# Laboratory Hydrogen-Beta Emission Spectroscopy for Analysis of Astrophysical White Dwarf Spectra

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{⊙}, and radius, R

_{⊙}, in solar units,

## 2. Results

#### 2.1. Laboratory Experiments

#### 2.2. Astrophysical White Dwarf Spectra

## 3. Discussion

## 4. Materials and Methods

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

c | speed of light |

CMa | Canis Majoris—big dog |

CMi | Canis Minoris—little dog |

$\Delta {{\mathrm{w}}_{\mathrm{H}}}_{{\scriptscriptstyle \mathsf{\alpha}}}$ | hydrogen alpha FWHM |

$\Delta {{\lambda}_{\mathrm{H}}}_{{\scriptscriptstyle \mathsf{\alpha}}}$ | hydrogen alpha redshift |

$\Delta {{\mathrm{w}}_{\mathrm{H}}}_{{\scriptscriptstyle \mathsf{\beta}}}$ | hydrogen beta FWHM |

$\Delta {\mathsf{\lambda}}_{\mathrm{ps}}$ | hydrogen beta peak separation |

$\Delta {\mathsf{\lambda}}_{\mathrm{ds}}$ | hydrogen beta central dip-shift |

DA | dwarf A—hydrogen lines are present |

DAH | dwarf A with magnetic field |

DQZ | dwarf Q and Z—carbon and metal rich atmosphere |

FWHM | full width at half maximum |

G | gravitational constant |

H${}_{\mathsf{\beta}}$ | hydrogen beta line |

H${}_{\mathsf{\alpha}}$ | hydrogen alpha line |

HIRES | high resolution spectrometer |

ICCD | intensified charge coupled device |

JA | Jarrel-Ash |

JY | Jobin Yvon |

KOA | Keck observatory archive |

${\mathsf{\lambda}}_{\mathsf{\alpha}}$ | wavelength of H${}_{\mathsf{\alpha}}$ |

${\mathsf{\lambda}}_{\mathsf{\beta}}$ | wavelength of H${}_{\mathsf{\beta}}$ |

M | WD mass |

M_{⊙} | WD mass in solar units |

N${}_{\mathrm{H}\phantom{\rule{4pt}{0ex}}\mathrm{I}}$ | electron interstellar density of the hydrogen column |

N${}_{\mathrm{e}}$ | electron density |

R | WD radius |

R_{⊙} | WD radius in solar units |

${\mathsf{\sigma}}_{\mathsf{\alpha}}$ | blue- and red-peak Zeeman separation for H${}_{\mathsf{\alpha}}$ |

${\mathsf{\sigma}}_{\mathsf{\beta}}$ | blue- and red-peak Zeeman separation for H${}_{\mathsf{\beta}}$ |

SDSS | Sloan digital sky survey |

$\mathsf{\tau}$ | time delay from optical breakdown |

v${}_{\mathrm{g}}$ | gravitational redshift |

WD | white dwarf |

$\chi \left(\mathsf{\beta}\right)$ | integral describing predominant Holtsmark contribution |

$\Lambda \left(\mathsf{\beta}\right)$ | integral describing predominant Lorentz contribution |

$\Delta \left(\mathsf{\beta}\right)$ | integral describing predominant Doppler contribution |

$\mathsf{\beta}$ | ration of electric and normal field strength |

F | electric field strength |

F${}_{0}$ | normal field strength |

CGP | Christian Gerhard Parigger |

CMH | Christopher Matthew Helstern |

KAD | Kyle Anthony Drake |

GG | Ghaneshwar Gautam |

## Appendix A. Formulae for Determination of Electron Density from H_{β} and H_{α} Profiles

**Table A1.**Average H${}_{\mathsf{\beta}}$ central dip-shifts, widths, and electron densities, N${}_{\mathrm{e}}$, from dip-shifts and widths for selected time delays of optical breakdown in 0.76 × 10${}^{5}$ Pa hydrogen gas [22].

Time Delay (ns) | H${}_{\mathsf{\beta}}$ Dip-Shift (nm) | H${}_{\mathsf{\beta}}$ Width (nm) | N${}_{\mathbf{e}}$ from Dip-Shift (10${}^{17}$ cm${}^{-3}$) | N${}_{\mathbf{e}}$ from Width (10${}^{17}$ cm${}^{-3}$) |
---|---|---|---|---|

25 | 1.02 ± 0.15 | - | 20 (15–24) | - (H${}_{\mathsf{\alpha}}$: 17) |

50 | 0.83 ± 0.15 | - | 14 (11–19) | - (H${}_{\mathsf{\alpha}}$: 14) |

75 | 0.77 ± 0.1 | ∼25 | 13 (10–15) | ∼11 |

100 | 0.65 ± 0.1 | ∼23 ± 4 | 10 (7.7–12) | ∼10 |

125 | 0.58 ± 0.1 | ∼21 ± 4 | 8.4 (6.3–11) | ∼8.8 |

150 | 0.50 ± 0.1 | ∼15 ± 3 | 6.7 (4.8–8.9) | ∼5.5 |

175 | 0.42 ± 0.1 | 11 ± 2 | 5.2 (3.5–7.2) | 3.5 |

200 | 0.37 ± 0.1 | 9 ± 1 | 4.3 (2.7–6.1) | 2.7 |

225 | 0.32 ± 0.1 | 8 ± 0.5 | 3.5 (2.0–5.2) | 2.3 |

250 | 0.26 ± 0.05 | 7.5 ± 0.5 | 2.5 (1.8–3.3) | 2.1 |

275 | 0.24 ± 0.05 | 7.0 ± 0.5 | 2.2 (1.6–3.0) | 1.9 |

## Appendix B. Typical Laboratory Spectra of H_{β} and H_{α}

**Figure A1.**H${}_{\mathsf{\beta}}$ spectra for $\mathsf{\tau}$ = 250 ns: (

**a**) H${}_{\mathsf{\beta}}$ map; and (

**b**) scaled average.

**Figure A2.**H${}_{\mathsf{\alpha}}$ spectra for $\mathsf{\tau}$ = 250 ns: (

**a**) H${}_{\mathsf{\alpha}}$ map; and (

**b**) scaled average.

## Appendix C. Line Shapes

**Figure A3.**Comparison of measured data at a time delay of 175 ns, one-dimensional Holtsmark, Lorentz, and Doppler line shapes with the same full-width-half-maxima: (

**a**) H${}_{\mathsf{\beta}}$; and (

**b**) H${}_{\mathsf{\alpha}}$.

**Table A2.**Characteristic functions, $\mathsf{\phi}\left(\mathrm{x}\right)$ for one-dimensional Holtsmark, Lorentz, and Doppler profiles.

Holtsmark | Lorentz | Doppler |
---|---|---|

$exp\{\mathrm{it}{\mathsf{\mu}}_{\mathrm{h}}-|{\mathrm{c}}_{\mathrm{h}}\mathrm{t}{|}^{3/2}\}$ | $exp\{\mathrm{it}{\mathsf{\mu}}_{\mathrm{l}}-{\mathrm{c}}_{\mathrm{l}}\mathrm{t}\}$ | $exp\{\mathrm{it}{\mathsf{\mu}}_{\mathrm{d}}-{({\mathrm{c}}_{\mathrm{d}}\mathrm{t})}^{2}\}$ |

^{®}Characteristic Functions Toolbox [73]. In other words, the spectral line is evaluated as the real part of the Fourier transform as function of x,

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**Figure 1.**H${}_{\mathsf{\beta}}$ spectra for $\mathsf{\tau}$ = 400 ns and inferred electron densities of 2.3 × 10${}^{17}$ cm${}^{-3}$: (

**a**) line-of-sight at the vertical center of the 4-mm plasma; and (

**b**) Abel-inverted spectrum 0.5 mm from the horizontal center [23].

**Figure 2.**H${}_{\mathsf{\beta}}$ line-of-sight spectra: (

**a**) $\mathsf{\tau}$ = 650 ns, N${}_{\mathrm{e}}$ = 1.4 × 10${}^{17}$ cm${}^{-3}$, H${}_{2}$ gas pressure of $1.06\phantom{\rule{4pt}{0ex}}\times \phantom{\rule{4pt}{0ex}}{10}^{5}\phantom{\rule{4pt}{0ex}}\mathrm{Pa}$ [23]; and (

**b**) $\mathsf{\tau}$ = 100 ns, spectral resolving power 24,000 or resolution of 0.02 nm, N${}_{\mathrm{e}}$ = 8.6 × 10${}^{17}$ cm${}^{-3}$, H${}_{2}$ gas pressure of 1.32 × 10${}^{5}$ Pa [17].

**Figure 3.**Sirius B and Procyon white dwarf spectra [5]: (

**a**) Sirius B at 26 kK; and (

**b**) Procyon B at 8 kK.

**Figure 5.**Magnetic white dwarf HS 1031 + 0343 spectra [56]: (

**a**) H${}_{\mathsf{\beta}}$; and (

**b**) H${}_{\mathsf{\alpha}}$.

**Figure 6.**Hyades cluster white dwarf HG 7-85, recorded with a resolving-power 40,000 Echelle-spectrometer: (

**a**) H${}_{\mathsf{\beta}}$ expanded region; and (

**b**) H${}_{\mathsf{\beta}}$ center portion.

**Table 1.**Sirius B parameters [45].

Temperature [K] | Gravity [cm/s${}^{2}$] | Mass${}_{\mathbf{Sirius}\phantom{\rule{4pt}{0ex}}\mathbf{B}}$/Mass${}_{\mathbf{sun}}$ | Radius${}_{\mathbf{Sirius}\phantom{\rule{4pt}{0ex}}\mathbf{B}}$/Radius${}_{\mathbf{sun}}$ |
---|---|---|---|

24,790 ± 100 | $log\left(\mathrm{g}\right)=8.57\pm 0.06$ | 0.984 ± 0.074 | 0.0084 ± 0.00025 |

White Dwarf Star | Mass/Mass${}_{\mathbf{sun}}$ | Radius/Radius${}_{\mathbf{sun}}$ | M/R | ${\mathbf{v}}_{\mathbf{g}}$[km/s] |
---|---|---|---|---|

Sirius B | 1.03 ± 0.015 | 0.0111 ± 0.0007 | 92.79 | 59.02 |

Procyon B | 0.594 ± 0.012 | 0.0096 ± 0.0005 | 61.88 | 39.35 |

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**MDPI and ACS Style**

Parigger, C.G.; Drake, K.A.; Helstern, C.M.; Gautam, G. Laboratory Hydrogen-Beta Emission Spectroscopy for Analysis of Astrophysical White Dwarf Spectra. *Atoms* **2018**, *6*, 36.
https://doi.org/10.3390/atoms6030036

**AMA Style**

Parigger CG, Drake KA, Helstern CM, Gautam G. Laboratory Hydrogen-Beta Emission Spectroscopy for Analysis of Astrophysical White Dwarf Spectra. *Atoms*. 2018; 6(3):36.
https://doi.org/10.3390/atoms6030036

**Chicago/Turabian Style**

Parigger, Christian G., Kyle A. Drake, Christopher M. Helstern, and Ghaneshwar Gautam. 2018. "Laboratory Hydrogen-Beta Emission Spectroscopy for Analysis of Astrophysical White Dwarf Spectra" *Atoms* 6, no. 3: 36.
https://doi.org/10.3390/atoms6030036