The Mihalas-Hummer-Däppen (MHD) equation of state is a part of the Opacity Project (OP), where it mainly provides ionization equilibria and level populations of a large number of astrophysically relevant species. Its basic concept is the idea of perturbed atomic and ionic states. At high densities, when many-body effects become dominant, the concept of perturbed atoms loses its sense. For that reason, the MHD equation of state was originally restricted to the plasma of stellar envelopes, that is, to relatively moderate densities, which should not exceed ρ < 10-2 g cm-3. However, helioseismological analysis has demonstrated that this restriction is much too conservative. The principal feature of the original Hummer & Mihalas paper is an expression for the destruction probability of a bound state (ground state or excited) of a species (atomic or ionic), linked to the mean electric microfield of the plasma. Hummer & Mihalas assumed, for convenience, a simplified form of the Holtsmark microfield for randomly distributed ions. An improved MHD equation of state (Q-MHD) is introduced. It is based on a more realistic microfield distribution that includes plasma correlations. Comparison with an alternative post-Holtsmark formalism (APEX) is made, and good agreement is shown. There is a clear signature of the choice of the microfield distribution in the adiabatic index γ1, which makes it accessible to present-day helioseismological analysis. However, since these thermodynamic effects of the microfield distribution are quite small, it also follows that the approximations chosen in the original MHD equation of state were reasonable. A particular feature of the original MHD papers was an explicit list of the adopted free energy and its first- and second-order analytical derivatives. The corresponding Q-MHD quantities are given in the Appendix.

Line emissivities, effective recombination coefficients, opacity factors, departure coefficients and total recombination coefficients are calculated for hydrogenic ions with Z less-than-or-equal-to 8. Results are obtained for Cases A and B for n less-than-or-equal-to 50. Collisional transitions among individual n and l states are fully treated. Calculations were made for log N(e) = 2(1)14 for Case B and log N(e) = 2(1)10 for Case A. The electron temperature takes between nine and 12 values, lying within the range 500 to 100 000 K, depending on the ion. All results are available in the form of machine-readable files.Secondary files containing only effective emissivities for transitions for n less-than-or-equal-to 25 and total recombination coefficients are also available for use with an interactive data server. The server produces tables of relative intensities of any two specified transitions or emissivities for any transition at all temperatures and densities in the data set. Extensive facilities for two-dimensional interpolation of relative intensities, emissivities and total recombination coefficients are provided.

The formation of a spectral line in a uniformly expanding infinite medium is investigated in the Sobolev approximation with particular attention to the various mechanisms for frequency redistribution. Numerical and analytic solutions of the transfer equation are presented for a number of redistribution functions and their approximations, including type I and type II partial redistribution, coherent scattering and complete redistribution, and the Fokker-Planck and uncorrelated approximation to the R(II) function. The solutions for the mean intensity are shown to depend very much on the type of redistribution mechanism, while for the frequency-weighted mean intensity JBAR, which enters the rate equations, this dependence is weak. This implies that use of Sobolev escape probabilities based on complete redistribution can be an adequate approximation for many calculations for which only the radiative excitation rates are needed. However, it is shown that the criteria for applicability of Sobolev theory may be difficult to meet when transfer occurs primarily in the Voigt wings, especially for complete redistribution.

Asymptotic expressions are obtained for the kernel functions M̃2(τ, α, β) and K̃2(τ, α, β) appearing in the theory of line formation with complete redistribution over a Voigt profile with damping parameter α, in the presence of a source of continuous opacity parameterized by β. For α > 0, each coefficient in the asymptotic series is expressed as the product of analytic functions of α and η ≡ βτ separately. For Doppler broadening, only the leading term can be evaluated analytically.

view Abstract Citations (100) References (13) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Equation of State for Stellar Envelopes. IV. Thermodynamic Quantities and Selected Ionization Fractions for Six Elemental Mixes Mihalas, Dimitri ; Hummer, D. G. ; Mihalas, Barbara Weibel ; Daeppen, Werner Abstract 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. Publication: The Astrophysical Journal Pub Date: February 1990 DOI: 10.1086/168383 Bibcode: 1990ApJ...350..300M Keywords: Equations Of State; Gas Ionization; Metallic Stars; Stellar Envelopes; Thermodynamics; Helium Ions; Hydrogen Ions; Temperature Dependence; Astrophysics; ATOMIC PROCESSES; EQUATION OF STATE; STARS: ATMOSPHERES full text sources ADS | Related Materials (3) Part 1: 1988ApJ...331..794H Part 2: 1988ApJ...331..815M Part 3: 1988ApJ...332..261D

view Abstract Citations (31) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Photospheres of Hot Stars. IV. Spectral Type O4 Bohannan, Bruce ; Voels, Stephen A. ; Hummer, David G. ; Abbott, David C. Abstract The basic stellar parameters of a supergiant (Zeta Pup) and two main-sequence stars, 9 Sgr and HD 46223, at spectral class O4 are determined using line profile analysis. The stellar parameters are determined by comparing high signal-to-noise hydrogen and helium line profiles with those from stellar atmosphere models which include the effect of radiation scattered back onto the photosphere from an overlying stellar wind, an effect referred to as wind blanketing. At spectral class O4, the inclusion of wind-blanketing in the model atmosphere reduces the effective temperature by an average of 10 percent. This shift in effective temperature is also reflected by shifts in several other stellar parameters relative to previous O4 spectral-type calibrations. It is also shown through the analysis of the two O4 V stars that scatter in spectral type calibrations is introduced by assuming that the observed line profile reflects the photospheric stellar parameters. Publication: The Astrophysical Journal Pub Date: December 1990 DOI: 10.1086/169526 Bibcode: 1990ApJ...365..729B Keywords: Early Stars; O Stars; Photosphere; Radiative Transfer; Stellar Atmospheres; Supergiant Stars; Abundance; Helium; Line Spectra; Signal To Noise Ratios; Stellar Luminosity; Stellar Mass; Stellar Spectra; Astrophysics; LINE PROFILES; RADIATIVE TRANSFER; STARS: ATMOSPHERES; STARS: EARLY-TYPE; STARS: SUPERGIANTS full text sources ADS | data products SIMBAD (7) Related Materials (3) Part 1: 1985ApJ...294..286A Part 2: 1986ApJ...308..728B Part 3: 1989ApJ...340.1073V

view Abstract Citations (109) References (49) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Photospheres of Hot Stars. III. Luminosity Effects at Spectral Type O9.5 Voels, Stephen A. ; Bohannan, Bruce ; Abbott, David C. ; Hummer, D. G. Abstract 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. Publication: The Astrophysical Journal Pub Date: May 1989 DOI: 10.1086/167459 Bibcode: 1989ApJ...340.1073V Keywords: Hot Stars; O Stars; Photosphere; Stellar Luminosity; Stellar Spectra; Absorption Spectra; Abundance; Line Spectra; Stellar Atmospheres; Stellar Winds; Astrophysics; LINE PROFILES; STARS: ABUNDANCES; STARS: ATMOSPHERES; STARS: EARLY-TYPE full text sources ADS | data products SIMBAD (5) Related Materials (3) Part 1: 1985ApJ...294..286A Part 2: 1986ApJ...308..728B Part 4: 1990ApJ...365..729B

The intensities of recombination lines of C VI, N VII and O VIII are calculated, accounting for both electron and heavy particle collisions and assuming Case B of Baker & Menzel. The intensities of lines formed by transitions |$n_\text u\rightarrow n_l$|⁠, are tabulated for nu≼50, nl≼29, at log Ne=4(l)13 and at 10 values of electron temperature in the interval 10 000 K≼Te≼500 000 K.

view Abstract Citations (76) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Fast and Accurate Method for Evaluating the Nonrelativistic Free-free Gaunt Factor for Hydrogenic Ions Hummer, D. G. Abstract A two-dimensional Chebyshev expansion of the nonrelativistic hydrogenic free-free Gaunt factor is presented which can be evaluated extremely rapidly and which gives a maximum relative error of 0.7% over a very wide range of temperatures and frequencies. This expansion is obtained from numerical values computed primarily from the exact expressions of Karzas and Latter, augmented by certain analytic approximations that are valid in regions of the energy plane where the series given by Karzas and Latter become computationally intractable. A brief table of thermally averaged Gaunt factors is given. Publication: The Astrophysical Journal Pub Date: April 1988 DOI: 10.1086/166210 Bibcode: 1988ApJ...327..477H Keywords: Electron Transitions; Hydrogen Ions; Nonrelativistic Mechanics; Numerical Analysis; Stellar Atmospheres; Stellar Envelopes; Electron Energy; Error Analysis; Frequency Ranges; Stellar Winds; Atomic and Molecular Physics; ATOMIC PROCESSES; NUMERICAL METHODS full text sources ADS |

view Abstract Citations (26) References (21) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Failure of Continuum Methods for Determining the Effective Temperature of Hot Stars Hummer, D. G. ; Abbott, David C. ; Voels, Stephen A. ; Bohannan, Bruce Abstract The authors demonstrate that for hot stars (Teff > 30,000K) methods based on the integrated continuum flux are completely unreliable discriminators of the effective temperature. They show that the observed continuum flux distribution of ζ Pup (O4f) can be fitted to within the observational errors by photospheric models having a wide range of effective temperatures and gravities. In contrast, absorption line profiles provide much more accurate values of these parameters. As an example, profiles of weak lines of H I, He I, and He II are computed from standard non-LTE models at the effective temperature and surface gravity inferred by Underhill from the continuous energy flux. These profiles show a severe lack of agreement with observed profiles. Publication: The Astrophysical Journal Pub Date: May 1988 DOI: 10.1086/166328 Bibcode: 1988ApJ...328..704H Keywords: Early Stars; Hot Stars; Line Spectra; Stellar Atmospheres; Stellar Temperature; Energy Distribution; Error Analysis; O Stars; Thermodynamic Equilibrium; Astrophysics; LINE PROFILES; SPECTROPHOTOMETRY; STARS: ATMOSPHERES; STARS: EARLY-TYPE full text sources ADS | data products SIMBAD (4)

A partir des travaux de Cherepashchuk et al. (1984) sur les courbes de lumiere de V444 Cyg et de la theorie des vents stellaires induits radiativement, on a calcule des modeles qui conduisent a l'existence d'une photosphere etendue a expension supersonique, les valeurs du rayon photospherique, du taux de perte de masse et de la vitesse terminale

view Abstract Citations (71) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Sobolev approximation for line formation with continuous opacity Hummer, D. G. ; Rybicki, G. B. Abstract The Sobolev approximation for line-formation problems in atmospheres with high-speed flows is generalized to include the effects of continuum absorption and emission in the region of the line. The result is very simple, being expressed entirely in terms of known functions with the exception of one quantity of order unity, which is tabulated. Comparison with accurate numerical solutions for simple problems in plane-parallel geometry shows the approximation to be quite accurate in those regions of the atmosphere where the conditions for the validity of the approximation are satisfied. A three-dimensional version of the theory is given that applies to general geometries. Publication: The Astrophysical Journal Pub Date: June 1985 DOI: 10.1086/163232 Bibcode: 1985ApJ...293..258H Keywords: Approximation; Line Spectra; Opacity; Stellar Atmospheres; Stellar Spectra; Limits (Mathematics); Stellar Models; Velocity Distribution; Astrophysics full text sources ADS |

The moments Mn(r) ≡ 1/2 ∝2π0 dθ sinn θ I(r,θ) of the intensity I(r, θ) in free space surrounding a spherical object emitting radiation with an arbitrary directional dependence are shown to be exactly proportional to r-(n+1), n = 0, 1,….

view Abstract Citations (64) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A unified treatment of escape probabilities in static and moving media. I - Plane geometry. Hummer, D. G. ; Rybicki, G. B. Abstract An expression giving the escape probability for photons in a spectral line formed in a planar atmosphere with an arbitrary monotonic velocity law is derived and evaluated. For a small velocity gradient, the usual static result is recovered; for large velocity gradients the Sobolev result is obtained, but only at optical depths sufficiently large that the static part of the escape probability is negligible. Extensive numerical results for the escape-probability function for a constant velocity gradient are given for Doppler, Voigt (a = 0.001, 0.01) and Lorentz profiles. The use of these results for flows with nonconstant gradients is discussed. Publication: The Astrophysical Journal Pub Date: March 1982 DOI: 10.1086/159788 Bibcode: 1982ApJ...254..767H Keywords: Atmospheric Optics; Line Spectra; Radiative Transfer; Spectral Emission; Transition Probabilities; Doppler Effect; Escape Velocity; Flow Velocity; Integral Equations; Kernel Functions; Photons; Planar Structures; Voigt Effect; Astrophysics full text sources ADS |

view Abstract Citations (25) References (30) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Second-order escape probability approximations in radiative transfer Hummer, D. G. ; Rybicki, G. B. Abstract Second-order escape probability approximations make some allowance for the transfer of radiation between the point where a photon is created and that where it escapes or is absorbed. An approximation of this kind has recently been formulated by Puetter et al. (1982) for planar atmospheres of finite thickness, in the form of a first-order differential equation relating the integrated mean intensity to the source function. Two alternative normalizations to the one proposed by these authors are given, the first of these enforces global conservation of photons in each transition, and the second gives reasonably accurate results with much less computational effort than the first. These results, along with the first-order approximation and a second-order result of Ivanov (1972), are compared with accurate numerical results for static isothermal planar atmospheres. The maximum error for the photon-conserving approximation is less than 25 percent for Doppler and 7 percent for Lorentz broadening. Publication: The Astrophysical Journal Pub Date: December 1982 DOI: 10.1086/160560 Bibcode: 1982ApJ...263..925H Keywords: Line Spectra; Photons; Probability Theory; Radiative Transfer; Stellar Atmospheres; Stellar Radiation; Approximation; Escape; Optical Thickness; Astrophysics full text sources ADS | Related Materials (1) Erratum: 1983ApJ...271..888H

view Abstract Citations (39) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The effect of reflected and external radiation on stellar flux distributions Hummer, D. G. Abstract The effect of radiation emitted or scattered by circumstellar material, such as a stellar wind, into the stellar photosphere is investigated on the basis of a gray model atmosphere generalized to include the effects of an external radiation field and a surface boundary condition describing the reflection of a specified fraction, depending on the frequency, of the outgoing radiation. Substantial modifications both to the temperature and flux distributions are found. Publication: The Astrophysical Journal Pub Date: June 1982 DOI: 10.1086/160027 Bibcode: 1982ApJ...257..724H Keywords: Radiant Flux Density; Radiative Transfer; Stellar Atmospheres; Stellar Envelopes; Stellar Radiation; Stellar Winds; Atmospheric Models; Backscattering; Photosphere; Reflectance; Step Functions; Astrophysics full text sources ADS |

Analytical expressions are given for the coefficients, as a function of the Voigt parameter a, in the asymptotic expansions of the kernel functions K1(π), K2(π), M1(π), and M2(π) that describe the transfer of radiation scattered with complete redistribution over a Voigt profile.

We describe the techniques and the approximations used in extensive calculations for cross sections and reaction-rate parameters for electron-impact excitation of a number of heliumlike ions. All transitions involving the ground state and the $n=2$ states are considered. Calculations are made in the distorted-wave approximation using configuration-interaction wave functions to represent the target states. Autoionizing resonances in the scattering cross sections are included through bound-channel correlation-type functions and through quantum-defect-theory analysis of the reactance matrices. The resonances are shown to make considerable contributions to the cross sections and thereby, in many cases, to enhance the excitation-rate coefficients by a significant factor. This should have important consequences for practical applications in the analysis of laboratory and astrophysical plasmas. The accuracy of our approximations is also discussed.

Rational approximations for the kernel functions K1(τ), K2(τ), M1(τ) and M2(τ) that describe the transfer of radiation scattered with complete redistribution over Doppler and Lorentz profiles have been obtained from their series and asymptotic expressions by the techniques of Padé approximants with a maximum relative error of less than 10-4.