Doppler parameter

The Doppler parameter, or Doppler broadening parameter, usually denoted as $$b$$, is a parameter commonly used in astrophysics to characterize the width of observed spectral lines of astronomical objects. It is defined as
 * $$ b = \sqrt{2} \sigma $$,

where $$\sigma$$ is the one-dimensional velocity dispersion. Given this parameter, the velocity distribution of the line-emitting/absorbing atoms and ions proximated by a Gaussian can be rewritten as
 * $$ p = \frac{1}{\sqrt{2\pi}}\frac{1}{\sigma}e^{-(v-v_0)^2/2\sigma^2} = \frac{1}{\sqrt{\pi}}\frac{1}{b}e^{-(v-v_0)^2/b^2}$$,

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

The line width is also often specified in terms of the FWHM (full width at half maximum), which is
 * $$ \mathrm{FWHM} = 2\sqrt{2\ln 2}\sigma = 2\sqrt{\ln 2} b \approx 1.665b $$.

Distribution
The Doppler parameters of Lyman-alpha forest absorption lines are in the range 10–100 km s−1, with a median value around $$b_m = 36\ \mathrm{km\ s}^{-1}$$ that decrease with redshift. Analyses of the HST/COS dataset of low-redshift quasars gives a median $$b$$ parameter of around $$33\ \mathrm{km\ s}^{-1}$$.