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Doppler parameter

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

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

where σ {\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

p = 1 2 π 1 σ e − ( v − v 0 ) 2 / 2 σ 2 = 1 π 1 b e − ( v − v 0 ) 2 / b 2 {\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 p d v {\displaystyle p\mathrm {d} v} is the probability of the velocity along the line of sight being in the interval [ v , v + d v ] {\displaystyle } .

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

F W H M = 2 2 ln ⁡ 2 σ = 2 ln ⁡ 2 b ≈ 1.665 b {\displaystyle \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, with a median value around b m = 36   k m   s − 1 {\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 b {\displaystyle b} parameter of around 33   k m   s − 1 {\displaystyle 33\ \mathrm {km\ s} ^{-1}} (Danforth et al. 2016, Gaikwad et al. 2017).

See also

References

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