Spectral Line Broadening Hubeny & Mihalas Chap. 8 Gray Chap. 11 Natural Broadening Doppler Broadening Collisional Broadening: Impact, Statistical, Quantum Theories

Broadening of Absorption Profile Natural – energy uncertainty due to finite lifetime Doppler – thermal motion of gas Pressure – perturbations in energy levels due to collisions (encounters) with charged particles [important in transfer equation] Stellar rotation – Doppler shifts across disk Stellar turbulence – Doppler shifts from motion [important in line synthesis] Instrumental – projected slit of spectrograph [always important]

Natural Broadening Uncertainty principle level j depopulated by spontaneous emission, rate Aji (Einstein coeff.) Lifetime for j to i Lifetime for all downward transitions FWHM j i

Natural Broadening Damping constant Lorentzian profile Small, important in low density gas

Doppler Broadening by Thermal Motion Profile at Doppler shifted frequency by speed ξ Integrate over Maxwellian velocity distribution along the line of sight

Doppler Broadening by Thermal Motion Substitute Then final profile has form H(a,V) = Voigt profile

Voigt Profile Gaussian in core and Lorentzian in wings IDL version: IDL> u=findgen(201)/40.-2.5 IDL> v=voigt(0.5,u) IDL> plot,u,v

Collisional Broadening: Classical Impact – Phase Shift Theory Suppose encounter happens quickly and atom emits as an undisturbed oscillator between collisions but ceases before and after Frequency content of truncated wave from FT Power spectrum (observed)

Collisional Broadening: Classical Impact – Phase Shift Theory Probability number occurring in time dT at T where T0 = average time between collisions Mean energy spectrum is then Lorentzian profile damping constant Γ=2/T0

Collisional Broadening: Classical Impact – Phase Shift Theory Frequency of collisions = 1/T0 Suppose collisions occur if particles pass within distance = impact parameter ρ0 N = #perturbers/cm3, v = relative velocity cm/s Then damping parameter is

Weisskopf approximation perturber is a classical particle path is a straight line no transitions caused in atom interaction creates a phase shift or frequency shift given by

p exponents of astronomical interest p = 2 linear Stark effect (H + charged particle) p = 3 resonance broadening (atom A + atom A) p = 4 quadratric Stark effect (non-hydrogenic atom + charged particle) p = 6 van der Waals force (atom A + atom B) Cp from experiment or quantum theory

Weisskopf approximation Total phase shift Atom r(t) t = 0 v=constant p ψp 2 π 3 4 π/2 6 3π/8 perturber path

Weisskopf approximation Assume that only collisions that produce a phase shift > η0 are effective in broadening Weisskopf assumed η0 =1 , yields damping depends on ρ, T Ignores weak collisions η < η0

Better Impact Model: Lindholm-Foley Includes effects of multiple weak collisions, which introduce a phase shift Δω0 ; ΓLF > ΓW Impact theory fails for: small ρ, large broadening time overlap of collisions nonadiabatic collisions p 3 4 6 Γ 2π2C3N 11.37 C42/3 v1/3 N 8.08 C62/5 v3/5 N Δω0 9.85 C42/3 v1/3 N 2.94 C62/5 v3/5 N

Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation Imagine atom sitting in a static sea of perturbers (OK for slow moving ions) that produces a relative probability of perturbing electric field and Δω Close to atom, consider probability that nearest neighbor is located at a distance in the range (r,r+Δr) = W(r) dr Corresponding frequency profile

Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation Probability proportional to (1) % that do not occur at <r (2) increasing numbers at increasing distance Differentiate wrt r

Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation Consider frequency shifts relative to that for mean interparticle distance r0 #particles x volume for each = total volume Insert into expression W(r) Express with relative frequency shift

Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation Replace W(r) with W(Δω) Probability that atom will experience a perturbing field to give a frequency shift Δω

Apply to Linear Stark effect p=2 Express in terms of normal field strength Change of variables

Apply to Linear Stark effect p=2 Then probability in terms of field strength is [note missing minus sign in Hubeny & Mihalas] Final expression for profile

Holtsmark Statistical Theory Ensemble of perturbers instead of single more particles, more chances for strong field e- attracted to ions, reduce perturbation by Debye shielding in stellar atmospheres density is low, number of perturbers is large, and Holtsmark distribution is valid

Hydrogen: Linear Stark Effect each level degenerate with 2n2 sublevels perturbing field will separate sublevels observed profile is a superposition of components weighted by relative intensities and shifted by field probability function

Hydrogen: Linear Stark Effect each component shifted by profile is a sum over all components density dependent shift (N) statistical theory OK for interactions H + protons impact theory ~OK for interactions H + electron, but electron collisions are non-adiabatic

Quantum Calculations for the Linear Stark effect of Hydrogen unified theory for electron and proton broadening for Lyman and Balmer series: Vidal, Cooper, & Smith 1973, ApJS, 25, 37 IR series Lemke 1997, A&AS, 122, 285 Model Microfield Method (not static for ions) Stehle & Hutcheon 1999, A&AS, 140, 93

Summary final profile is a convolution of all the key broadening processes convolution of Lorentzian profiles: Γtotal=ΣΓi convolution of Lorentzian and Doppler broadening yields a Voigt profile convolution of Stark profile with Voigt (for H) calculate as a function of depth in atmosphere because broadening depends on T, N (Ne)