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 Template:Harv. Given this parameter, the velocity distribution of the line-emitting/absorbing atoms and ions proximated by a Gaussian can be rewritten as

$p=\frac{1}{\sqrt{2\pi }}\frac{1}{\sigma }{e}^{-(v-v_0{)}^2/2{\sigma }^2}=\frac{1}{\sqrt{\pi }}\frac{1}{b}{e}^{-(v-v_0{)}^2/b^2}$,

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

$\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}=2\sqrt{2\ln 2}\sigma =2\sqrt{\ln 2}b\approx 1.665b$.

Distribution

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

See also

• Doppler broadening
• Doppler spectroscopy

References

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