D. G. Hummer

The equation of state for stellar envelopes. I - An occupation probability formalism for the truncation of internal partition functions

An equation of state for material in stellar envelopes, subject to the limits of temperature less than about 10 to the 7th K and density less than about .01 g/cu cm is presented. The equation makes it possible to express free energy as the sum of several terms representing effects such as partial degeneracy of the electron, Coulomb interactions among charged particles, finite-volume, hard sphere repulsion, and van der Waals attraction. An occupation probability formalism is used to represent the effects of the plasma in establishing a finite partition function. It is shown that the use of the static screened Coulomb potential to calculate level shifts and to estimate the cutoff of the internal partition function is invalid. For most of the parameter space relevant to stellar envelopes, perturbations arising from the plasma ions are shown to be dominant in establishing the internal partition function.

The equation of state for stellar envelopes. II - Algorithm and selected results

A free-energy-minimization method for computing the dissociation and ionization equilibrium of a multicomponent gas is discussed. The adopted free energy includes terms representing the translational free energy of atoms, ions, and molecules; the internal free energy of particles with excited states; the free energy of a partially degenerate electron gas; and the configurational free energy from shielded Coulomb interactions among charged particles. Internal partition functions are truncated using an occupation probability formalism that accounts for perturbations of bound states by both neutral and charged perturbers. The entire theory is analytical and differentiable to all orders, so it is possible to write explicit analytical formulas for all derivatives required in a Newton-Raphson iteration; these are presented to facilitate future work. Some representative results for both Saha and free-energy-minimization equilibria are presented for a hydrogen-helium plasma with N(He)/N(H) = 0.10. These illustrate nicely the phenomena of pressure dissociation and ionization, and also demonstrate vividly the importance of choosing a reliable cutoff procedure for internal partition functions.

Stellar atmospheres : beyond classical models

Section 1..- Recent Advances in Computational Methods.- Acceleration of Convergence.- Fast Solution of Radiative Transfer Problems with a Multi-Grid Method.- Line Blanketing without LTE: Simple and Complex Spectra.- Global and Local Methods for 1-D Problems Implementation Aspects and CPU-Time and Memory Scalings.- 2-D Axisymmetric Line Transport.- NLTE Spectral Line Formation in Three Dimensions.- Iteration with Approximate Lambda Operators, and its Application to the Expanding Atmospheres of WR Stars.- Analytical Methods of Line Formation Theory: Are They Still Alive?.- Iteration Factors in the Solution of the NLTE Line Transfer Problem.- Analysis of Ultraviolet P Cygni Profiles in the Spectra of O-Type Stars.- Computer Codes for Stellar Atmospheric Modeling.- Section 2..- The Quest for Physical Realism in Stellar Atmospheric Modeling.- Unified NLTE Model Atmospheres Including Spherical Extension and Stellar Winds: Euv-Fluxes and the HE II Discrepancy in Central Stars of Planetary Nebulae.- Non-LTE Model Atmosphere Calculations with Approximate Lambda Operators.- NLTE Model Atmospheres for Hot Stars.- Radiative Transfer in Expanding Atmospheres - Radiative Acceleration of Wolf-Rayet Envelopes?.- Non-LTE Analysis of Hot Stars Including Line Blanketing.- Time-Dependent Two-Dimensional Radiation Hydrodynamics of Accreting Matter onto Highly Magnetized Neutron Stars: The Evolution of Photon Bubbles.- The Origin and Development of Instabilities in Radiatively- Driven Stellar Winds.- A Smooth Source Function Method for Including Scattering in Radiatively Driven Wind Simulations.- 2-D Radiation Hydrodynamics Models of the Solar Photosphere.- Dynamics of and Radiative Transfer in Inhomogeneous Media.- Atmospheres of Late-Type Giants.- Fe II Emission Line Diagnostics of the Sun and Stars.- Chromospheric Inhomogeneities in Cool Stars: Possible Effect on Hydrogen Line Profiles.- Numerical Simulation of Photospheric Convection: Hydrodynamical Test Calculations.- Section 3..- Stellar Atmosphere Theory as a Spectroscopic Tool. The Example of Hot Stars.- The Winds of O-Stars. V: Tests of the Accuracy of the Radiation Driven Wind Models.- Diagnostics of Wolf-Rayet Atmospheres.- Central Stars of PN: Spectral Diagnostics Based on Model Atmospheres VS. Diagnostics Based on the Classical Nebular Approach.- Spectral Diagnostics of Hot Subdwarfs: Successes and Problems.- Temperatures, Gravities and Abundances of B Stars: Recent Progress and Remaining Problems.- Line Blanketing Without LTE: The Effect on Diagnostics for B-Type Stars.- Particle Transport in Magnetic Stellar Atmospheres.- Spectroscopic Tests of Late-Type Model Atmospheres of Dwarf Stars.- Dust in the Shells of Cool Giants and Supergiants.- NLTE Analysis of Massive OB Stars.- Formation of the K I 7699 A Line in Sunspots.- On the Influence of Multi-Dimensional Radiative Transfer on the Energetic Contribution of the Ca K Line.- Special Session.- The Opacity Project and the Practical Utilization of Atomic Data.- New Opacity Calculations.- 40 Years Numerical Stellar Atmospheres: Concluding Remarks and Personal Considerations.

The equation of state for stellar envelopes. III - Thermodynamic quantities

A method is described for deriving general expressions for all thermodynamic quantities of interest of a partially ionized multicomponent gas in terms of derivatives of the free energy. Explicit analytical formulas for all derivatives required in the evaluation of these quantities are given. Representative results for a hydrogen-helium mixture are shown.

Electron impact excitation of positive ions

Non-relativistic Coulomb-Born-Oppenheimer reactance matrices and cross-sections are given for all transitions between the Is, 2s and 2p states in He+ and in hydrogen-like ions of large nuclear charge. From these results some cross-sections for intercombination transitions in highly charged non-hydrogenic ions are estimated.

The equation of state for stellar envelopes. IV - Thermodynamic quantities and selected ionization fractions for six elemental mixes

The free-energy minimization technique in the form developed in the preceding papers in this series is employed to evaluate thermodynamic quantities and ionization fractions on a fine temperature and density grid for six astrophysical mixtures of 15 elements. The mixtures range from that appropriate to super-metal-rich stars, through solar abundance, to that for extreme Population II objects. In this paper, the results for solar abundances are summarized in a form that is illustrative and which facilitates comparison with the results from other equation of state calculations. 16 refs.

Atoms in Astrophysics

1. Low-Energy Electron Collisions with Complex Atoms and Ions.- 1. Introduction.- 2. Theory of Electron Collisions with Atoms and Ions.- 2.1. Expansion of the Collision Wave Function.- 2.2. Expansion of the Target Wave Function.- 2.3. Variational Principles and the Derivation of the Coupled Integro-Differential Equations.- 2.4. Derivation of the Cross Section.- 2.5. Inclusion of Relativistic Effects.- 2.5.1. Use of the Breit-Pauli Hamiltonian.- 2.5.2. Use of the Dirac Hamiltonian.- 3. Numerical Solution of the Coupled Integro-Differential Equations.- 3.1. Early Work.- 3.1.1. Iterative Methods.- 3.1.2. Reduction to a System of Coupled Differential Equations.- 3.2. Reduction to a System of Linear Algebraic Equations.- 3.3. R-Matrix Method.- 3.4. Matrix Variational Method.- 3.5. Noniterative Integral Equations Method.- 3.6. New Directions.- 3.7. Illustrative Results.- 4. Computer Program Packages.- 4.1. Structure Packages.- 4.2. Collision Packages.- References.- 2. Numerical Methods for Asymptotic Solutions of Scattering Equations.- 1. Introduction.- 2. Specification of Asymptotic Forms.- 3. Travels in Intermedia.- 3.1. Numerical Integration of the Differential Equations.- 3.2. Noniterative Integration of the Phase-Amplitude Equations.- 4. At the Border of Asymptopia.- 4.1. Asymptopic Expansions.- 4.2. Iterative Techniques.- 4.2.1. The Iterated WBK (IWBK) Method.- 4.2.2. The Generalized Matricant.- 5. Concluding Remarks.- References.- 3. Collisions between Charged Particles and Highly Excited Atoms.- 1. Introduction.- 2. The Impact Parameter (IP) Method.- 3. The Sudden Approximation.- 4. Transitions between Levels with Quantum Defects.- 5. Transitions within the Degenerate Sea.- References.- 4. Proton Impact Excitation of Positive Ions.- 1. Introduction.- 2. Excitation of Fine-Structure Transitions.- 2.1. Semiclassical Theory.- 2.2. Quantal Theory.- 3. Excitation of Metastable Transitions.- 4. Charge-Transfer Ionization.- References.- 5. Long-Range Interactions in Atoms and Diatomic Molecules.- 1. Introduction.- 2. General Form of the Model Hamiltonian.- 3. Form of the Model Potential for Atomic Systems.- 3.1. The Exact Interaction between the Valence Electrons and the Core.- 3.2. Second-Order Perturbation Theory: The Static Contribution.- 3.3. The First Nonadiabatic Correction.- 3.4. The Second Nonadiabatic Correction.- 3.5. Nonadiabatic Corrections of Higher Order.- 3.6. Third-Order Perturbation Theory: The Static Contributions.- 3.7. Summary of Results.- 4. Form of the Model Potential for Diatomic Systems.- 4.1. The Exact Interaction between the Valence Electrons and the Cores.- 4.2. Second-Order Perturbation Theory: The Static Contributions.- 4.3. The First Nonadiabatic Correction.- 4.4. The Second Nonadiabatic Correction.- 4.5. Nonadiabatic Corrections of Higher Order.- 4.6. Third-Order Perturbation Theory: The Static Contribution.- 4.7. Summary of Results and the Separated Atom Limit.- 5. Addition Theorems for Solid Harmonics.- References.- 6. Applications of Quantum Defect Theory.- 1. Historical Survey.- 2. Mathematical Background to Quantum Defect Theory.- 2.1. Properties of Coulomb Wave Functions.- 2.2. Solutions of the Coupled Equations.- 2.2.1. All Channels Closed.- 2.2.2. Some Channels Open.- 3. Single-Channel Quantum Defect Methods: General Formulas in the Independent-Particle Approximation.- 3.1. Expressions for the Wave Functions.- 3.2. General Formulas for Radiative Transition Probabilities.- 3.3. Collision Cross Sections.- 3.3.1. Use of Extrapolated Quantum Defects.- 3.3.2. Use of Adjusted Calculated Quantum Defects.- 3.3.3. Use of Observed Quantum Defects in the Bethe Approximation.- 3.4. Summary.- 4. Applications to Simple Multichannel Problems.- 4.1. The Spectrum of Calcium.- 4.1.1. Bound States.- 4.1.2. Autoionizing States.- 4.2. Bound States of Complex Ions by Extrapolation of Calculated Scattering Parameters: Configurations 1s22s22pqnl.- 4.3. The Spectrum of the H2 Molecule.- 5. Extrapolation of the Generalized Reactance Matrix.- 5.1. Discussion of Extrapolation Methods.- 5.1.1. Restrictions on the Validity of Extrapolation Methods.- 5.1.2. Fitting Techniques.- 5.2. Applications.- 5.2.1. Collision Strengths in LS-Coupling.- 5.2.2. Collision Strengths for Excitations between Fine-Structure Levels.- 5.2.3. Photoionization Gross Sections.- 5.2.4. Electron Impact Ionization Cross Sections.- 6. Conclusions.- References.- 7. Electron-Ion Processes in Hot Plasmas.- 1. Introduction.- 2. Line Intensities.- 2.1. Level Populations.- 2.2. Forbidden Lines.- 2.3. Satellite Lines.- 3. Electron-Ion Processes.- 3.1. Introduction.- 3.2. The A+z + e System.- 3.3. Coupling between Open Channels.- 3.4. Resonance Contribution to Collisional Excitation.- 3.5. Approximate Methods for Strong Allowed Transitions.- 3.6. Relativistic Effects.- 3.7. Relativistic Effects in Autoionization.- 4. Conclusion.- A.1. Derivation of am(E) and b?i(E).- A.2. Resonances.- A.3. Gailitis Formula.- References.- 8. The University College Computer Package for the Calculation of Atomic Data: Aspects of Development and Application.- 1. Introduction.- 2. The Growth of Astronomical Observations.- 3. Some Aspects of the Genesis of the Programs.- 4. The C III Challenge.- 4.1. Collision Strengths and Transition Probabilities for the Interpretation of the Solar Spectrum.- 4.2. Excitation of C III by Recombination.- 4.2.1. Observations.- 4.2.2. Rate Coefficients from Detailed Balance Arguments.- 4.2.3. Improved Low-Temperature Coefficients.- References.- 9. Planetary Nebulae.- 1. Introduction.- 2. Observations.- 2.1. Optical.- 2.2. Infrared.- 2.3. Radio.- 2.4. Ultraviolet.- 2.4.1. NGC 7662.- 2.4.2. IC 418.- 2.4.3. The C/O Ratio in Planetary Nebulae.- 3. Models: Atomic Data.- 3.1. Charge Transfer.- 3.1.1. O+.- 3.1.2. O2+.- 3.1.3. Ne2+.- 3.2. Dielectronic Recombination.- 3.3. Photoionization and Radiative Recombination.- 3.4. Electron Collisional Excitation.- 3.5. Radiative Transition Probabilities.- References.- 10. Forbidden Atomic Lines in Auroral Spectra.- 1. Introduction.- 2. Beginnings.- 3. Seatons Work.- 4. Refinement of Classical Theory.- 5. Advent of In Situ Measurements.- 6. N2(A3?u+)-O Excitation Transfer.- 7. Quenching.- 8. Coordinated Rocket and Satellite Measurements.- 9. ?3466 and ?10,400 of N I.- References.

Fast computer evaluation of radiative properties of hydrogenic systems

Abstract Three subroutines are described for the very fast calculation of bound-bound, bound-free, and free-free cross-sections for nonrelativistic hydrogenic systems of arbitrary nuclear charge and reduced mass. The first two are essentially exact, being based on recursion relations which are known to be stable. The third evaluates the thermally-averaged free-free Gaunt factor by means of a two-dimensional Chebyshev expansion calculated by numerical evaluation of the cross-sections expressed as hypergeometric functions, augmented by other analytical approximations.

Photospheres of hot stars. III - Luminosity effects at spectral type 09.5

Hydrogen and helium line profiles with high signal-to-noise ratios were obtained for four stars of spectral type 09.5 (Alpha Cam, Xi Ori A, Delta Ori A,AE Aur) that form a sequence in luminosity: Ia, Ib, II, V. The basic stellar parameters of these stars are determined by fitting the observed line profiles of weak photospheric absorption lines with profiles from models which include the effect of radiation scattered back onto the photosphere from their stellar winds, an effect referred to as wind blanketing. For these stars, the inclusion of wind blanketing is significant only for the most luminous star, Alpha Cam, for which the effective temperature was shifted about -2000 K relative to an unblanketed model.

A fast and accurate method for evaluating the nonrelativistic free-free Gaunt factor for hydrogenic ions

A fast method for obtaining a two-dimensional Chebyshev expansion of the nonrelativistic hydrogenic free-free Gaunt factor is presented which gives a maximum relative error of 0.7 percent over a wide range of temperatures and frequencies. The expansion derived by Karzas and Latter (1961) is used to derive primary values of the free-free Gaunt factor as a function of the initial and final electron energies. Thermal averages have been numerically computed and are presented in tabular form in intervals of 0.25 dex.