Line Broadening Mechanisms

A spectral line is the fingerprint an atom leaves on light, and no fingerprint is a single sharp point. The idealized transition between two levels would emit at one frequency $\nu_0=(E_2-E_1)/h$ , but every real line has a finite width and shape, encoded in a normalized profile $\phi(\nu)$ with $\int\phi(\nu)\,\mathrm{d}\nu=1$ . That profile is not a nuisance to deconvolve away — it is the message. Because $\alpha_\nu\propto\phi(\nu)$ , a line's shape turns a smear of starlight into a thermometer, a barometer, a tachometer, and a magnetometer at once. Four mechanisms dominate, each writing its own signature. Natural broadening is quantum: a finite-lifetime state has a fuzzy energy, and its decaying wave train Fourier-transforms into a Lorentzian. Doppler broadening is thermal: line-of-sight motions from the Maxwell–Boltzmann distribution smear the line into a Gaussian of width $\propto\sqrt{T/m}$ . Pressure broadening is collisional: encounters interrupt the radiating atom, giving a density-dependent Lorentzian. Rotational broadening is macroscopic: a spinning star's approaching and receding limbs convolve the line with a kernel of half-width $v\sin i$ . Because the intrinsic (Lorentzian) and thermal (Gaussian) contributions are statistically independent, the emergent atomic profile is their convolution — the Voigt function, Gaussian in the core and Lorentzian in the wings. That two-regime structure is what lets an observer pull temperature apart from pressure, turbulence apart from rotation, and abundance apart from gravity, all from a single fitted line.

Key takeaways

A spectral line is never infinitely sharp: quantum, thermal, collisional, and dynamical effects give it a finite width and shape — the profile function $\phi(\nu)$ , normalized to unit area.
Natural broadening from finite state lifetimes gives a Lorentzian of FWHM $\gamma/2\pi$ — for the classical oscillator a wavelength-independent $\approx1.2\times10^{-4}$ Å.
Doppler broadening from Maxwellian velocities gives a Gaussian core of FWHM $\propto\sqrt{T/m}$ ; solar Hα at 6000 K broadens to about 0.36 Å, roughly 3000 times the natural width.
Pressure broadening is also Lorentzian and scales with density, so the damping wings of strong lines measure photospheric pressure and hence surface gravity.
A Gaussian core convolved with Lorentzian wings is the Voigt profile; the wings win far from centre, falling as $\Delta\lambda^{-2}$ while the Gaussian dies as $e^{-x^2}$ .
Rotational broadening convolves the line with a semi-elliptical kernel of half-width $v\sin i$ , reading out projected stellar spin.
Key idea: each mechanism imprints a characteristic shape and width scaling — natural and pressure give Lorentzian damping wings, thermal motion gives a Gaussian core, their convolution is the Voigt profile — so a line's shape separates temperature, density, and dynamics.

The core insight

Each broadening mechanism imprints a characteristic profile shape and width scaling: natural and pressure broadening give Lorentzians (damping wings), thermal motion gives a Gaussian core, and their convolution is the Voigt profile — so the shape of a line separates temperature, density, and dynamics.

Where this lives today

Line broadening is the daily working tool of quantitative spectroscopy. Stellar-atmosphere codes (MARCS, ATLAS) and radiative-transfer suites (SME, MOOG, Turbospectrum) fit Voigt profiles to thousands of lines at once to extract temperature, surface gravity, microturbulence, and abundances; the Gaia-ESO, APOGEE, and GALAH surveys apply this to millions of stars to map the chemical history of the Milky Way. The pressure-broadened wings of the hydrogen Balmer lines remain the standard gravity (luminosity-class) diagnostic, while projected rotation $v\sin i$ feeds stellar angular-momentum studies and the exoplanet Rossiter–McLaughlin effect. In cosmology, the Lyman-α forest — intergalactic hydrogen clouds imprinted on quasar spectra — is fit cloud-by-cloud with Voigt profiles, the Doppler $b$ -parameter reading temperature-plus-turbulence and the damping wings of rare damped systems reading neutral-hydrogen column densities that trace the cosmic baryon budget. And a Zeeman-split broadened line becomes a stellar magnetometer. One profile function and a spectrograph: that is how a smear of starlight becomes a number.

The physical picture

Start with a single radiating atom. It does not emit forever: spontaneous decay makes the wave train decay as $e^{-\gamma t/2}$ , and a finite-duration oscillation cannot be perfectly monochromatic. Fourier analysis of that decaying cosine spreads the power into a bell-with-heavy-tails Lorentzian whose width is the inverse of how long the atom radiated undisturbed. Anything that shortens the coherence time — spontaneous decay (natural) or a phase-scrambling collision (pressure) — widens the line by the same mechanism and to the same shape. The line width is a clock reading in disguise. Now put many atoms into a warm gas. They share a Maxwellian distribution of line-of-sight velocities, each Doppler-shifting its emission by $\Delta\lambda/\lambda=v_r/c$ . Summing over that Gaussian velocity spread smears the line into a Gaussian of width $\propto\sqrt{T/m}$ — hotter gas and lighter atoms broaden more, which is why hydrogen lines are far fatter than iron lines in the same photosphere. The first figure holds the two origins side by side: a decaying wave train becoming a Lorentzian, a velocity spread becoming a Gaussian.

Figure

Two shapes two origins — a finite-lifetime wave train on the left Fourier-transforms into a Lorentzian of FWHM γ/2π ≈ 1.2×10⁻⁴ Å for Hα while the Maxwellian spread of thermal velocities on the right smears the line into a Gaussian of FWHM ≈ 0.36 Å at 6000 K, so the thermal core outweighs the natural width by a factor of about 3000

The Voigt profile marries these two. Near line centre the fast-falling Gaussian sets the shape — the thermal core — but far in the wings the Lorentzian, decaying only as $(\nu-\nu_0)^{-2}$ , always wins, since an exponential eventually loses to any power law. That is why weak lines look purely Gaussian while strong lines grow broad damping wings whose strength measures pressure. Rotation and turbulence sit on top, blending many Doppler-shifted local profiles across an unresolved disk — rotation convolving the line with a distinctive semi-elliptical kernel. The second figure makes the two-regime structure quantitative for a real line.

The core equations

Every broadening result is a statement about the profile function, so we begin there. A line profile is a probability density over frequency — non-negative, peaked at centre, normalized to unit area — so it redistributes the line's strength without changing it, making $\phi(\nu)$ the object separating shape from strength.

\phi(\nu)\ge 0,\qquad \int_0^\infty \phi(\nu)\,\mathrm{d}\nu = 1 \qquad (1)

where: $\phi(\nu)$ — line profile function, units of inverse frequency; $\nu$ — frequency; $\nu_0$ — line-centre frequency. Reading: the area is fixed, so a broader profile is shorter. Natural and pressure broadening produce the same functional form — a Lorentzian — because both arise from a wave train truncated in time. The damping constant $\gamma$ collects every rate that shortens the coherence, decay and collisions adding linearly into it.

\phi_L(\nu)=\frac{\gamma/4\pi^2}{(\nu-\nu_0)^2+(\gamma/4\pi)^2} \qquad (2)

where: $\gamma$ — damping constant (sum of decay and collision rates), s⁻¹; FWHM $=\gamma/2\pi$ . Reading: the wings fall only as $(\nu-\nu_0)^{-2}$ , so a Lorentzian always has more far-wing strength than an equal-width Gaussian. Doppler broadening comes from the thermal velocity distribution, not any single atom's radiation, so its shape is a Gaussian — the Maxwellian projected onto the line of sight and mapped into frequency. The Doppler width $\Delta\nu_D$ carries temperature and atomic mass.

\phi_D(\nu)=\frac{1}{\Delta\nu_D\sqrt{\pi}}\exp\!\left[-\frac{(\nu-\nu_0)^2}{\Delta\nu_D^2}\right],\qquad \Delta\nu_D=\frac{\nu_0}{c}\sqrt{\frac{2kT}{m}} \qquad (3)

where: $\Delta\nu_D$ — Doppler width, the $1/e$ half-width in frequency; $T$ — kinetic temperature; $m$ — atomic mass; $k$ — Boltzmann constant; $c$ — speed of light. Reading: $\Delta\nu_D\propto\sqrt{T/m}$ — hotter, lighter atoms give broader cores. The corresponding wavelength FWHM follows from the half-max condition and $\Delta\lambda/\lambda=\Delta\nu/\nu$ : $\Delta\lambda_{1/2}=(2\lambda_0/c)\sqrt{2kT\ln2/m}$ , with microturbulence entering as $2kT/m\to2kT/m+\xi^2$ — the mass dependence separates temperature from turbulence. Because the damping displacement and the thermal shift are independent contributions to the observed frequency, the emergent atomic profile is their convolution — the Voigt profile, with no closed form but universally tabulated.

\phi_V(\nu)=(\phi_L * \phi_D)(\nu)=\int_{-\infty}^{\infty}\phi_L(\nu-\nu')\,\phi_D(\nu')\,\mathrm{d}\nu' \qquad (4)

where: $*$ — convolution; $\phi_V$ — Voigt profile. Reading: Gaussian core plus Lorentzian wings — one profile carrying temperature and damping. In practice the Voigt profile is written through the dimensionless Voigt–Hjerting function $H(a,u)$ , packaging the shape into a damping parameter $a$ (the ratio of Lorentzian to Doppler width) and a scaled offset $u$ .

H(a,u)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-y^2}}{(u-y)^2+a^2}\,\mathrm{d}y,\qquad a=\frac{\gamma}{4\pi\Delta\nu_D},\quad u=\frac{\nu-\nu_0}{\Delta\nu_D} \qquad (5)

where: $a$ — damping parameter; $u$ — frequency offset in Doppler units; $\phi_V=H(a,u)/(\Delta\nu_D\sqrt\pi)$ . Reading: $a\ll1$ gives a near-pure Gaussian core with faint wings; $a\sim1$ means wings everywhere. Finally, the profile enters observable spectra through the line absorption coefficient, where broadening couples to radiative transfer. The classical constant $\pi e^2/m_ec$ times the oscillator strength sets the total strength; the profile distributes it over frequency.

\alpha_\nu = \frac{\pi e^2}{m_e c}\,f\,n_{\rm low}\,\phi(\nu)\left(1-e^{-h\nu/kT}\right) \qquad (6)

where: $f$ — oscillator strength; $n_{\rm low}$ — number density of lower-level absorbers; $e,\,m_e$ — electron charge and mass (Gaussian units); $(1-e^{-h\nu/kT})$ — stimulated-emission correction. Reading: $\alpha_\nu\propto\phi(\nu)$ , so measuring line shape measures the profile.

The core derivations

Derivation AThe Doppler width from the Maxwell-Boltzmann distribution

The goal is to build the Gaussian line profile and its width entirely from thermal motion, with no quantum input. Each atom emits at its rest frequency in its own frame, but the observer sees it Doppler-shifted by the line-of-sight velocity; averaging that sharp emission over the Maxwellian distribution produces a frequency Gaussian whose width fixes the $\sqrt{T/m}$ scaling. We take the one-dimensional velocity distribution, map velocity to frequency, change variables, and read off the width.

Step 1. Project the three-dimensional Maxwellian onto the line of sight to get $f(v_r)\,\mathrm{d}v_r=\sqrt{m/2\pi kT}\,e^{-mv_r^2/2kT}\,\mathrm{d}v_r$ , a Gaussian in $v_r$ with variance $\sigma_v^2=kT/m$ . Why: only the line-of-sight velocity component Doppler-shifts the observed frequency.

Step 2. Insert the non-relativistic Doppler relation $\nu-\nu_0=(\nu_0/c)\,v_r$ , so $v_r=c(\nu-\nu_0)/\nu_0$ and a receding atom is redshifted. Why: this bridges a distribution over velocity to one over frequency.

Step 3. Change variables with the Jacobian $|\mathrm{d}v_r/\mathrm{d}\nu|=c/\nu_0$ , giving $\phi(\nu)=\sqrt{m/2\pi kT}\,(c/\nu_0)\,\exp[-mc^2(\nu-\nu_0)^2/2kT\nu_0^2]$ . Why: probability is conserved, so the velocity Gaussian maps directly into a frequency Gaussian.

Step 4. Define $\Delta\nu_D\equiv(\nu_0/c)\sqrt{2kT/m}$ , so $\phi(\nu)=(\Delta\nu_D\sqrt\pi)^{-1}\exp[-(\nu-\nu_0)^2/\Delta\nu_D^2]$ ; note $\Delta\nu_D=\sqrt2\,\sigma_\nu$ is the $1/e$ half-width, not the standard deviation. Why: the $1/e$ width is the natural unit that keeps the exponent dimensionless.

Step 5. Convert to a wavelength FWHM: setting $\phi=\tfrac12\phi_{\max}$ gives $e^{-x^2}=\tfrac12$ , $x=\sqrt{\ln2}$ , so $\mathrm{FWHM}=2\sqrt{\ln2}\,\Delta\nu_D$ and $\Delta\lambda_{1/2}=(2\lambda_0/c)\sqrt{2kT\ln2/m}$ . Why: the half-max width is what an observer measures, and $2\sqrt{\ln2}\approx1.665$ bridges the $1/e$ width to the FWHM. *Note: microturbulence enters by replacing $2kT/m$ with $2kT/m+\xi^2$ ; solar Hα at 6000 K gives 0.36 Å.*

Go deeper · Derivation B · 1 minThe Lorentzian profile from a damped oscillator

Derivation BThe Lorentzian profile from a damped oscillator

The aim is to show that the finite lifetime of a radiating state forces a Lorentzian shape, and to pin its width to the damping rate $\gamma$ . We model the atom as a classical electron oscillating at $\omega_0$ but losing energy — to radiation (natural) or collisions (pressure) — so its amplitude decays as $e^{-\gamma t/2}$ ; the spectrum is the squared Fourier transform of that decaying wave train, a Lorentzian of FWHM $\gamma/2\pi$ . The same shape re-emerges from the energy–time uncertainty relation, which is why natural and pressure broadening share one functional form.

Step 1. Write the equation of motion of a damped bound electron, $\ddot x+\gamma\dot x+\omega_0^2 x=0$ , with $\omega_0=2\pi\nu_0$ and $\gamma\ll\omega_0$ . Why: any linear oscillator with a small energy-loss term reduces to this, independent of whether the loss is radiative or collisional.

Step 2. Solve the underdamped case as $x(t)=x_0\,e^{-\gamma t/2}\cos(\omega_0 t)$ , $t\ge0$ — a cosine in a slowly decaying envelope. Why: the amplitude decays at rate $\gamma/2$ , so intensity $\propto|x|^2$ decays at $\gamma$ ; this factor of two recurs throughout.

Step 3. Fourier-transform the field, keeping the resonant term near $+\omega_0$ , to get $\tilde x(\omega)\propto[(\gamma/2)-i(\omega-\omega_0)]^{-1}$ . Why: a truncated oscillation cannot be monochromatic — sharper decay, wider transform, the heart of "shorter lifetime, broader line."

Step 4. Square it: $I(\omega)\propto|\tilde x(\omega)|^2=[(\omega-\omega_0)^2+(\gamma/2)^2]^{-1}$ , a Lorentzian of half-width $\gamma/2$ . Why: intensity is amplitude squared, which produces the symmetric $(\omega-\omega_0)^{-2}$ wings.

Step 5. Convert with $\omega=2\pi\nu$ and normalize to unit area, giving $\phi_L(\nu)=(\gamma/4\pi^2)/[(\nu-\nu_0)^2+(\gamma/4\pi)^2]$ with FWHM $\gamma/2\pi$ . Why: $\gamma/2\pi$ is the single number a measured Lorentzian core hands you directly.

Step 6. Interpret $\gamma$ quantum-mechanically as $\gamma=1/\tau_i+1/\tau_j$ , consistent with $\Delta E\,\Delta t\sim\hbar$ ; collisions add $2/\tau_{\rm coll}$ , giving pressure broadening of identical shape. Why: natural broadening is the collisionless floor, and pressure broadening simply enlarges $\gamma$ as density rises. *Note: the classical oscillator gives a wavelength-independent natural FWHM $\approx1.2\times10^{-4}$ Å.*

Go deeper · Derivation C · 1 minWhy independent broadenings convolve into a Voigt profile

Derivation CWhy independent broadenings convolve into a Voigt profile

The purpose is to justify the single most useful fact in line analysis: the thermal and damping mechanisms combine not by adding widths but by convolving profiles, producing the Voigt function. The argument is purely statistical — it rests only on the observed frequency being a sum of independent random shifts — so it applies to any pair of independent broadeners, and shows why widths cannot simply be added.

Step 1. Recognize that the observed frequency displacement is a sum of two independent random pieces, $\Delta\nu=\Delta\nu_{\rm damp}+\Delta\nu_{\rm therm}$ — the damping shift from the truncated wave train and the thermal shift from the atom's bulk velocity. Why: the collision or decay setting $\Delta\nu_{\rm damp}$ is independent of the velocity setting $\Delta\nu_{\rm therm}$ .

Step 2. Apply the theorem that the density of a sum of independent variables is the convolution of their densities, so $\phi_V=\phi_L*\phi_D$ . Why: this turns "independent shifts add" into "profiles convolve" — exactly why widths cannot simply be added.

Step 3. Read off the two regimes: near centre the Gaussian ( $\propto e^{-x^2}$ ) fixes the core, while far out the Lorentzian ( $\propto(\nu-\nu_0)^{-2}$ ) fixes the wings. Why: an exponential eventually loses to a power law, so damping wings are inevitable on strong lines even with a thermal core. *Note: numerical convolution of a Gaussian and a Lorentzian of equal FWHM reproduces the second figure — the wing holds about 12 percent of peak at 0.5 Å where the pure Gaussian has collapsed below 1 percent.*

Limiting cases

The four mechanisms scale differently with temperature, density, mass, and rotation; the skill is knowing which dominates a given line.

Case	Result	Meaning
Natural (radiative)	$\Delta\nu_{\rm nat}=\dfrac{\gamma}{2\pi}=\dfrac{1}{2\pi}\left(\dfrac{1}{\tau_i}+\dfrac{1}{\tau_j}\right)$	Set by state lifetimes; classical FWHM $\approx1.2\times10^{-4}$ Å, wavelength-independent. The irreducible Lorentzian floor; dominates the far wings when pressure is negligible.
Doppler (thermal)	$\Delta\lambda_{1/2}=\dfrac{2\lambda_0}{c}\sqrt{\dfrac{2kT\ln 2}{m}}$	Gaussian FWHM $\propto\sqrt{T/m}$ ; dominates the core in most photospheres. Microturbulence via $2kT/m\to 2kT/m+\xi^2$ . Solar Hα at 6000 K: 0.36 Å.
Pressure (collisional)	$\Delta\nu_{\rm coll}\approx\dfrac{1}{\pi\,\tau_{\rm coll}}\propto n\,\sigma\,\bar v$	Lorentzian $\propto$ density $n$ ; strong in high-gravity stars. The classic indicator separating giants from dwarfs of the same temperature.
Rotational	$\Delta\lambda_{\rm rot}\approx\dfrac{2\lambda_0}{c}\,v\sin i$	A convolution with a semi-elliptical kernel, not a simple width; its shape lets $v\sin i$ separate from thermal and pressure broadening.

Common trapYou cannot add a Gaussian and a Lorentzian width in quadrature

The most common error in line fitting is combining widths the wrong way. Only Gaussian widths add in quadrature, because independent Gaussians convolve to a Gaussian whose variances sum — so thermal, microturbulent, and instrumental broadenings combine as $\sigma^2=\sigma_1^2+\sigma_2^2+\cdots$ . Lorentzian widths instead add linearly: $\gamma=\gamma_{\rm nat}+\gamma_{\rm coll}$ . But a Gaussian and a Lorentzian combine into neither — their convolution is a Voigt profile with no exact closed-form width (the Olivero–Longbothum formula only approximates it). A related trap: never confuse FWHM with the standard deviation, since $\mathrm{FWHM}=2\sqrt{2\ln2}\,\sigma\approx2.355\,\sigma$ , and dropping that factor quietly ruins every temperature you derive.

Worked examples

1 · The Doppler width of Hα in the Sun (confidence builder)

Estimate the thermal (Doppler) FWHM of the hydrogen Hα line, $\lambda_0=6562.8$ Å, in the solar photosphere at $T=6000$ K. Take $m_{\rm H}=1.67\times10^{-27}$ kg, $k=1.38\times10^{-23}$ J K⁻¹, $c=3.00\times10^8$ m s⁻¹.

Step 1. Find the characteristic thermal speed $\sqrt{2kT/m}=\sqrt{2(1.38\times10^{-23})(6000)/(1.67\times10^{-27})}=\sqrt{9.9\times10^{7}}=\mathbf{9.9\ km\,s^{-1}}$ — the most-probable line-of-sight speed that Doppler-smears the line.

Step 2. Convert to a $1/e$ Doppler width: $\Delta\lambda_D=(\lambda_0/c)\sqrt{2kT/m}=6562.8\ \text{Å}\times(9.9\times10^3)/(3.00\times10^8)=\mathbf{0.217\ Å}$ , about a part in $3\times10^4$ of the wavelength.

Step 3. Multiply by $2\sqrt{\ln2}=1.665$ for the FWHM: $\Delta\lambda_{1/2}=1.665\times0.217\ \text{Å}=\mathbf{0.36\ Å}$ . The thermal core of solar Hα is about 0.36 Å wide — readable straight off a solar spectrum, and the anchor number in both figures.

2 · Hydrogen versus iron, and the natural-width floor

At the same $T=6000$ K, compare the Doppler width of Hα to that of an iron line, and to the natural (lifetime) width. Iron has $m_{\rm Fe}\approx56$ amu.

Step 1. Because $\Delta\lambda_D\propto1/\sqrt{m}$ at fixed $T$ , the ratio of thermal widths is $\sqrt{m_{\rm Fe}/m_{\rm H}}=\sqrt{56/1.0}=\mathbf{7.5}$ . Hydrogen's core is 7.5 times broader than iron's, so an iron line at 6000 K has a Doppler FWHM of only about $0.36/7.5\approx\mathbf{0.048\ Å}$ .

Step 2. Now the natural width. The classical electron oscillator gives a wavelength-independent $\Delta\lambda_{\rm nat}=(4\pi/3)r_e=(4\pi/3)(2.82\times10^{-13}\ \text{cm})=\mathbf{1.2\times10^{-4}\ Å}$ , with $r_e$ the classical electron radius — the irreducible Lorentzian floor.

Step 3. Compare: the thermal core (0.36 Å) exceeds the natural width by a factor of about $\mathbf{3000}$ . So the core of solar Hα is overwhelmingly thermal — yet the natural (and, at photospheric densities, collisional) Lorentzian still governs the far damping wings, which fall as $\Delta\lambda^{-2}$ and eventually rise above the vanishing Gaussian. Core: thermal. Wings: damping. That division is the diagnostic power of the Voigt profile.

3 · Reading off a projected rotation speed

A hot star shows lines broadened well beyond their thermal width, with a symmetric, semi-elliptical shape. The extra full width of a line at $\lambda_0=5000$ Å is $\Delta\lambda_{\rm rot}\approx1.7$ Å. Estimate $v\sin i$ .

Step 1. Rotation spans the full range of line-of-sight speeds across the disk, so the total width is $\Delta\lambda_{\rm rot}\approx(2\lambda_0/c)\,v\sin i$ , giving $v\sin i\approx c\,\Delta\lambda_{\rm rot}/(2\lambda_0)$ .

Step 2. Insert the numbers: $v\sin i\approx(3.00\times10^5\ \text{km s}^{-1})(1.7\ \text{Å})/(2\times5000\ \text{Å})=\mathbf{51\ km\,s^{-1}}$ — modest for a hot star, where hundreds of km/s are common near the top of the main sequence.

Step 3. Sanity check: this far exceeds the thermal width (a 5000 Å hydrogen line at 10000 K has a Doppler FWHM near 0.3 Å), and the shape is semi-elliptical rather than Gaussian, so rotation dominates. The line shape alone reads out $v\sin i\approx50$ km/s.

Physics of the portrait

The Voigt profile is named for the physicist who first wrote the emergent line as the convolution of a Gaussian and a Lorentzian — the marriage of thermal motion and radiative damping that spectroscopists now fit nightly.

Origins · 1 minWoldemar Voigt and how this was discovered

Woldemar Voigt

1850–1919 · spectral line shapes and the physics of anisotropic media

Woldemar Voigt was a German theoretical physicist at Göttingen whose name attaches to a remarkable spread of physics — the Voigt effect in magneto-optics, Voigt notation in elasticity, and an 1887 coordinate transformation that anticipated part of special relativity two decades early. But to astronomers he is the man behind the line profile. Studying how a gas shapes spectral lines, Voigt recognized in 1912 that two broadening processes act at once: thermal motion spreads the line into a Gaussian, while the finite duration of each emission — the radiative and collisional damping worked out by Lorentz — spreads it into a Lorentzian. Since the two shifts are independent, the observed profile must be their convolution: a Gaussian core with Lorentzian wings and no elementary closed form. That integral stalled quantitative work until Josef Hjerting tabulated it in 1938 and stellar-atmosphere modellers built the curve of growth on it. Today the Voigt profile is the default line model from photospheric abundance analysis to the Lyman-α forest — early-twentieth-century line physics quietly underlying the measurement of stellar compositions and the cosmic distribution of hydrogen.

History

1906

— Lorentz models collisional (pressure) broadening as interrupted wave trains

1912

— Voigt convolves a Gaussian and a Lorentzian into the profile that bears his name

1930

— Weisskopf and Wigner give the quantum theory of natural line width

1938

— Hjerting tabulates the Voigt function for stellar-atmosphere work

1955

— the curve of growth matures as a quantitative tool for stellar abundances

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 9–10 · Gray, The Observation and Analysis of Stellar Photospheres, 3rd ed., Ch. 11 and 18 · Rybicki and Lightman, Radiative Processes in Astrophysics, Ch. 10 · Mihalas, Stellar Atmospheres, 2nd ed. · Draine, Physics of the Interstellar and Intergalactic Medium, Ch. 6 and 9. Key numerical checks: (1) the Doppler width $\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}$ and FWHM $\Delta\lambda_{1/2}=(2\lambda_0/c)\sqrt{2kT\ln2/m}$ , giving 0.36 Å for solar Hα at 6000 K; (2) the natural width $\Delta\nu=\gamma/2\pi$ from the Fourier transform of a damped oscillator, and the classical FWHM $(4\pi/3)r_e\approx1.2\times10^{-4}$ Å; (3) the mass ratio $\sqrt{m_{\rm Fe}/m_{\rm H}}\approx7.5$ ; (4) the Gaussian factors $2\sqrt{\ln2}\approx1.665$ and $2\sqrt{2\ln2}\approx2.355$ ; (5) the rotational estimate $v\sin i\approx c\,\Delta\lambda_{\rm rot}/2\lambda_0$ .

Prerequisites. Spectral Line Formation · Opacity and Absorption Coefficients · The Boltzmann Excitation Equation · The Equation of Radiative Transfer. Leads to. The Saha Ionization Equation · Local Thermodynamic Equilibrium · Specific Intensity and Radiative Flux.

Ask the tutor: Want to see how the same Voigt fit yields temperature from the core and surface gravity from the wings of a single Balmer line — or how a curve of growth turns equivalent widths into abundances? Ask and we will work one line end to end.

Why is a spectral line broadened at all if the transition has a single energy?

Because "a single energy" is an idealization no real state satisfies. An excited level has a finite lifetime, and $\Delta E\,\Delta t\sim\hbar$ forces a finite energy spread — natural broadening, present even for one motionless atom in a vacuum. On top of that, real atoms move (Doppler), collide (pressure), and spin with the stellar surface (rotation), each adding width. Even a frozen, isolated atom keeps its natural width as an irreducible floor, so an infinitely sharp line is physically impossible.

What is the difference between the Doppler width and the standard deviation of the line?

They differ by fixed factors that are easy to confuse. The Doppler width $\Delta\nu_D=(\nu_0/c)\sqrt{2kT/m}$ is the $1/e$ half-width — where the Gaussian falls to $1/e$ of its peak. The standard deviation is $\sigma_\nu=\Delta\nu_D/\sqrt2$ , and $\mathrm{FWHM}=2\sqrt{\ln2}\,\Delta\nu_D=2\sqrt{2\ln2}\,\sigma_\nu\approx2.355\,\sigma_\nu$ . Mixing these up is a leading source of wrong temperatures: since $T\propto(\text{width})^2$ , a 2.355 slip becomes a factor-5.5 error in $T$ . Always state which width you mean.

Why can't I just add the Gaussian and Lorentzian widths to get the Voigt width?

Because widths only add when the profiles stay in the same family, and these do not. Two Gaussians convolve into a Gaussian whose variances add (widths in quadrature); two Lorentzians convolve into a Lorentzian whose $\gamma$ values add linearly. But a Gaussian convolved with a Lorentzian is a Voigt profile — a new shape, Gaussian core and Lorentzian wings — with no exact closed-form width. You fit it through the damping parameter $a=\gamma/4\pi\Delta\nu_D$ , or an approximation like Olivero–Longbothum, but never the naive sum.

How do astronomers separate thermal broadening from microturbulence?

By comparing lines of different atomic mass. The thermal contribution to the squared Doppler width is $2kT/m$ , mass-dependent, while microturbulence adds a mass-independent $\xi^2$ , giving $2kT/m+\xi^2$ . Heavy elements have a suppressed thermal term and are turbulence-dominated; light elements like hydrogen are thermally dominated. Fitting many lines of several elements at once — curve-of-growth or spectrum-synthesis analysis — disentangles the two, yielding both the kinetic temperature and the microturbulent velocity.

What does the strength of a line's damping wings actually measure?

Pressure, and through it surface gravity. The far wings are Lorentzian, set by $\gamma=\gamma_{\rm nat}+\gamma_{\rm coll}$ . The natural part is a fixed floor, but the collisional part grows with the density of perturbers, $\gamma_{\rm coll}\propto n\,\sigma\,\bar v$ . A high-gravity star (dwarf or white dwarf) has a dense, high-pressure photosphere and broad Balmer wings; a low-gravity supergiant has sharp lines. Wing strength thus yields the pressure and hence $\log g$ — the physical basis of the luminosity dimension of the MK classification.

Why does the Lyman-alpha forest use Voigt profiles and not simple Gaussians?

Because intergalactic hydrogen clouds span a huge range of column densities. Most forest lines are weak and unsaturated, dominated by the thermal-plus-turbulent Gaussian core summarized by the Doppler $b$ -parameter, $b=\sqrt{2kT/m+\xi^2}$ . But the rare, high-column damped Lyman-α systems are so optically thick that their cores saturate and the line strength comes from the Lorentzian wings, whose extent scales with neutral-hydrogen column density. Only a full Voigt profile captures both regimes — Gaussian core for weak lines, damping wings for strong — so it is the standard fitting function for the whole forest.