Ly-\(\alpha\) Forest from Cholla Simulations

Single Absorber

The opacity \(\tau_{\nu}\) for cold interstellar gas can be approximated by:

\[\tau_{\nu}=\frac{\pi e^{2}}{m_{e} c} f_{\ell u} N_{\ell} \phi_{\nu}\]

The line profile function \(\phi_{\nu}\) is generally a Voigt profile, meaning that it has a Gaussian core and Lorentzian damping wings. If we focus on the Gaussian core, we can approximate \(\phi_{\nu}\) by a pure Gaussian,

\[\phi_{\nu}=\frac{1}{\sqrt{\pi} b} e^{-\left(1-\nu / \nu_{0}\right)^{2} /(b / c)^{2}} ; \quad b \equiv \sqrt{2} \sigma_{v}\]

where \(b\) is the Doppler broadening parameter. This has a maximum value of \(1/\sqrt{\pi b}\) at line center \(\nu=\nu_0\) , so the (maximum) line center optical depth is:

\[\tau_{0}=\sqrt{\pi} \frac{e^{2}}{m_{e} c} \frac{f_{\ell u} \lambda_{u \ell} N_{\ell}}{b}=0.758\left(\frac{N_{\ell}}{10^{13} \mathrm{cm}^{-2}}\right)\left(\frac{f_{\ell u}}{0.4162}\right)\left(\frac{\lambda_{u \ell}}{1215.7 \mathrm{A}}\right)\left(\frac{10 \mathrm{km} \mathrm{s}^{-1}}{b}\right)\]

where \(\lambda_{u \ell} = c/\nu_0\) , and the normalization quantities are those appropriate for the Lyman-\(\alpha\) line of hydrogen. The optical depth in the Gaussian part of the line profile is then

\[\tau_{\nu}=\tau_{0} e^{-u^{2} / b^{2}}\]

where \(u=c(\nu_0 - \nu)/\nu\) is the velocity shift required to produce a frequency shift \(\nu\).