:
  

  $$\phi_{\lambda} = \frac {1}{\pi}\ \int_{-\infty}^{+\infty} \ \... ... [\lambda - \lambda_{lk}(1 + \frac {v}{c})]^{2}} \ \ \psi(v)dv$$ (8.1)

  
  

• In the program the velocity is assumed to be gaussian, thus:

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

$$b = \ \sqrt \frac {2kT}{m}$$

v_o = 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 $\phi_{\lambda}$

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

$$\frac {w}{c} = \frac {\nu - \nu_{lk}}{\nu_{lk}}$$

$$S = \frac {\pi e^{2}}{m_{e}c} \ f_{lk}$$

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:

$$\tau = \sum_{i = 1}^{N} \ \tau_{i}$$

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:

$$Za = \frac {\lambda_{a}}{\lambda_{o}} \ -1$$

The velocity of the cloud relative to the source is:

$$v_{rel} = \ c \ \frac {R^{2} -1}{R^{2} +1}$$

$$R = \frac {1 + Z}{1 + Z_{a}}$$

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