Doppler parameter

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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} σ

,

where $σ$ is the one-dimensional velocity dispersion Template:Harv. 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 π}} \frac{1}{σ} e^{- (v - v_{0})^{2} / 2 σ^{2}} = \frac{1}{\sqrt{π}} \frac{1}{b} e^{- (v - v_{0})^{2} / b^{2}}

,

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

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

F W H M = 2 \sqrt{2 \ln 2} σ = 2 \sqrt{\ln 2} b \approx 1.665 b

.

Distribution

The Doppler parameters of Lyman-alpha forest absorption lines are in the range 10–100 km s⁻¹, with a median value around $b_{m} = 36 {k m s}^{- 1}$ that decrease with redshift Template:Harv. Analyses of the HST/COS dataset of low-redshift quasars gives a median $b$ parameter of around $33 {k m s}^{- 1}$ (Template:Harvnb, Template:Harvnb).