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Low-temperature gas opacity - AESOPUS: a versatile and quick computational tool

Paola Marigo, Bernhard Aringer

arXiv:0907.3248v2astro-ph.SR

TL;DR

Low-temperature Rosseland mean opacity calculations need to accommodate diverse chemical mixtures beyond standard scaled-solar prescriptions. AESOPUS combines equation-of-state and opacity calculations for roughly 800 species and provides user-defined opacity tables, with documented sensitivity to mixture construction and molecular-data treatments.

  • Problem

    Low-temperature opacity resources need broader access to the diverse chemical patterns found in stellar atmospheres and primordial gas.

  • Method

    AESOPUS computes the gas equation of state and Rosseland mean opacity while allowing users to specify temperature-density grids and arbitrary chemical mixtures.

  • Results

    The tool models approximately 800 species and includes atomic, molecular, scattering, and collision-induced opacity sources across varied mixture cases.

  • Takeaways & Limitations

    AESOPUS enables on-demand opacity tables for scaled-solar, CNO-variable, α-enhanced, abundance-anticorrelated, and metal-free mixtures.

  • Takeaways & Limitations

    Opacity differences can reach approximately −0.15 to −0.20 in molecular-dominated regions because molecular line-list and opacity-treatment choices differ.

Abstract

from arXiv · show

We introduce a new tool - AESOPUS: Accurate Equation of State and OPacity Utility Software - for computing the equation of state and the Rosseland mean (RM) opacities of matter in the ideal gas phase. Results are given as a function of one pair of state variables, (i.e. temperature T in the range 3.2 <= log(T) <= 4.5, and parameter R= rho/(T/10^6 K)^3 in the range -8 <= log(R) <= 1), and arbitrary chemical mixture. The chemistry is presently solved for about 800 species, consisting of almost 300 atomic and 500 molecular species. The gas opacities account for many continuum and discrete sources, including atomic opacities, molecular absorption bands, and collision-induced absorption. Several tests made on AESOPUS have proved that the new opacity tool is accurate in the results,flexible in the management of the input prescriptions, and agile in terms of computational time requirement. We set up a web-interface (http://stev.oapd.inaf.it/aesopus) which enables the user to compute and shortly retrieve RM opacity tables according to his/her specific needs, allowing a full degree of freedom in specifying the chemical composition of the gas. Useful applications may regard RM opacities of gas mixtures with i) scaled-solar abundances of metals, choosing among various solar mixture compilations available in the literature; ii) varying CNO abundances, suitable for evolutionary models of red and asymptotic giant branch stars and massive stars in the Wolf-Rayet stages; iii) various degrees of enhancement in alpha-elements, and C-N, Na-O and Mg-Al abundance anti-correlations, necessary to properly describe the properties of stars in early-type galaxies and Galactic globular clusters; iv) zero-metal abundances appropriate for studies of gas opacity in primordial conditions.

1. Introduction

Rosseland mean opacity is a frequency-integrated measure relevant under LTE and diffusion conditions, but existing low-temperature resources provide limited flexibility for arbitrary stellar chemical mixtures. AESOPUS addresses this gap by computing equation-of-state and opacity results on demand for user-specified compositions.

  • Rosseland mean opacity emphasizes weakly absorbing spectral regions because it is a harmonic frequency average.Under LTE and the diffusion approximation, it is commonly used to describe radiative transport in stellar interiors.
  • Rosseland mean opacity depends on temperature, density or pressure, and the gas chemical composition.
  • Existing low-temperature opacity resources include static tables for scaled-solar and α-enhanced mixtures, while other resources provide data or codes for calculating opacities.
  • Modern stellar applications require opacity data for diverse abundance patterns, including AGB, globular-cluster, elliptical-galaxy, and primordial compositions.
  • AESOPUS provides accurate, flexible low-temperature Rosseland mean opacity tables on demand with full freedom to specify the chemical mixture.
  • AESOPUS combines an equation-of-state calculation with monochromatic opacity-source evaluation and offers an interactive web interface for user-defined input parameters.The EOS includes approximately 800 chemical species, including neutral atoms, ions, and molecules.

2.1. Equation of state

AESOPUS solves the ideal-gas equation of state under thermodynamic and instantaneous chemical equilibrium by combining equilibrium relations with conservation constraints. Its implementation covers roughly 800 atomic and molecular species and supports arbitrary mixtures, including zero-metallicity and hydrogen-free cases.

  • The EOS assumes an ideal gas in thermodynamic and instantaneous chemical equilibrium, so species abundances depend on local temperature and density.
  • Chemical-equilibrium calculations combine a defined physical-chemical system, reduced governing equations, and a numerical solution method.
  • Dissociation-recombination and ionisation relations describe equilibrium among atoms, molecules, ions, and electrons.The formulation includes positive and negative ionisation, molecular ionisation, and combined dissociation-ionisation paths.
  • Equilibrium constants are expressed through temperature-dependent molecular relations and Saha equations using partition functions, energies, and electron properties.
  • The species concentrations are closed by conserving atomic nuclei, enforcing charge neutrality, and conserving the total number of nuclei.
  • The database contains approximately 800 species, including approximately 300 atoms and 500 molecules, and supports arbitrary mixtures including Z = 0 and X = 0.
  • When density is specified instead of pressure, an external root-finding iteration reaches δρ = 10^-8 typically after 3–4 iterations.

2.2. Opacity

AESOPUS combines continuum and line opacity sources into Rosseland mean opacities using opacity sampling and a finite frequency grid. Tests show that 944 frequency points provide a practical accuracy–computing-time balance, enabling rapid on-demand table generation.

  • Opacity sources: AESOPUS includes scattering, continuum absorption, atomic bound-bound absorption, and molecular band absorption in its opacity treatment.The listed continuum processes include Rayleigh and Thomson scattering, bound-free and free-free absorption, and collision-induced absorption.
  • Opacity sources: The monochromatic true-absorption and scattering opacities are constructed from process cross sections, particle number densities, gas density, and stimulated-emission corrections.Atomic cross sections are interpolated from the Opacity Project database, while molecular data are prepared as opacity-sampling files.
  • Molecular opacity: Molecular absorption is represented with opacity-sampling files generated from line lists and interpolated during AESOPUS calculations.The interpolation errors are described as marginal compared with other uncertainty sources, including molecular data and microturbulent velocity.
  • Frequency grid: 944 frequency points were selected as the reference grid after tests with grids from 149 to 5488 points.The grid is chosen as a compromise between integration precision and computational speed.
  • Computing time: 45 s is the reported time to generate a 1273-value table at fixed composition with 944 frequency points on a 2.0 GHz processor.The corresponding times are approximately 200 s for 5488 points and 15 s for 149 points.
  • Computing time: Beyond a frequency-resolution threshold, Rosseland mean opacities vary negligibly while computing time continues to scale almost linearly with frequency-point count.This supports the stated favourable accuracy/computing-time ratio of the adopted approach.

3. Opacity tables: basic parameters

AESOPUS tables are parameterized by chemical composition and a two-dimensional state-variable grid using log(T) and log(R). The chosen domain covers ideal-gas conditions and extends into radiation-pressure-dominated regions while excluding several high-density non-ideal regimes.

  • Input parameters: Users specify the gas composition and the two-dimensional state-variable space over which opacity values are calculated.The composition is represented through a reference solar mixture, metallicity, hydrogen abundance, reference mixture, and element-specific enhancement or depression factors.
  • State variables: AESOPUS commonly uses log(T) and log(R), where R = ρ/(T6)^3 and T6 = T/(10^6 K), as the independent variables.The framework assumes an ideal gas for this parameterization.
  • State variables: Using R allows opacity tables to cover rectangular regions of the (R,T)-plane without voids caused by equation-of-state transitions.This avoids gaps that could arise when switching between ideal and degenerate-gas regimes.
  • Table domain: The default table domain is 3.2 ≤ log(T) ≤ 4.5 and −8 ≤ log(R) ≤ 1.The domain lies mainly in the ideal-gas region and reaches radiation-pressure dominance for log(R) ≤ −4.5.
  • Table domain: Electron degeneracy, Coulomb coupling, and pressure ionisation are expected to dominate outside the table boundaries in high-density plasmas.The tables can be extended to higher temperatures using OPAL and OP data, with good agreement reported in the overlap region 3.9 ≲ log(T) ≲ 4.5.
  • Grid specification: The web interface lets users choose effective log(T) and log(R) ranges and grid spacings within the supported domain.Suggested sampling is Δlog(T) = 0.05 above log(T) = 3.7, Δlog(T) = 0.01 at lower temperatures, and Δlog(R) = 0.5.

4. Results

The paper presents applications of the new opacity calculations to stellar modelling and compares the results with other opacity data available in the literature.

  • Applications and comparisons: The study discusses applications of the opacity calculations relevant to stellar models and compares the results with opacity data from the literature.The cited passage introduces the application and comparison scope without reporting a specific result.

4.1. Scaled-solar mixtures

For scaled-solar mixtures, ÆSOPUS produces smoothly interpolable opacity tables whose dominant Rosseland-mean sources change across temperature and density regimes. The results are sensitive to the adopted solar mixture, while comparisons with established databases are generally close, with larger discrepancies concentrated in molecular-opacity regions.

  • Reference scaled-solar grid: The opacity grid uses Δlog(T) = 0.01 below log(T) = 3.5, Δlog(T) = 0.05 above it, and Δlog(R) = 0.5 to support smooth interpolation.The reference mixture has X = 0.7, Z = Zref = 0.02, with all metal abundances scaled-solar.
  • Temperature-dependent opacity sources: At log(T) ≈ 3.2–3.6, molecular absorption dominates, with H2O strongest below log(T) ≈ 3.4 and smaller contributions from metal oxides and CN.At log(R) = −3, a CN opacity bump appears near log(T) ≈ 3.5 despite oxygen-bearing molecules dominating for C/O < 1.
  • Sensitivity to solar abundances: Changing the reference solar mixture can lower the opacity near log(T) ≈ 3.3 by roughly 50%, because the H2O bump depends strongly on the oxygen-to-carbon ratio.Holweger (2001) gives the lower value through its higher C/O ratio, while other mixtures with similar C/O ratios produce closely grouped curves.
  • Comparisons with other databases: ÆSOPUS agrees with reference opacity data mostly within ±0.05 dex, with differences reaching about ±0.10–0.20 dex only in narrow regions.Agreement with Wichita State data is mostly within ±0.05 dex for 3.4 ≤ log(T) ≤ 4.5; larger low-temperature deviations occur where molecular line data and treatments differ.

4.2. Varying C-N-O mixtures

ÆSOPUS shows that low-temperature Rosseland mean opacities are extremely sensitive to the C/O ratio because molecular chemistry changes sharply near C/O = 1. These transitions alter both the dominant molecules and the opacity structure across temperature and density.

  • Motivation: TP-AGB surface abundance changes require opacity tables that account explicitly for varying CNO abundances and C/O ratios.Correct interpolation should use both carbon abundance and C/O, especially near the narrow opacity dip around C/O = 1.
  • Opacity sensitivity: At log(T) = 3.3, C/O = 1.3 produces much larger RM opacities than C/O = 0.49 at low densities, but the trend reverses for log(R) > −3.The contrast is attributed to the strong sensitivity of molecular absorption to the chemical regime.
  • Molecular chemistry: Near C/O = 1, CO and SiO trap most carbon and oxygen, suppressing other O-bearing and C-bearing molecules in a narrow transition region.The transition is bracketed by abrupt abundance changes near C/O ≈ 0.93 and 1.0, with the critical range depending on the reference mixture.
  • Opacity sources: C-bearing molecules, especially CN, dominate important opacity changes above log(T) ≈ 3.4, while molecular effects become negligible for log(T) > 3.7.At higher temperatures, hydrogen bound-free and free-free transitions control the opacity.
  • Opacity sensitivity: The RM opacity drops by more than two orders of magnitude near C/O = 0.95 as the H2O feature disappears, then rises sharply once C/O exceeds unity.Higher C/O values strengthen C-bearing molecular absorption bands after the transition to carbon-dominated chemistry.
  • Interpolation: At fixed log(R) = −3, the opacity transition appears as a narrow strip spanning approximately 0.95 <∼ C/O <∼ 1.00 for 3.2 ≤ log(T) <∼ 3.35.This narrow structure should be resolved when constructing and interpolating opacity tables.

4.3. α-enhanced mixtures

The paper shows that α-enhancement does not uniquely define a chemical mixture: different prescriptions for redistributed elements and total metallicity produce distinct C/O ratios and Rosseland mean opacities. ÆSOPUS therefore compares multiple mixture constructions and traces their opacity differences to molecular chemistry, electron density, and metallicity.

  • Mixture definitions: The user can select alternative elements and positive or negative [Xi/Fe] prescriptions through the ÆSOPUS interface.The formalism is not restricted to enhancing all α-elements by the same amount.
  • Mixture definitions: A given [α/Fe] value does not uniquely specify the chemical mixture because the enhanced, depressed, and fixed element groups can differ.The paper defines mixtures A, B, and C with different constraints on total metallicity and non-α elements.
  • Chemical consequences: At [α/Fe] = 0.4, mixture A lowers C/O from approximately 0.49 to 0.19, whereas mixture B changes it only to approximately 0.46.Mixture C shares A’s elemental ratios but reaches Z ≃ 2 Zref at the same enhancement.
  • Opacity effects: For mixture A, increasing α-enhancement strengthens the low-temperature opacity bump near log(T) ≃ 3.3 while slightly smoothing the knee near log(T) ≃ 3.55.The two temperature intervals show opposite opacity trends.
  • Opacity mechanisms: At log(T) = 3.55, [α/Fe] = 0.6 reduces electron density by only approximately 6%, while weakened CN absorption explains most of mixture A’s opacity-knee depression.Electron contributions from depressed and enhanced groups largely counterbalance each other.
  • Opacity effects: Mixture C produces systematically larger RM opacities across 3.2 <∼ log(T) <∼ 3.75 because its total metallicity increases with α-enhancement.Its opacity variations are larger than those of mixtures A and B.
  • Practical caution: Opacity tables must be interpreted with attention to how α-enhanced mixtures assign depressed and fixed elements and set total metallicity.These construction choices can materially affect the resulting Rosseland mean opacity.

4.4. Other peculiar mixtures: C-N-O-Na-Mg-Al abundance anti-correlations

ÆSOPUS models globular-cluster-like C-N-O-Na-Mg-Al anti-correlations and shows that composition changes reshape low-temperature Rosseland mean opacity, especially through C/O and H2O absorption.

  • Mixture construction: The modeled mixture combines α-enhancement with C-N-O-Na-Mg-Al anti-correlations representative of extreme abundance patterns measured in Galactic globular-cluster stars.The perturbations are log(fC) = −0.6, log(fN) = +1.8, log(fO) = −0.8, log(fNa) = +0.8, log(fMg) = −0.4, and log(fAl) = +1.0.
  • Opacity effects: Reducing C/O from ≃0.49 to ≃0.19 in the α-enhanced mixture increases the H2O-driven opacity peak below log(T) ≲ 3.4.The total metallicity is preserved in this comparison.
  • Opacity effects: Despite nearly doubling metallicity to Z = 1.97 10^-3, the anti-correlated mixture has lower opacity in the H2O-bump region because C and O abundances fall substantially.Its C/O ratio is ≃0.31 and ε(C) + ε(O) decreases by ≃83%.
  • Opacity effects: Between 3.4 ≲ log(T) ≲ 3.6, opacity differences among the three mixtures are small and mainly reflect altered electron-donor abundances affecting H− opacity.This temperature interval differs from the lower-temperature region dominated by the H2O bump.

4.5. Metal-free mixtures

For primordial, metal-free gas, ÆSOPUS identifies how density and temperature shift the dominant opacity sources and shows that trace lithium and H3+ chemistry can materially affect the Rosseland mean opacity.

  • Primordial composition: The primordial mixture assumes X = 0.7521, εLi/εH = 4.15 × 10^-10, Y = 1 − X − Li, and Z = 0.These abundances follow the paper’s adopted standard Big Bang nucleosynthesis composition.
  • Opacity regimes: At lower densities, H− and H2 remain scarce, H3+ is negligible, and scattering dominates the Rosseland mean opacity.Thomson scattering dominates at higher temperatures, while hydrogen-atom scattering dominates at lower temperatures.
  • Opacity regimes: At intermediate densities, hydrogen-atom scattering dominates 3.2 ≲ log(T) ≲ 3.6, H− contributes from 3.6 < log(T) ≲ 3.85, and continuous H absorption dominates above log(T) ≃ 3.85.Free electrons are supplied by H+ and Li+ in these regimes.
  • Opacity regimes: At the highest densities, collision-induced absorption from mainly H2-H2 collisions controls 3.2 ≲ log(T) ≲ 3.5, H− produces a prominent bump up to log(T) ≲ 4.0, and H processes dominate at higher temperatures.Increasing density raises the abundances of H2, H−, and H3+.
  • Trace-element effects: Neglecting H3+ underestimates opacity by weakening electron-mediated H− and Thomson contributions as well as omitting H3+ free-free absorption.The omitted H3+ line opacity is small in most cases, contributing a few percent and peaking at 15% for some temperatures and densities.
  • Trace-element effects: Including lithium can change the opacity by Δlog(κR) ≃ 1.6 at log(T) = 3.2 and log(R) = −8 by increasing free electrons and long-wavelength absorption.The effect is stronger at lower temperatures and densities.

5. Final remarks

ÆSOPUS provides accurate, flexible low-temperature Rosseland mean opacity tables through a public interface, with rapid computation and applications across varied stellar and primordial mixtures.

  • Tool and access: ÆSOPUS computes ideal-gas Rosseland mean opacities over 3.2 ≤ log(T) ≤ 4.5 and exposes the calculations through an interactive web interface.Users can specify the state-variable grid, reference solar composition, total metallicity, and abundance changes from H to U.
  • Performance: A default 67 × 19 grid containing 1273 opacity values takes less than 50 s at fixed composition on a 2.0 GHz processor.The speed comes from optimized opacity sampling for molecular lines and pre-tabulated metal cross-sections.
  • Accuracy: Tests find that the procedure produces fairly accurate Rosseland mean opacities because fine spectral details are smoothed by the harmonic average.The reported accuracy is comparable to that of other opacity codes.
  • Applications: Opacity-data choices change giant-star effective temperatures by a few tens of degrees, usually below or comparable to the typical semi-empirical red-giant temperature-scale uncertainty.The comparison concerns evolutionary models using different opacity datasets for the same chemical composition.

Appendix A: EOS under ICE conditions: numerical details

The EOS solver formulates conservation and equilibrium constraints as a nonlinear system for atomic, total-atom, and electron densities, then iteratively corrects the solution with Newton-Raphson steps.

  • Nonlinear system: ÆSOPUS solves Nel + 2 nonlinear equations for neutral-atom densities, total atom density Na, and electron density ne.The equations include elemental conservation, charge neutrality, and total-number-density conservation.
  • Numerical stabilization: The solver uses logarithmic forms for inherently non-negative unknowns to prevent physically unrealistic estimates during iteration.This transformation is applied to the number-density variables.
  • Conservation equations: Elemental conservation sums particle densities across atoms, ions, and molecules using stoichiometric coefficients νA,α.The coefficient counts how many atoms of element α each species contains.
  • Iteration: Newton-Raphson corrections are obtained from the Jacobian system, solved by LU decomposition, and iterated until the maximum relative density change is typically below 10^-5.Each correction updates the densities as nnew = nold + δn.
  • Species reconstruction: After the primary densities are determined, concentrations of other ionized and molecular species are calculated from equilibrium and ionization relations.The system therefore solves a reduced set of unknowns before deriving the remaining species densities.

Appendix B: The frequency distribution

The appendix constructs frequency distributions for Rosseland-mean opacity calculations, then tests how reducing the frequency-grid size affects accuracy across two chemical mixtures. Differences are generally small, while molecular-band contributions make low-temperature results more sensitive to sampling.

  • Frequency-grid construction: A relatively low number of frequency points can provide good Rosseland-mean opacity results while reducing computational time.The opacity is a frequency-integrated mean, so fewer points can still be adequate.
  • Frequency-grid construction: The seed distribution is optimized by maintaining constant normalized Planckian energy density over frequency intervals and taking the upper envelope across temperatures.The resulting distribution is sharply peaked at lower frequencies and declines exponentially at longer frequencies.
  • Test design: Tests compare grids containing 5488, 1799, 944, 510, and 149 points for scaled-solar and carbon-rich chemical mixtures.The densest grid, with 5488 points, is used as the reference for evaluating lower-resolution grids.
  • Accuracy assessment: Differences remain within approximately ±0.05 dex across most of the log(T)–log(R) space for most tested frequency grids.The comparisons use the 5488-point grid as the reference and include both tested mixtures.
  • Accuracy assessment: The largest deviations occur at lower temperatures, where molecular-band opacity is more sensitive to frequency sampling.Even the smallest frequency set produces a loss in accuracy described as not dramatic.
  • Accuracy assessment: Frequency-distribution uncertainties are comparable to or smaller than typical opacity differences produced by different computational codes.This comparison is reported as an assessment of the tested sampling uncertainties.

Appendix C: Chemical mixtures with non-solar

The appendix formalizes non-solar chemical mixtures by separating selected, fixed, balancing, and non-selected metal groups. The resulting equations preserve or modify reference metallicity according to the chosen mixture construction and abundance constraints.

  • Mixture parameterization: Non-scaled-solar mixtures with preserved reference metallicity divide metals with Zi ≥ 3 into selected, fixed, and balancing groups.The balancing elements compensate for abundance changes in the selected elements.
  • Mixture parameterization: Selected-element ratios [Xs_i/XFe] can be chosen freely, while balancing elements share a common variation factor to preserve metallicity.The abundance variation of the balancing elements compensates for the selected-element changes.
  • Mixture parameterization: The formalism sets up Nsel + 1 equations for the abundance-variation factors and metallicity constraint.The system is solved for the selected-element and balancing-element factors.
  • Specific mixture cases: Mixture A has no fixed group, whereas mixture B uses Fe-group elements as the balancing elements.These cases are treated as specializations of the general mixture formalism.
  • Specific mixture cases: Mixture C allows actual metallicity to vary as Z = fZ Zref and distinguishes selected from non-selected elements.Mixture C shares mixture A’s non-solar metal partitions, but its metallicity differs by a factor fZ.
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