Doppler parameter

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The Doppler parameter, or Doppler broadening parameter, usually denoted as ${\displaystyle b}$, is a parameter commonly used in astrophysics to characterize the width of observed spectral lines of astronomical objects. It is defined as

${\displaystyle b={\sqrt {2}}\sigma }$,

where ${\displaystyle \sigma }$ is the one-dimensional velocity dispersion (Draine 2011, p. 58). Given this parameter, the velocity distribution of the line-emitting/absorbing atoms and ions proximated by a Gaussian can be rewritten as

${\displaystyle p={\frac {1}{\sqrt {2\pi }}}{\frac {1}{\sigma }}e^{-(v-v_{0})^{2}/2\sigma ^{2}}={\frac {1}{\sqrt {\pi }}}{\frac {1}{b}}e^{-(v-v_{0})^{2}/b^{2}}}$,

where ${\displaystyle p\mathrm {d} v}$ is the probability of the velocity along the line of sight being in the interval ${\displaystyle [v,v+\mathrm {d} v]}$.

The line width is also often specified in terms of the FWHM (full width at half maximum), which is

${\displaystyle \mathrm {FWHM} =2{\sqrt {2\ln 2}}\sigma =2{\sqrt {\ln 2}}b\approx 1.665b}$.

The Doppler parameters of Lyman-alpha forest absorption lines are in the range 10–100 km s⁻¹, with a median value around ${\displaystyle b_{m}=36\ \mathrm {km\ s} ^{-1}}$ that decrease with redshift (Kim et al. 1997). Analyses of the HST/COS dataset of low-redshift quasars gives a median ${\displaystyle b}$ parameter of around ${\displaystyle 33\ \mathrm {km\ s} ^{-1}}$ (Danforth et al. 2016, Gaikwad et al. 2017).

• Doppler broadening
• Doppler spectroscopy