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 (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

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

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 (Kim et al. 1997). Analyses of the HST/COS dataset of low-redshift quasars gives a median $b$ parameter of around $33\ \mathrm {km\ s} ^{-1}$ (Danforth et al. 2016, Gaikwad et al. 2017).

See also

• Doppler broadening
• Doppler spectroscopy

References

• Danforth, Charles W.; Keeney, Brian A.; Tilton, Evan M.; Shull, Michael (2016). "An HST/COS Survey of the Low-redshift Intergalactic Medium. I. Survey, Methodology, and Overall Results". Astrophysical Journal. 817 (2): 111. arXiv:1402.2655. Bibcode:2016ApJ...817..111D. doi:10.3847/0004-637X/817/2/111. S2CID 118631947.

• Draine, Bruce (2011). Physics of the Interstellar and Intergalactic Medium. Princeton University Press. Bibcode:2011piim.book.....D. ISBN 9780691122144.

• Gaikwad, Prakash; Srianand, Raghunathan; Choudhury, Tirthankar Roy; Khaire, Vikram (2017). "VoIgt profile Parameter Estimation Routine (viper): H I photoionization rate at z \< 0.5". MNRAS. 467 (3): 3172–3187. arXiv:1610.06572. Bibcode:2017MNRAS.467.3172G. doi:10.1093/mnras/stx248.

• Kim, Tae-Sun; Hu, Esther M.; Cowie, Lennox L.; Songaila, Antoinette (1997). "The Redshift Evolution of the Ly alpha Forest". Astronomical Journal. 114: 1–13. arXiv:astro-ph/9704184. Bibcode:1997AJ....114....1K. doi:10.1086/118446. S2CID 119387231.