Line Profiles

If the line width is greater than the spectral resolution one can discuss the variation of the intensity of the line as a function of frequency. This description, called the line profile, can be denoted by $\phi(\nu)$. If the reason for the line width is thermal broadening or turbulent broadening, the line profile will have a gaussian profile such that $\phi(\nu) \propto e^{-(\nu-\nu_{l})^{2}/(\delta\nu)^{2}}$, where $\nu_{l}$ is the frequency at the line center and $\delta\nu$ is the rms value of the gaussian. The width of the line refers to the full-width at half-maximum and is equal to $\sim$ 2.35 $\delta\nu$. The observed width of the line ($\delta\nu_{o}$) and the true width of the line ($\delta\nu_{l}$) are related by $\delta\nu^{2}_{o} = \delta\nu_{l}^{2} + \delta\nu_{r}^{2}$, where, $\delta\nu_{r}$ is the width of each channel (spectral resolution). This simple relation is strictly true only when the spectral channels have a gaussian response. In addition, this is relevant if the widths of the spectral line and the spectral channel are comparable.

Pressure broadened lines show Voigt profiles. This will have a Doppler (gaussian) profile in the center of the line whereas the wings are dominated by the Lorentz profile. Obviously an analysis of the line profile is crucial in understanding the physical conditions of the system producing the spectral line.

Acknowledgments: I would like to thank A.A. Deshpande for a critical reading of the manuscript and for useful comments to improve its clarity.