The true magnitude, functional form, and dependence on atomic number, transition type, temperature, and electron density of Stark line broadening in hot dense plasmas at stellar-interior conditions is unknown and experimentally unvalidated, yet bound-bound line absorption is a major component of the radiative opacity that governs the structure of the Sun and stars. A 2016 sensitivity study showed that plausible inter-code discrepancies in Stark widths produce an opacity enhancement qualitatively consistent with the opacity missing from solar models near the base of the convection zone, and that an unphysical uniform multiplication of all line widths by a factor of about 100 recovers it quantitatively. Whether realistic, physically motivated line-shape corrections — together with quantified plasma-environment uncertainties — can supply the missing opacity, or can be quantitatively excluded, remains open. This problem asks for a validated determination of line broadening at stellar-interior conditions and a definitive accounting of its role in the solar opacity problem.

Background

Below the solar convection zone, energy is carried outward by radiation, and the rate at which photons diffuse through the plasma is set by the radiative opacity — most often the Rosseland mean opacity, a frequency-integrated average of the absorption coefficient over the thermal photon spectrum, weighted by the temperature derivative of the Planck function. The total absorption at solar-interior conditions (temperatures of order 10610^6–10710^7 K and electron densities up to roughly 102310^{23} cm−3^{-3}) receives contributions from bound–bound transitions (photon absorption driving an electron between two bound states of an ion), bound–free photoionization, and free–free absorption. For iron-group elements, millions of closely spaced bound–bound lines form unresolved transition arrays — statistical groupings of many lines sharing a configuration pair into blended spectral features — so the frequency distribution of line absorption, and hence the Rosseland mean, depends directly on line widths.

Stark broadening is the broadening of spectral lines caused by the electric microfields and electron impacts of the surrounding plasma. At stellar-interior densities, line shapes and widths are computed from approximations (impact approximations, semiclassical or parametric electron-impact formulae, ion microfield models) whose validity in this regime has never been tested experimentally, and atomic codes disagree: the study motivating this problem found sizeable discrepancies between several codes and the Opacity Project — a major international collaboration computing atomic data for stellar opacities — in computed K-shell line widths.

The astrophysical stakes come from the solar opacity problem: solar models built with modern, lower photospheric abundances of heavy elements disagree with helioseismology — the analysis of the Sun's acoustic oscillation frequencies, which constrains the sound-speed profile and the depth of the convection zone — and the discrepancy is widely suspected to reflect missing opacity in the radiative interior. Independently, the Sandia Z-pinch experiment (a pulsed-power machine that magnetically compresses plasma to hot-dense conditions) measured spectrally resolved iron opacity at conditions close to the base of the solar convection zone (T≈2.11×106T \approx 2.11 \times 10^6 K, ne≈3×1022n_e \approx 3 \times 10^{22} cm−3^{-3}) and found it 30–400% higher than the predictions of all participating opacity codes.

Krief, Feigel and Gazit (2016) used their STAR atomic code to study how sensitive the solar opacity profile is to Stark line widths. Increasing the widths in line with the inter-code discrepancies produces an opacity enhancement that agrees qualitatively with the missing-opacity profile inferred from helioseismic and neutrino constraints and recovers about half of it near the convection-zone base, while an admittedly unphysical uniform multiplication of all line widths by a factor of about 100 recovers the missing opacity quantitatively. Because the true uncertainty of Stark widths at these conditions is unknown — experimental data exist only at much lower temperatures and densities than the solar interior — the possibility could be neither confirmed nor excluded, motivating better theoretical characterization of line broadening at stellar-interior conditions and propagation of these uncertainties into solar opacity predictions.

Problem Statement

Determine the line-shape physics of hot dense plasmas at stellar-interior conditions, and settle whether it can account for the missing solar opacity.

Concretely, the question left open by Krief, Feigel and Gazit (2016) comprises three jointly linked parts serving this single objective: (1) the true magnitude, atomic-number, transition-type, temperature, and density dependence, and functional form of Stark (electron-impact and microfield) line broadening at stellar-interior conditions, where current approximations are untested, codes disagree, and no experiment has measured line widths or shapes; (2) whether realistic, physically motivated line-shape corrections — as opposed to implausible uniform width multiplications such as the factor-of-about-100 sensitivity probe — can reproduce the required radially varying opacity increase imposed by helioseismic and neutrino observations, including whether they can reconcile the laboratory iron (and chromium) opacity measurements with atomic calculations, or whether other plasma-environment and atomic-structure uncertainties (ionization balance, occupation-number distributions, excited spectator electrons, autoionizing states, ion-sphere boundary conditions, short-range ion–ion and electron–ion correlations) dominate; and (3) a quantified propagation of these line-broadening and plasma-environment uncertainties into the Rosseland mean opacity and the solar opacity profile.

The problem is complete when the line-broadening contribution to the missing solar opacity is determined: either realistic line shapes supply the required enhancement within the combined uncertainties of solar models and the helioseismic and neutrino observations that constrain them, or line broadening is quantitatively excluded as sufficient, with bounds covering the relevant elements, transitions, temperatures, and densities.

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