Author name code: hummer ADS astronomy entries on 2022-09-14 author:"Hummer, David G." ------------------------------------------------------------------------ Title: The MHD Equation of State with Post-Holtsmark Microfield Distributions Authors: Nayfonov, Alan; Däppen, Werner; Hummer, David G.; Mihalas, Dimitri Bibcode: 1999ApJ...526..451N Altcode: 1999astro.ph..1360N 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. Title: Recombination of helium-like ions - I. Photoionization cross-sections and total recombination and cooling coefficients for atomic helium Authors: Hummer, D. G.; Storey, P. J. Bibcode: 1998MNRAS.297.1073H Altcode: A new calculation of photoionization cross-sections is described for the ground and excited states of atomic helium up to principal quantum number n=25 and angular momentum quantum number l=5. These cross-sections are used to calculate total recombination and cooling coefficients for atomic helium for electron temperatures given by log(T)=1(0.2)4.4. A comparison of the threshold photoionization cross-sections obtained here with extrapolations of the highly accurate bound-bound oscillator strength calculations by Drake shows that the new calculations are in error by no more than 1 per cent. The accuracy of the photoionization cross-sections used by previous workers to derive recombination coefficients is also discussed. Title: VizieR Online Data Catalog: Recombination line intensities for hydrogenic ions (Storey+ 1995) Authors: Storey, P. J.; Hummer, D. G. Bibcode: 1995yCat.6064....0S Altcode: Line emissivities, effective recombination coefficients, opacity factors, departure coefficients and total recombination coefficients are calculated for hydrogenic ions with Z<=8.

There are 162 primary files labelled rZCTTTT.d, where Z=1,2,..,8 is the ionic charge C=a or b is the Case in the sense of Baker and Menzel (1938ApJ....88...52B), i.e. -- A for a nebula transparent to Lyman line radiation, -- B for an opaque nebula TTTT=0.01Te is a reduced temperature variable.

There are 9 values of electron densities (log(Ne)=2(1)10) for case A, and 13 (log(Ne)=2(1)14) for case B. The structure of these files is detailed in section 4 of the MNRAS paper.

There are 16 secondary files labelled eZC.d, where Z=1,2,..,8 is the ionic charge C=a or b is the Case in the sense of Baker and Menzel (1938ApJ....88...52B) which contain emissivities and total recombination coefficients.

These files are accompanied by an interactive data server intrat.f. Please refer to section 5 of the MNRAS paper which details the usage of the program.

CGS units are used, meaning that => densities are expressed in cm-3 => emissivities are expressed in erg.s-1.cm-3 (10-7W.cm-3) => effective recombination coefficients are expressed in cm3.s-1

(1 data file). Title: Radiative Transfer Authors: Hummer, D. Bibcode: 1995aelm.conf...35H Altcode: No abstract at ADS Title: Recombination line intensities for hydrogenic ions-IV. Total recombination coefficients and machine-readable tables for Z=1 to 8 Authors: Storey, P. J.; Hummer, D. G. Bibcode: 1995MNRAS.272...41S Altcode: Line emissivities, effective recombination coefficients, opacity factors, departure coefficients and total recombination coefficients are calculated for hydrogenic ions with Z<=8. Results are obtained for Cases A and B for n<=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 logN_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<=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. Title: An accelerated lambda iteration method for multilevel radiative transfer. III. Noncoherent electron scattering Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1994A&A...290..553R Altcode: 1994astro.ph..4019R Since the mass of the electron is very small relative to atomic masses, Thomson scattering of low-energy photons (hν<<m_e_c^2^) produces thermal Doppler frequency shifts that are much larger than atomic Doppler widths. A method is developed here to evaluate the electron scattering emissivity from a given radiation field which is considerably faster than previous methods based on straightforward evaluation of the scattering integral. This procedure is implemented in our multilevel radiative code (MALI), which now takes full account of the effects of noncoherent electron scattering on level populations, as well as on the emergent spectrum. Calculations using model atmospheres of hot, low-gravity stars display not only the expected broad wings of strong emission lines but also effects arising from the scattering of photons across continuum edges. In extreme cases this leads to significant shifts of the ionization equilibrium of helium. Title: Total Recombination and Energy Loss Coefficients for Hydrogenic Ions at Low Density for 10<T/E/Z/2<10/7K Authors: Hummer, D. G. Bibcode: 1994MNRAS.268..109H Altcode: Total recombination and energy-loss coefficients, including the free-free component, are calculated for Cases A and B from extract hydrogenic photoionization and free-free cross-sections, and are tabulated for log( Te/Z2)=1(0.2)7. Title: Algorithms for rapid evaluation of band transfer functions for linear molecules. Authors: Hummer, D. G.; Kutepov, A. A. Bibcode: 1994JQSRT..51..729H Altcode: Efficient algorithms are developed to compute transfer functions for bands in linear molecules and their application to the determination of non-LTE populations and cooling rates in planetary atmospheres are discussed. Title: NLTE model stellar atmospheres with line blanketing near the series limits. Authors: Hubeny, I.; Hummer, D. G.; Lanz, T. Bibcode: 1994A&A...282..151H Altcode: In this paper we study the influence of line-merging regions at the intermediate long-wavelength side of a continuum threshold on the computed model atmosphere structure and predicted spectrum. In order to model these regions sufficiently accurately, we have developed two concepts. First, we have extended the occupation probability formalism of Hummer and Mihalas to non-local thermodynamic equilibrium (non-LTE) plasmas. Second, in order to treat the very complicated opacity in the line merging region, we have generalized the concept of opacity distribution functions to treat non-LTE situations. All Rydberg states are consistently included within this framework, so that no arbitrary cutoff of high (LTE) levels is made. We have calculated several pure hydrogen models atmospheres for two effective temperatures, Teff = 20000 and 35000 K, and discussed the differences between models calculated with various treatments of the line merging. In particular, we have shown that the error in the predicted profiles of Balmer lines resulting from the neglect of line merging is typically of the order of 3-4%, while the errors in the far-UV portion of the Balmer continuum reaches 15-35%. The errors generally decrease with increasing effective temperature. At the same time, the internal accuracy of the models is shown to be about or below 0.5% for all predicted spectral features. We conclude that for interpreting current high-accuracy spectrophotometric observations models including the line merging are necessary, and that the formalism developed in this paper is capable of providing a sufficiently accurate and robust modeling technique. Title: Atomic data from the IRON project. I. Goals and methods. Authors: Hummer, D. G.; Berrington, K. A.; Eissner, W.; Pradhan, Anil K.; Saraph, H. E.; Tully, J. A. Bibcode: 1993A&A...279..298H Altcode: The IRON Project has the goal of computing on a large scale electron excitation cross sections and rates of astrophysical and technological importance, using the most reliable procedures curently available. Radiative transition probabilities and photoionization cross sections not known from other sources e.g. from the Opacity Project, will also be presented. Although the major effort will be for ions of the iron-group elements, other ions of astrophysical interest will also be included. In this introductory paper models and procedures to be used are summarized and the approximations are discussed. As an example of our computational procedures, typical results for fine structure transitions involving electron collisions with Fe XVIII ions and radiative data from Fe XVII are presented. Title: FOS Spectra of OB-type Stars in the LMC Authors: Voels, S. A.; Butler, K.; Gabler, R.; Groth, H. G.; Lennon, D. J.; Haser, S.; Hummer, D. G.; Husfeld, D.; Kudritzki, R. P.; Pauldrach, A.; Puls, J.; Baade, D.; Walborn, N. R.; Heap, S. R.; Garmay, C. D.; Conti, P. S.; Bohannan, B. Bibcode: 1992AAS...181.1908V Altcode: 1992BAAS...24.1151V As part of continuing project to determine the intrinsic stellar parameters (temperature, gravity, luminosity, chemical composition, radius, mass-loss rate) of early OB-type stars, we are observing several stars in the Large and Small Magellanic Clouds using the HST/FOS high resolution gratings (G130H, G190H) in the UV (1200 to 2200 Angstroms). This part of the UV spectrum contains several line profiles of C, N, O, and Si which when combined with ground based observations in the optical and NLTE stellar atmosphere models allow us to accurately determine the intrinsic stellar parameters of the star. In this poster, we will present the FOS spectra and preliminary model fits for AV-232 (Sk-80) and AV-488 (Sk-159) which are to be obtained Fall 1992. We will also present our latest work on the three stars (Sk-70D69, Sk-66D100, and Sk-68D137) obtained last summer. Title: An accelerated lambda iteration method for multilevel radiative transfer. II. Overlapping transitions with full continuum. Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1992A&A...262..209R Altcode: The ALI method has been generalized to include treatment of overlapping, active continuum transfer and overlapping lines. The linearity of the iterative equations is maintained in this method by consistent use of the psi operator rather than the lambda operator. The method has been applied to a sample problem of pure helium, which includes 23 levels, 31 lines, 22 continua, three stages of ionization, and electron scattering. Velocity fields of order of a few Doppler widths were also included by means of an observer's frame formulation. The convergence of the solutions was found to be comparable to that achieved in the previous pure line method. Title: Radiation-driven winds of hot luminous stars. X. The determination of stellar masses, radii and distances from terminal velocities and mass-loss rates. Authors: Kudritzki, R. -P.; Hummer, D. G.; Pauldrach, A. W. A.; Puls, J.; Najarro, F.; Imhoff, J. Bibcode: 1992A&A...257..655K Altcode: A new, purely spectroscopic method to determine masses, radii, and distances of massive, luminous hot stars is presented. This method is based on the theory of radiation-driven winds and uses terminal velocity, mass-loss rate and effective temperature as observational quantities determined from the spectrum. It is demonstrated that in situations where the distance is already known from other methods, masses can be determined from vinfinity and T(eff) with an accuracy of +/- 25 percent, which is a factor of two better than the classical method using the information obtainable from the quantitative analysis of photospheric absorption lines. These masses, which agree with those obtained from the spectroscopic values of log g, are systematically somewhat smaller than masses found from evolutionary calculations. An independent determination of radii and distances is possible, if good measurements of mass-loss rates can be carried out. Title: The Sobolev Approximation for Line Formation with Partial Frequency Redistribution Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1992ApJ...387..248H Altcode: Attention is given to the formation of a spectral line in a uniformly expanding infinite medium in the Sobolev approximation, with emphasis on the various mechanisms for frequency redistribution. Numerical and analytic solutions of the transfer equation are presented of 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 RII 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, which enters the rate equations, this dependence is weak. It is inferred 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. Title: Hot Stars and the Hubble Space Telescope Authors: Kudritzki, R. P.; Lennon, D. J.; Becker, S. R.; Butler, K.; Gabler, R.; Haser, S.; Hummer, D. G.; Husfeld, D.; Pauldrach, A. W. A.; Puls, J.; Voels, S.; Walborn, N. R.; Heap, S. R.; Bohannan, B.; Conti, P.; Garmany, C. D.; Baade, D. Bibcode: 1992ESOC...44..279K Altcode: 1992swhs.conf..279K No abstract at ADS Title: Recombination line intensities for hydrogenic ions. III. Effects of finite optical depth and dust. Authors: Hummer, D. G.; Storey, P. J. Bibcode: 1992MNRAS.254..277H Altcode: The effect on the recombination spectrum of hydrogen arising from: (1) finite optical thickness in the Lyman lines; (2) the overlapping of Lyman lines near the series limit; (3) the absorption of Lyman lines by dust or photoionization, and (4) the long-wave radiation emitted by dust is examined. Full account is taken of electron and heavy particle collisions in redistributing energy and angular momentum. It is seen that each of these deviations from the classical Case B leads to observable effects, and that dust influences the recombination spectrum in characteristic ways that may make possible new observational constraints on dust properties in nebulosities. On the basis of these calculations it is believed that the uncertainty in the determination of the helium-to-hydrogen abundance ratio in the universe may be larger than currently claimed. Title: Book-Review - Stellar Atmospheres Beyond Classical Models Authors: Crivellari, L.; Hubeny, I.; Hummer, D. G. Bibcode: 1991JBAA..101..364C Altcode: No abstract at ADS Title: The solution of radiative transfer problems in molecular bands without the LTE assumption by accelerated lambda iteration methods. Authors: Kutepov, A. A.; Kunze, D.; Hummer, D. G.; Rybicki, G. B. Bibcode: 1991JQSRT..46..347K Altcode: An iterative method based on the use of approximate transfer (or Λ) operators, which was designed initially to solve multilevel NLTE line formation problems in stellar atmospheres, is adapted and applied to the solution of the NLTE molecular band radiative transfer in planetary atmospheres. The matrices to be constructed and inverted are much smaller than those used in the traditional Curtis matrix technique, which makes possible the treatment of more realistic problems (including rotational NLTE, overlapping of lines in the bands and overlapping of bands with continuua) using relatively small computers. This technique converges much more rapidly than straightforward iteration between the transfer equation and the equations of statistical equilibrium (Λ-iteration). A test application of this new technique to the solution of NLTE radiative transfer problems for optically-thick and thin bands (the 4.3 μm CO2 band in the Venusian atmosphere and the 4.7 and 2.3 μm CO bands in the Earth's atmosphere) is described. Title: An accelerated lambda iteration method for multilevel radiative transfer. Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1991A&A...245..171R Altcode: A method is presented for solving multilevel transfer problems when nonoverlapping lines and background continuum are present and active continuum transfer is absent. An approximate lambda operator is employed to derive linear, 'preconditioned', statistical-equilibrium equations. A method is described for finding the diagonal elements of the 'true' numerical lambda operator, and therefore for obtaining the coefficients of the equations. Iterations of the preconditioned equations, in conjunction with the transfer equation's formal solution, are used to solve linear equations. Some multilevel problems are considered, including an eleven-level neutral helium atom. Diagonal and tridiagonal approximate lambda operators are utilized in the problems to examine the convergence properties of the method, and it is found to be effective for the line transfer problems. Title: Asymptotic expansions of the kernel functions for line formation with continuous absorption. Authors: Hummer, D. G. Bibcode: 1991JQSRT..45..211H Altcode: Asymptotic expressions are obtained for the kernel functions M∼2(τ,α,β) and K∼2(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. Title: Stellar atmospheres. Beyond classical models Authors: Crivellari, Lucio; Hubeny, I.; Hummer, D. G. Bibcode: 1991ASIC..341.....C Altcode: 1991QB809.N38......; 1991sabc.conf.....C No abstract at ADS Title: The Opacity Project and the Practical Utilization of Atomic Data Authors: Hummer, D. G. Bibcode: 1991ASIC..341..431H Altcode: 1991sabc.conf..431H No abstract at ADS Title: Computer Codes for Stellar Atmospheric Modeling Authors: Hummer, D. G.; Hubeny, I. Bibcode: 1991ASIC..341..119H Altcode: 1991sabc.conf..119H No abstract at ADS Title: Photospheres of Hot Stars. IV. Spectral Type O4 Authors: Bohannan, Bruce; Voels, Stephen A.; Hummer, David G.; Abbott, David C. Bibcode: 1990ApJ...365..729B Altcode: 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. Title: The Equation of State for Stellar Envelopes. IV. Thermodynamic Quantities and Selected Ionization Fractions for Six Elemental Mixes Authors: Mihalas, Dimitri; Hummer, D. G.; Mihalas, Barbara Weibel; Daeppen, Werner Bibcode: 1990ApJ...350..300M Altcode: 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. Title: Comparison of Stellar Parameters from Line Profile Analysis with those from Stellar Evolution Calculations Authors: Voels, S. A.; Bohannan, B.; Abbott, D. C.; Hummer, D. G. Bibcode: 1990BAAS...22..743V Altcode: No abstract at ADS Title: Quantitative spectroscopy of hot stars. Authors: Kudritzki, R. P.; Hummer, D. G. Bibcode: 1990ARA&A..28..303K Altcode: A review on the quantitative spectroscopy (QS) of hot stars is presented, with particular attention given to the study of photospheres, optically thin winds, unified model atmospheres, and stars with optically thick winds. It is concluded that the results presented here demonstrate the reliability of Qs as a unique source of accurate values of the global parameters (effective temperature, surface gravity, and elemental abundances) of hot stars. Title: Photospheres of Hot Stars. III. Luminosity Effects at Spectral Type O9.5 Authors: Voels, Stephen A.; Bohannan, Bruce; Abbott, David C.; Hummer, D. G. Bibcode: 1989ApJ...340.1073V Altcode: 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. Title: The Equation of State for Stellar Envelopes. III. Thermodynamic Quantities Authors: Daeppen, Werner; Mihalas, Dimitri; Hummer, D. G.; Mihalas, Barbara Weibel Bibcode: 1988ApJ...332..261D Altcode: The authors derive general formulae for the computation of the thermodynamic properties of a partially ionized (and/or dissociated) multicomponent gas in terms of second derivatives of the free energy with respect to temperature, volume, and occupation numbers. For the free energy used in previous work the authors give explicit analytical expressions for all derivatives required to construct the thermodynamic quantities. Representative results for several different thermodynamic properties of a hydrogen-helium plasma with N(He)/N(H) = 0.10 are presented as color plots. Title: The Equation of State for Stellar Envelopes. I. an Occupation Probability Formalism for the Truncation of Internal Partition Functions Authors: Hummer, D. G.; Mihalas, Dimitri Bibcode: 1988ApJ...331..794H Altcode: The authors have formulated a new equation of state for material in stellar envelopes, subject to the limits T ⪉ 107K and ρ ⪉ 10-2g cm-3. Under these conditions, the method of free energy minimization can be used to allow for nonideal effects; this method ensures thermodynamic consistency among the material properties. A chemical picture is used in which atomic and molecular particles retain their identities. An occupation probability formalism is used to represent the effects of the plasma in establishing a finite partition function, and attention is paid to representing these effects in a way that is consistent from the point of view of statistical mechanics. The use of the static screened Coulomb potential to calculate level shifts and to estimate the cutoff of the internal partition function is shown by elementary arguments to be 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 functions. Title: The Equation of State for Stellar Envelopes. II. Algorithm and Selected Results Authors: Mihalas, Dimitri; Dappen, Werner; Hummer, D. G. Bibcode: 1988ApJ...331..815M Altcode: 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. Title: Failure of Continuum Methods for Determining the Effective Temperature of Hot Stars Authors: Hummer, D. G.; Abbott, David C.; Voels, Stephen A.; Bohannan, Bruce Bibcode: 1988ApJ...328..704H Altcode: 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. Title: Recombination line intensities for hydrogenic ions. II. Case B calculations for C VI, N VII and O VIII. Authors: Storey, P. J.; Hummer, D. G. Bibcode: 1988MNRAS.231.1139S Altcode: The intensities of recombination lines formed in extended, optically thin, photoionized plasmas such as those found in PN, H II regions, and winds of certain hot stars are an important source of information on chemical abundances and can sometimes provide estimates of electron temperature. In this paper, the intensities of the 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 and Menzel. The computational procedure is explained. The intensities of lines formed by transitions n(u) - n(l) are tabulated for n(u) of 50 or less, n(l) of 29 or less, at log N(e) = 4(1)13 and 10 values of electron temperature in the interval 10,000 K to 500,000 K. Title: A Fast and Accurate Method for Evaluating the Nonrelativistic Free-free Gaunt Factor for Hydrogenic Ions Authors: Hummer, D. G. Bibcode: 1988ApJ...327..477H Altcode: 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. Title: Collisional-radiative switching - A powerful technique for converging non-LTE calculations Authors: Hummer, D. G.; Voels, S. A. Bibcode: 1988A&A...192..279H Altcode: A very simple technique has been developed to converge statistical equilibrium and model atmospheric calculations in extreme non-LTE conditions when the usual iterative methods fail to converge from an LTE starting model. The proposed technique is based on a smooth transition from a collision-dominated LTE situation to the desired non-LTE conditions in which radiation dominates, at least in the most important transitions. The proposed approach was used to successfully compute stellar models with He abundances of 0.20, 0.30, and 0.50; Teff = 30,000 K, and log g = 2.9. Title: C/He abundances in WC stars. Authors: Smith, Lindsey F.; Hummer, David G. Bibcode: 1988MNRAS.230..511S Altcode: Recombination theory has been used to determine the C/He abundances in 17 southern WC stars from spectra in the H and K bands. An LTE treatment of C II provides C/He values in the 0.04-0.3 range, by number, and maximum allowance for possible non-LTE effects yields values in the 0.1-0.7 range, by number. C/He is shown to decrease from WC4 to WC7. The ionization balance is found to be lower than in previous reports. He(+)/He(2+) values are between 2.6 and 3.0 for WC5-8 stars, increasing to 7 for WC9 stars. Continuum slopes and corrections for the emission line contributions to the H and K magnitudes are evaluated. Title: The Sobolev approximation for the line force and line source function in a spherically-symmetrical stellar wind with continuum opacity Authors: Puls, J.; Hummer, D. G. Bibcode: 1988A&A...191...87P Altcode: An expression for the force arising from diffuse line radiation is derived in the context of the Sobolev approximation for spectral line formation, as generalized by Hummer and Rybicki (1985) to include the effects of continuous opacity and emissivity in a general three-dimensional gas flow. This theory is then specialized to spherical symmetry and used to calculate the line source function and radiative pressure gradient in model stellar winds simulating essential aspects of the outflows' characteristic of O-stars and Wolf-Rayet objects. These models are compared both with models based on the Sobolev approximation without continuum opacity, and with numerically exact solutions obtained by using the co-moving frame method. The properties and methods of evaluating the special functions, Z (τ, β) and U (τ, β), arising in this work and in that of Hummer and Rybicki (1985) are discussed in the appendices. Title: Effective Temperatures and Gravities for O-Type Stars Determined from High Precision Line Profiles and Wind-Blanketed Model Atmospheres Authors: Bohannan, B.; Abbott, D. C.; Voels, S. A.; Hummer, D. G. Bibcode: 1988IAUS..132..127B Altcode: No abstract at ADS Title: Enhanced Helium Abundance in O Supergiants Authors: Voels, S. A.; Bohannan, B.; Abbott, D. C.; Hummer, D. G. Bibcode: 1987BAAS...19.1025V Altcode: No abstract at ADS Title: Recombination-line intensities for hydrogenic ions - I. Case B calculations for H I and He II. Authors: Hummer, D. G.; Storey, P. J. Bibcode: 1987MNRAS.224..801H Altcode: The relative intensities of H I and He II recombination lines are calculated, including full collisional effects, for a considerably larger range of temperature, density and principal quantum numbers than before. Case B of Baker & Menzel (1938) is assumed, and tables of line opacities are also presented to enable the assumption of negligible optical depth in all but the Lyman series to be checked. Collisional excitation of the n = 3 levels from both n = 1 and n = 2 states is considered, and is found to invalidate Case B theory in some conditions which depend on electron density and the Lyman-α escape probability. The regimes of temperature and particle density for which Case B is valid are discussed. Newly calculated collision strengths for the n = 1, 2 and 3 states of He+ are tabulated. Title: Luminosity Effects of Wind Blanketed Models for Late O-Type Stars Authors: Voels, S. A.; Bohannan, B.; Abbott, D. C.; Hummer, D. G. Bibcode: 1986BAAS...18..953V Altcode: No abstract at ADS Title: Photospheres of Hot Stars. II. an Analysis of Zeta Puppis Authors: Bohannan, B.; Abbott, D. C.; Voels, S. A.; Hummer, D. G. Bibcode: 1986ApJ...308..728B Altcode: High signal-to-noise ratio line profiles of ζ Puppis (O4f) obtained with a CCD camera are compared with theoretical model atmospheres that include radiation scattered back from the stellar wind onto the photosphere, an effect referred to as wind blanketing. The best-fit wind-blanketed model for the observed mass loss rate of 5×10-6M_sun;yr-1 yields the following properties of ζ Puppis: Teffective = 42,000±1500K, log g = 3.5±0.1, helium abundance by number, [Y] = 0.17±0.03. If one assumes a distance modulus of 8.2±0.4 mag, ζ Puppis has a bolometric magnitude of -10.1±0.4, a radius of 18±4 R_sun;, and current mass of 36 (+21, -15) M_sun;. Title: Rational approximations for the Holtsmark distribution, its cumulative and derivative. Authors: Hummer, D. G. Bibcode: 1986JQSRT..36....1H Altcode: The convergent series expansions of the Holtsmark distribution P(β), its cumulative Q(β), its derivative R(β) and the semiconvergent asymptotic series for these functions are used to calculate rational approximations for P, Q and R, which are valid for all positive β and have maximum errors of approximately 10-8, 10-9 and 10-7, respectively. Title: Non-Lte Analysis of Massive Stars in the Magellanic Clouds Authors: Gehren, T.; Husfeld, D.; Kudritzki, R. P.; Conti, P. S.; Hummer, D. G. Bibcode: 1986IAUS..116..413G Altcode: The massive stars of the Magellanic Clouds are of considerable current interest with regard to questions of initial mass function, star formation mechanisms, stellar evolution with mass loss and the chemical evolution of galaxies. The effective temperatures, surface gravities and helium abundances of 6 main sequence O-type stars, obtained by fitting non-LTE model atmospheres to high quality spectra, are presented. Title: Steps to a New Calibration of the Spectral Type - Effective Temperature Relationship for Early-Type Stars Authors: Bohannan, B.; Abbott, D. C.; Voels, S. A.; Hummer, D. G. Bibcode: 1986IAUS..116..111B Altcode: The spectral type - effective temperature calibration is a cornerstone of the understanding of massive stars and their environment. The authors have begun a new determination of the basic stellar parameters of massive stars with an analysis of ζ Pup (O4f). Title: Atoms in astrophysics. Authors: Burke, P. G.; Eissner, W. B.; Hummer, D. G.; Percival, I. C. Bibcode: 1986atas.book.....B Altcode: No abstract at ADS Title: Intrinsic parameters of hot blue stars Authors: Kudritzki, R. P.; Hummer, D. G. Bibcode: 1986IAUS..116....3K Altcode: Problems involved in the determination of the intrinsic parameters of hot blue stars are discussed. Discrepancies between effective temperatures obtained by calibrating against spectral type and temperatures obtained using measured angular diameters and flux distributions are examined using Zeta Puppis as an example. The problems posed by wind blanketing, metal opacity and non-LTE line blanketing, and radiation-driven winds are addressed, and methods to handle these problems are discussed. The results of applying some of these techniques to a number of Galactic O3 stars in the Eta Car region are presented and discussed. Title: Rotational relaxation of the 0001 level of CO2 including radiative transfer in the 4.3-μm band of planetary atmospheres. Authors: Kutepov, A. A.; Hummer, D. G.; Moore, C. B. Bibcode: 1985JQSRT..34..101K Altcode: The paper contains accurate numerical solutions of the problem of rotational relaxation of the C12O216 molecules in the 0001 level in conjunction with the transfer of radiation in the lines of the 0001-0000 transition of the 4.3-μm band in a plane-parallel isothermal atmosphere consisting of pure carbon dioxide. This model atmosphere is illuminated by the sun, and has a pressure profile given by the barometric formula. The band lines are assumed to be nonoverlapping, with line shapes described by Voigt profiles depending on temperature and pressure. The transfer problem has the form of a multiplet with a large number of lines with a common lower level. The most recent data on the rotational transition probabilities due to molecular collisions have been used. This work may be viewed as the first attempt to model the formation of the rotational populations of a vibrational level of CO2 in the upper atmospheres of Venus and Mars. Title: A radiation driven stellar wind model atmosphere for the Wolf-Rayet binary V444 Cygni. Authors: Pauldrach, A.; Puls, J.; Hummer, D. G.; Kudritzki, R. P. Bibcode: 1985A&A...148L...1P Altcode: Using the stellar parameters of the WN5 component of the eclipsing binary V 444 Cygni determined by Cherepashchuk et al. (1984) from multi-color light curves, and employing an improved theory of radiatively-driven stellar winds, models have been calculated which yield an extended, supersonically expanding photosphere, with values close to those observed for the photospheric radius, the mass-loss rate and the terminal velocity. The radial distributions of velocity and density are also in close agreement with those obtained by Cherepashchuk et al. This is regarded as strong evidence that the basic observational features of WR-stars can be reproduced by radiatively-driven wind theory if some previous simplifications in the theory are dropped and correctly determined stellar parameters are used. Title: Photospheres of hot stars. I. Wind-blanketed model atmospheres. Authors: Abbott, D. C.; Hummer, D. G. Bibcode: 1985ApJ...294..286A Altcode: Preliminary to an extensive and detailed comparison of improved non-LTE photospheric models with observations of hot stars made with high photometric accuracy, non-LTE stellar atmospheres are constructed which account for the radiation reflected back onto the photosphere by line and electron scattering from the wind. The effects of this 'wind blanketing' on the spectrum and internal structure of the atmosphere are given for an example with an effective temperature T(eff) of 42,000 K, and a wide range of wind density, gravity, and model assumptions. Particular attention is given to the problem of determining T(eff). Careful analysis of methods currently used to determine T(eff) from continuous flux distributions, with and without interferometric angular diameters, shows them to be unreliable in practice. Line profiles continue to provide a legitimate means of determining T(eff) but only when their dependence on gravity and mass loss is included. For the more luminous OB stars spectral classification is truly three-dimensional, with the mass loss rate, gravity, and effective temperature all playing nearly equal roles in specifying the observed spectrum. Title: The Sobolev approximation for line formation with continuous opacity Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1985ApJ...293..258H Altcode: 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. Title: The Line Spectrum of Zeta Puppis With High Photometric Accuracy Authors: Hummer, D. G.; Abbott, D. C.; Bohannan, B. Bibcode: 1984BAAS...16..509H Altcode: No abstract at ADS Title: A family of angle-moments proportional to r-n, n = 1,2,..., in free space. Authors: Hummer, D. G. Bibcode: 1984JQSRT..31..283H Altcode: The moments Mn(r) 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,.... Title: Book-Review - Atoms in Astrophysics Authors: Burke, P. G.; Eissner, W. B.; Hummer, D. G.; Percival, I. C.; Domke, H. Bibcode: 1984AN....305...58B Altcode: No abstract at ADS Title: The specific luminosity of a three-dimensional medium in terms of the escape probability Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1983ApJ...274..380R Altcode: Radiation emitted by a three-dimensional medium and received by a distant observer is expressed in terms of the specific luminosity, which can be evaluated in terms of escape probability functions for the medium. This approach is simpler in many cases than an integration over the plane of the sky, because it allows any symmetries of the problem to be more fully exploited. Integral theorems are derived involving averaged escape probabilities that generalize a theorem of Irons. Analogous results are presented for a three-dimensional line-emitting region in the Sobolev (or large velocity gradient) limit. To illustrate the theory, results for expanding spherical media and differently rotating thin disks are discussed. Computationally useful results for several related escape probability functions are given. Title: Erratum - Second-Order Escape Probability Approximations in Radiative Transfer Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1983ApJ...271..888H Altcode: No abstract at ADS Title: The generalized exponential-integral V(x,y) = ∫1 exp(-xt)ln(t+y)dt/t and computer algorithms for y = 0 and y = 1. Authors: Hummer, D. G. Bibcode: 1983JQSRT..30..281H Altcode: No abstract at ADS Title: Radiative transfer problems in planetary nebulae Authors: Hummer, D. G. Bibcode: 1983IAUS..103..211H Altcode: Models of radiative transfer in planetary nebulae (PN) based on UV observations such as those obtained with the IUE are surveyed. The formation of resonance lines is considered, taking the effects of frequency redistribution, dust absorption, the expansion velocity of the PN shell, and the inhomogeneous distribution of the PN gas into account. The casee of the He II Lyman-alpha resonance line, which is involved in the Bowen fluorescent mechanism, is explored in detail. The treatment of continuum transfer in PN models is discussed, and the significance of a quasi-diffusion (or variable-Eddington-factor) analysis of PN dust as (Petrosian and Dana, 1980) a bridge between UV spectra and IR features is indicated. Title: Atoms in Astrophysics, 1983 Authors: Burke, P. G.; Eissner, W. B.; Hummer, D. G.; Percival, I. C. Bibcode: 1983aia..conf.....B Altcode: 1983atas.book.....B Selected topics in the atomic and nuclear physics of astronomical objects are examined in reviews contributed in honor of M. J. Seaton. Topics explored include low-energy electron collisions with complex atoms and ions, numerical methods for asymptotic solutions of scattering equations, collisions between charged particles and highly excited atoms, proton-impact excitation of positive ions, long-range interactions in atoms and diatomic molecules, applications of quantum defect theory, and electron-ion processes in hot plasmas. Consideration is given to the development and application of the atomic-data-calculation computer system at University College, London, planetary nebulae, and forbidden atomic lines in auroral spectra. Title: Wind-Blanketed Stellar Atmospheres Authors: Abbott, D. C.; Hummer, D. G. Bibcode: 1983IAUS..103..546A Altcode: No abstract at ADS Title: Book-Review - Atoms in Astrophysics Authors: Burke, P. C.; Eissner, W. B.; Hummer, D. G.; Percival, I. C. Bibcode: 1983Sci...221..946B Altcode: No abstract at ADS Title: Atoms in Astrophysics Authors: Burke, P. G.; Eissner, W. B.; Hummer, D. G.; Percival, I. C.; Osterbrock, Donald E. Bibcode: 1983PhT....36h..67B Altcode: No abstract at ADS Title: Second-order escape probability approximations in radiative transfer Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1982ApJ...263..925H Altcode: 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. Title: On redistribution and the equations for radiative transfer Authors: Cooper, J.; Ballagh, R. J.; Burnett, K.; Hummer, D. G. Bibcode: 1982ApJ...260..299C Altcode: The derivation of the equations of statistical equilibrium are outlined, starting from the quantum density-matrix equations, drawing particular attention to the approximations and assumptions used in the development of tractable expressions. Then, using the quantum-fluctuation-regression theorem, emission and absorption coefficients are obtained for multilevel atomic systems which are nondegenerate except for m-substates. These coefficients are valid to first order in the incident intensity. Possible extensions to higher intensity broadband incoherent fields are suggested. Title: Wind-Blanketed Stellar Atmosphere Authors: Hummer, D. G.; Abbott, D. C. Bibcode: 1982BAAS...14..914H Altcode: No abstract at ADS Title: The effect of reflected and external radiation on stellar flux distributions Authors: Hummer, D. G. Bibcode: 1982ApJ...257..724H Altcode: 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. Title: A unified treatment of escape probabilities in static and moving media. I - Plane geometry. Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1982ApJ...254..767H Altcode: 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. Title: The WO Wolf-rayet stars. Authors: Barlow, M. J.; Hummer, D. G. Bibcode: 1982IAUS...99..387B Altcode: The intensities of all the recombination lines in the dereddened spectrum of Sand 3 are measured and compared with theoretical recombination line predictions for the relevant ions as calculated by Hummer and Storey (1982). The results for the case where the electron temperature is 200,000 K and the density is 10 to the 11th/cu cm are found to have given an excellent fit to the observed relative line intensities of transitions within He II, C IV, C V, O VI, O VII, and O VIII, covering upper levels as low as 3 and as high as 15 over the wavelength range 1170-7100 A. The derived relative abundances (by number), normalized to He(2+) = 100, are presented. The surface chemical composition of Sand 3 (by mass) is therefore H:He:C:O = 0:38:54:8. Since the mass fraction of oxygen in a ZAMS star is only 1%, these results suggest enhancement of oxygen through a alpha-particle capture by carbon in addition to the expected enhancement of carbon by the triple-alpha process. Title: The infrared recombination-line spectra of Wolf-Rayet stars Authors: Hummer, D. G.; Barlow, M. J.; Storey, P. J. Bibcode: 1982IAUS...99...79H Altcode: Effective recombination coefficients have recently been calculated for recombination lines of He I, He II and C IV (among other ions with up to three electrons) for densities and temperatures appropriate for Wolf-Rayet atmospheres. These have been applied to recently obtained infrared spectra of Gamma Vel in order to derive the He(+)/He(+2) and C(+4)/He(+) + He(+2) ratios. Title: High order asymptotic expansions of the four kernel functions for line formation with the Voigt profile. Authors: Hummer, D. G. Bibcode: 1982JQSRT..27..569H Altcode: No abstract at ADS Title: Rate coefficients for electron impact excitation of helium-like ions Authors: Pradhan, A. K.; Norcross, D. W.; Hummer, D. G. Bibcode: 1981ApJ...246.1031P Altcode: The distorted wave (DW) approximation is used to solve the e(+) ion scattering problem. It is found that for He-like ions the DW approximation, which neglects coupling between states other than the initial and the final, compares extremely well with the more elaborate close-coupling approximation. Rate parameters gamma(T) are presented for all 10 transitions involving the ground state and the n=2 states, for He-like Be, C, O, Ne, Si, Ca, and Fe. The rate parameters were obtained from collision strengths calculated and fitted to expressions in energy, separately in a number of different energy intervals. In terms of the variation with temperature T, the results gamma(T), may be divided into three categories corresponding to forbidden, intercombination, and allowed transitions. Title: Radiative Recombination Coefficients at Stellar Densities Authors: Hummer, D. G.; Storey, P. J. Bibcode: 1981BAAS...13..793H Altcode: No abstract at ADS Title: Line Formation in Accretion Disks Authors: Carroll, D.; Hummer, D. G.; Rybicki, G. B. Bibcode: 1981BAAS...13Q.818C Altcode: No abstract at ADS Title: Cross sections and excitation rates for electron collisions with heliumlike ions Authors: Pradhan, A. K.; Norcross, D. W.; Hummer, D. G. Bibcode: 1981PhRvA..23..619P Altcode: 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. Title: Expressions for the computer-evaluation of the four kernel functions for line formation with Doppler and Lorentz profiles. Authors: Hummer, D. G. Bibcode: 1981JQSRT..26..187H Altcode: No abstract at ADS Title: Optical observations of the ultrahigh-excitation Wolf-Rayet star Sanduleak 3. Authors: Barlow, M. J.; Blades, J. C.; Hummer, D. G. Bibcode: 1980ApJ...241L..27B Altcode: Recombination lines of O VII, O VIII, and C V have been identified in the optical spectrum of an O VI Wolf-Rayet star, representing the first non-X-ray detection of these ions in astronomical spectra and implying excitation energies in excess of 800 eV. Rapid variations on a time scale of about 150 s have been observed in the profile of one of the O VII lines. Title: Escape-Probability Method for Flows of Arbitrary Velocity Amplitude Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1980BAAS...12..798H Altcode: No abstract at ADS Title: Energy loss by resonance line photons in an absorbing medium Authors: Hummer, D. G.; Kunasz, P. B. Bibcode: 1980ApJ...236..609H Altcode: The mean path length of photons undergoing repeated scatterings in media of large optical thickness is calculated from accurate numerical solutions of the transfer equation including the effect of frequency redistribution characteristic of combined Doppler and natural broadening. Energy loss by continuous absorption processes, such as ionization or dust absorption, is discussed, and asymptotic scaling laws for the energy loss, the mean path length, and the mean number of scatterings are inferred from the numerical data. Title: A generalization of the Sobolev method for flows with nonlocal radiative coupling. Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1978ApJ...219..654R Altcode: The escape-probability technique of Sobolev for solving radiative transfer problems in moving atmospheres is extended to treat flows in which the line-of-sight component of the flow velocity is not monotonic. A completely general geometrical configuration and flow velocity field are considered; an integral equation is derived for configurations in which a surface is intersected an arbitrary number of times. For the case of just two intersections, it is shown that an iterative solution always converges rapidly. Numerical results for inverse power-law velocity fields demonstrate the magnitude of the radiative coupling between distant parts of the atmosphere. Title: Atmospheres of central stars. Authors: Hummer, D. G. Bibcode: 1978IAUS...76..171H Altcode: Atmospheric models that are of relevance to the central stars of planetary nebulae are summarized. The extent to which these models accord with the observations of both nebulae and central stars is examined. Particular attention is given to the significance of the very high Zanstra temperature implied by the nebular He II line at 4686 A, and to the discrepancy between the Zanstra He II temperature and the considerably lower temperatures suggested by the appearance of the visual stellar spectrum for some of these objects. Title: Erratum and Addendum: "Solution of the comoving-frame equation of transfer in spherically symmetric flows. IV. Frequency-dependent source functions for scattering by atoms and electrons [Astrophys. J., Vol. 210, p. 419 - 433 (1976)]. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1977ApJ...214..337M Altcode: No abstract at ADS Title: Solution of the comoving-frame equation of transfer in spherically symmetric flows. IV. Frequency-dependent source functions for scattering by atoms and electrons. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1976ApJ...210..419M Altcode: A numerical method is presented of solving the radiative transfer equation in the comoving frame of a spherically symmetric expanding atmosphere in which both the line and the electron-scattering source function can depend on frequency (i.e., when there is partial frequency redistribution in the scattering process). This method is used to assess the adequacy of various assumptions regarding frequency redistribution in the comoving frame and to discuss the effects of electron scattering more accurately than previously possible. The methods developed here can be used in realistic model atmospheres to account for the (major) effects of electron scattering upon emergent flux profiles. Title: Escape Probability Method for Line Transfer in Accelerating Inflows and Decelerating Outflows. Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1976BAAS....8R.546H Altcode: No abstract at ADS Title: Solution of the Comoving-Frame Equation of Transfer in Spherically Symmetric Flows. III. Effect of Aberration and Advection Terms Authors: Mihalas, Dimitri; Kunasz, P. B.; Hummer, D. G. Bibcode: 1976ApJ...206..515M Altcode: We investigate the importance of the advection and aberration terms, which are of order V/c, in the comoving-frame transfer equation in spherical geometry. Characteristic trajectories are found which reduce the spatial derivatives to a perfect differential, and a generalization of the numerical procedure developed in the earlier papers of this series that permits the integration of the transfer equation on these characteristics is presented. For cases in which V/c 1, a perturbation solution is developed which reduces the problem to that solved in the first paper in this series. For velocities of the form V(r) r (n = 0, 1, 2), it is shown that the magnitude of the effects arising from the advection and aberration terms is about 5 V/c relative to the solution with these terms omitted. In stellar winds V/c <% 0.01; hence we conclude that aberration and advection terms may safely be ignored, and that consideration of the Doppler-shift term alone is adequate in the computation of spectra from such expanding atmospheres. Subject headings: radiative transfer - stars: atmospheres Title: Resonance-line transfer with partial redistribution. VIII. Solution in the comoving frame for moving atmospheres. Authors: Mihalas, D.; Shine, R. A.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1976ApJ...205..492M Altcode: The effects of partial frequency redistribution in the scattering process for lines formed in moving atmospheres are analyzed using a general method that allows the transfer equation to be solved in the comoving frame of the gas. The same chromospheric and atomic model studied by Cannon and Vardavas (1974) is employed in the calculations, but a depth scale with logarithmically spaced points is adopted. It is found that in both static and moving atmospheres, the profiles obtained with complete and partial frequency redistribution are virtually identical. The large differences in profiles obtained by Cannon and Vardavas when they used complete and partial redistribution are shown to be spurious (and physically unreal) effects resulting from angle averaging in the observer's frame instead of the comoving frame. Title: Solution of the comoving-frame equation of transfer in spherically symmetric flows. II. Picket-fence models. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1976ApJ...203..647M Altcode: To examine the effect of the radial flow of atmospheric material on the temperature distribution in a stellar atmosphere, a picket-fence model with Gaussian lines is formulated and solved numerically in the comoving frame of the gas, which is assumed to move with a prescribed velocity law. Extensive results have been obtained for both static and dynamical models, with planar and moderately extended spherical geometries. For static models, the effect of lines on the temperature distribution is virtually independent of extension. When a large-scale velocity field is imposed, significant surface heating and additional back-warming are found; the magnitude of these effects increases with the extension of the atmosphere. If a significant flow velocity persists to sufficient depth, the enhanced escape probability can lead to a cooling in the deeper layers, which competes with the back-warming. The results obtained here suggest that the deposition of energy arising from the intrusion of line opacity into the continuum, caused by velocity gradients, could influence the dynamics of the flow.- Title: Line Formation in Expanding Atmospheres (review Paper) Authors: Hummer, D. G. Bibcode: 1976IAUS...70..281H Altcode: No abstract at ADS Title: Migration of excitation in transfer of spectral-line radiation. Authors: Hummer, D. G.; Kunasz, P. B. Bibcode: 1976JQSRT..16...77H Altcode: A simple mathematical model is developed for the transfer of energy through a gas by the combined effect of radiative transfer and migration of excitation. The excitation is carried through the gas by a succession of atoms which experience resonant excitation exchange; it thus appears to random walk through the medium. The theory developed here is valid when the distance traveled by an atom while excited is much larger than the typical distance at which two atoms can exchange excitation (roughly one millionth cm). The model is expressed in terms of a pair of coupled transport equations for the intensity of radiation and the density of excited atoms, which are solved by means of a generalized discrete-ordinate technique. Extensive numerical results are obtained and discussed in terms of characteristic lengths for the various transfer processes. Substantial effects of migration are seen in both the distribution of excited atoms near the cell windows and the line profile of the emergent radiation for typical laboratory conditions. Title: Solution of the comoving-frame equation of transfer in spherically symmetric flows. I. Computational method for equivalent-two-level-atom source functions. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1975ApJ...202..465M Altcode: A method for solving the line-formation problem using the full comoving-frame formulation of the radiative-transfer equation is presented for the case of spherically symmetric atmospheres expanding with arbitrarily large velocities. A stable differencing scheme and a frequency-by-frequency elimination procedure are developed to solve the partial differential equations that describe the radiation field in the comoving frame. It is noted that this method allows computation of the radiation field from a given model atmosphere which must specify the depth dependence of all the relevant physical variables. Numerical results obtained for several models involving line formation by two-level atoms, electron scattering, and continuous absorption are discussed which simulate situations in the stellar winds of hot stars and similar objects. The force exerted by radiation on the gas is examined in a number of situations, and flux profiles are described for very high-velocity flows with very weak or nonexistent continuum and electron-scattering opacities. It is concluded that the mechanism proposed by Noerdlinger and Rybicki (1974) for the destruction of radially driven envelopes in planar geometries becomes inoperative even in the case of slightly extended spherical configurations. Title: Theory of extended stellar atmospheres. II. A grid of static spherical models for O stars and planetary nebula nuclei. Authors: Kunasz, P. B.; Hummer, D. G.; Mihalas, D. Bibcode: 1975ApJ...202...92K Altcode: Spherical static non-LTE model atmospheres are presented for stars from 30 to 60 solar masses at various points on their evolutionary tracks, and for some nuclei of planetary nebulae at two points of a modified Harman-Seaton sequence. The method of Mihalas and Hummer was employed, which uses a parametrized radiation force multiplier to simulate the force of radiation arising from the entire line spectrum. However, in the present work the density structure computed in the LTE models was held fixed in the calculation of the corresponding non-LTE models; in addition, the opacity of an average light ion was taken into account. The effects of sphericity, as distinct from those arising from a density structure modified by a large radiation force, were investigated by computing a few planar models using the same parametrized radiation force multiplier as for the spherical models. Extensive tables are given of monochromatic magnitudes, continuum jumps and gradients, Stroemgren-system colors, monochromatic extensions, and the profiles and equivalent widths of the hydrogen lines for all models. Title: Solution of the Co-Moving Frame Equation of Transfer in Spherically Symmetric Flows. II. Picket-Fence Models. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1975BAAS....7..449M Altcode: No abstract at ADS Title: Solution of the Co-moving Frame Equation of Transfer in Spherically Symmetric Flows. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1975BAAS....7R.256M Altcode: No abstract at ADS Title: Solution of the co-moving frame equation of transfer in spherically symmetric flows. Authors: Mihalas, D.; Kunasz, P. B.; Hummer, D. G. Bibcode: 1975BAAS....7..256M Altcode: No abstract at ADS Title: A note on the ' peaking effect ' in spherical-geometry transfer problems Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1975MNRAS.170..423R Altcode: This note presents evidence that the claims advanced by Wilson, Tung & Sen regarding the adequacy of Wilson & Sen's half-range moment method for treating the outward peaking of the radiation field in a spherical system are unjustified. In particular, the emergent intensity obtained by Wilson et al. is shown to be negative for 0 < <0.5 and greatly in error for larger values of . A discussion is presented of the essential indeterminacy of the Wilson-Sen half-range method. It is suggested that the good values obtained by Wilson et al. for the mean intensity and the Eddington factor arise from their choice of the arbitrary function A(r) to include the known asymptotic forms of the source function. Title: Theory of extended stellar atmospheres. I. Computational method and first results for static spherical models. Authors: Mihalas, D.; Hummer, D. G. Bibcode: 1974ApJS...28..343M Altcode: A method is presented that makes possible, for the first time, the calculation of extended spherical non-LTE model stellar atmospheres in hydrostatic and radiative equilibrium. This method is a generalization of the complete-linearization technique of Auer and Mihalas. Models have been obtained for a star with 60 solar masses, 1,000,000 solar luminosities, and 24 solar radii, whose atmosphere is characterized by an effective temperature of 39,500 K and a surface gravity log g = 3.45, i.e., with a spectral type near O6. These models are differentiated by the magnitude and radial dependence of a radiation force multiplier that is inserted into the equation of hydrostatic equilibrium to simulate the effect of radiation force on opacity sources which have not been included explicitly in the calculation. Models have been obtained very close to the limit at which the radiation force and gravity balance; as this condition is approached, the atmospheres become more and more extended. Title: Rate Coefficients for Electron Excitation of the First Resonance Transition in h, Li, Na, Ca, Ca', and Ba' Calculated from Experimental Data Authors: Crandall, D. H.; Dunn, G. H.; Gallagher, A.; Hummer, D. G.; Kunasz, C. V.; Leep, D.; Taylor, P. O. Bibcode: 1974ApJ...191..789C Altcode: By fitting cubic splines augmented with special functional forms for low and high energies to cross-section data determined experimentally at discrete values of the electron energy, we obtain an interpolation that can be visually inspected and adjusted to prevent the appearance of spurious features. The familiar integral expressing the rate coefficient in terms of the cross-sections can then be evaluated with no further approximation. This procedure is applied here to cross-section data for the first resonance transition of H, Li, Na, Ca, Ca +, and Ba +. The resulting collisional de-excitation rate coefficient, from which the excitation-rate coefficient can be determined by the detailed balance relation, is tabulated in each case for electron temperatures in the interval from 1O to 1O K and is expressed in terms of a Chebyshev expansion valid for this range of temperature. Subject heading: transition probabilities Title: Light ions of astrophysical interest-radiative transition probabilities for C m, N iv, 0 V and NE VII Authors: Hummer, D. G.; Norcross, D. W. Bibcode: 1974MNRAS.168..263H Altcode: Multiconfiguration calculations of energy levels and bound-bound radiative transition probabilities for four astrophysically important ions of the Be isoelectronic sequence are presented. Convergence of the results with respect to the number of configurations included in the expansion is discussed. Results are presented for all ions from models with I I and 16 configurations including transition probabilities for 24 allowed electric dipole transitions. Some significant differences with previous calculations are obtained. The theoretical lifetimes are generally in good accord with those obtained by the beam-foil technique, and the results of other ab initio calculations. Title: Some Observational Implications of Extended Static O-Star Model Atmospheres Authors: Mihalas, Dimitri; Hummer, David G. Bibcode: 1974ApJ...189L..39M Altcode: Some results and observational implications are presented for the first extended spherical non-LTE model atmospheres in hydrostatic and radiative equilibrium. These models all correspond to a star with = 60 , L = 1.25 X 106 Lo and R = 24 , with an effective temperature Te 39,500 K and surface gravity log g 3.45 (spectral type near 06). They are differentiated by the magnitude and radial dependence of a radiation-force multiplier , inserted into the equation of hydrostatic equilibrium, to simulate the effect of radiation force on opacity sources (e.g., lines) that have not been included in the calculations. It has been possible to obtain models very close to the limit at which the radiation force balances the gravity. Hydrogen and helium (`7= 0.1) constitute the gas; siz hydrogen lines are treated explicitly. These models show La in emission, the lower Balmer lines in absorption, the Balmer jump in absorption, and both infrared and ultraviolet excesses relative to the visual. Continuum jumps and gradients, -system colors, and equlvalent widths of Ha, Hp, and H are tabulated and discussed briefly. Subject headings: atmospheres, stellar - radiative transfer Title: Radiative transfer in spherically symmetric systems-III. Fundamentals of line formation Authors: Kunasz, P. B.; Hummer, D. G. Bibcode: 1974MNRAS.166...19K Altcode: A generalization of the variable Eddington factor method is presented hat makes possible the solution of line formation problems in extended spherical atmospheres whose constitutive properties depend on radius in an arbitrary way. Extensive numerical results for Doppler broadening in models with power law opacities (n = 0, 2, 3) are presented and interpreted. Very substantial deviations are found from the solutions of sanalogous plane-parallel models. The single-flight escape probability is sderived for a general opacity law and is shown to exceed that for an sanalogous plane-parallel slab by no more than a factor of approximately stwo for Doppler broadening, or three- halves for Lorentz broadening. sHowever, it is shown that each time a photon is scattered, it has a sprobability greater than one-half of ending its flight at a radius slarger than that at which it was emitted. This effect is peculiar to sspherical geometries and may be important in aiding the escape of sphotons from optically thick systems. Finally the effects of dilution sare considered and some properties of the infinite radius, finite soptical depth models are inferred. An appendix contains the solution of sthe line transfer problem for a homogeneous sphere by the kernel-approximation method. Title: Radiative transfer in spherically symmetric systems-IV. Solution of the line transfer problem with radial velocity fields Authors: Kunasz, P. B.; Hummer, D. G. Bibcode: 1974MNRAS.166...57K Altcode: A numerical procedure is presented for solving the line transfer problem with complete redistribution in spherically symlnetric atmospheres in which the radial velocity is an arbitrary function of radius, limited by spractical considerations to maximum velocities a few times the mean thermal velocity. In this procedure the transfer equation, written in the observer's frame, is differenced along rays and the resulting very large set of coupled linear equations is cast into the novel form, proposed by Rybicki, that allows for extremely rapid solution of the system. Numerical results are discussed for three sequences of models, two with linear velocity laws and one with constant velocities, in which sthe effect of the transverse velocity gradient is demonstrated. It is sshown that velocities as small as of the mean thermal velocity produce sobservable asymmetries in the flux profile. Title: Informal remarks on a problem in astrophysics Authors: Hummer, D. Bibcode: 1973NucIM.110..421H Altcode: No abstract at ADS Title: Analyses of light-ion spectra in stellar atmospheres. Authors: Mihalas, D.; Hummer, D. G. Bibcode: 1973ApJ...179..827M Altcode: An analysis of the N iii emission lines in 0 stars has been carried out on the basis of a detailed solution of the coupled statistical-equilibrium and transfer equations for a multiline, multilevel, multi-ion ensemble. Our calculations, using static, plane-parallel models reproduce successfully the observed emission at AA4634, 4640, 4641 (3p 2P -3d 2D) and absorption at AA4097, 4103 (3s 2S-3p 2P ). The multiplet is found to come into emission at the observed temperature for both main-sequence and low-gravity objects. The equivalent widths of the emission lines agree very well with those measured for the class of Of stars thought to have compact atmospheres, i.e., those classified as O((f)) by Walborn. In these stars the basic physical mechanism responsible for this phenomenon is the overpopulation of 3d by means of dielectronic recombinations from the low-lying 2s2p(1P )3d autoionizing states with cascades 3d 3p. The 3p state is drained by the "two-electron jumps" coupling 3p to the 2s2p2 (25, 2p, 2D) states, thus preventing emission in the 3s-3p lines. The possible importance of the Swings mechanism to the fully developed Of stars (in Walborn's sense) is pointed out, and the irrelevance of the Bowen mechanism to all Of stars is firmly demonstrated. The fact that the N iii emission lines can be produced in static nonextended atmospheres in radiative equilibrium has the far-reaching significance that the presence of emission lines in a spectrum is not in itself sufficient evidence for the existence of a stellar chromosphere (i.e., an extended, nonradiatively heated region). Subject headings: atmospheres, stellar - emission-line stars line formation Of-type stars Title: Interpretation of the spectra of planetary nebulae. Authors: Hummer, D. G.; Seaton, M. J. Bibcode: 1973LIACo..18..225H Altcode: 1973MSRSL...5..225H No abstract at ADS Title: Formation of Spectral Lines in Spherical Stellar Atmospheres. Authors: Kunasz, P.; Hummer, D. G. Bibcode: 1973BAAS....5...11K Altcode: No abstract at ADS Title: Numerical evaluation of the formal solution of radiative transfer problems in spherical geometries. Authors: Hummer, D. G.; Kunasz, C. V.; Kunasz, P. B. Bibcode: 1973CoPhC...6...38H Altcode: No abstract at ADS Title: ON THE N III λλ4640, 4097 LINES IN Of STARS Authors: Mihalas, Dimitri; Hummer, D. G.; Conti, Peter S. Bibcode: 1972ApJ...175L..99M Altcode: Detailed calculations based on non-LTE plane-parallel model atmospheres show that the N iii emission lines at XX4634, 4640, 4641 observed in Of stars are produced primarliy by dielectronic recombination to 3d 3D followed by the 3d-3p transition in a compact atmosphere. The 3P state in turn is drained by two-electron transitions to the 2p2 levels. We find that X4640 is in emission for Teff < 37,0000 K for main-sequence objects (55,0000 K for giants), while X4097 remains strongly in absorption, in accordance with observations. The calculated equivalent widths are in substantial agreement with the observed values for those Of stars with He II X4686 in absorption, i.e., for those objects designated by Walborn as O((f)). Title: The Transfer of Linearly Polarized Radiation in Extended Atmospheres. Authors: Cassinelli, J. P.; Hummer, D. G. Bibcode: 1971BAAS....3Q.378C Altcode: No abstract at ADS Title: Numerical evaluation of the redistribution function RII-A(x, xÿ) and of the associated scattering integral. Authors: Adams, T. F.; Hummer, D. G.; Rybicki, G. B. Bibcode: 1971JQSRT..11.1365A Altcode: No abstract at ADS Title: Radiative transfer in spherically symmetricsystems-II. The non-conservative case and linearly polarized radiation Authors: Cassinelli, J. P.; Hummer, D. G. Bibcode: 1971MNRAS.154....9C Altcode: The method for the solution of transfer problems in spherically symmetric systems developed recently by Hummer and Rybicki is here generalized to the nonconservative case. This procedure, which depends on the iterative determination of the Eddington factor J = K/J, handles in a natural way the outward peaking of the radiation field which occurs in extended atmospheres. To illustrate the present extension of this method, solutions are obtained for the problem of scattering of linearly polarized radiation by an extended electron-scattering atmosphere. Although the transfer of radiation through such an atmosphere is conservative, each of the component equations is not. For opacity laws of the form K = r - ,o <r <R, n = a and 3' very large values of the polarization are found as a general feature arising from the strong peaking of the radiation field. It is found that the temperature distribution in such extended electron-scattering atmospheres differs negligibly from that computed on the assumption of isotropic scattering, with the neglect of polarization. The procedure used for the polarization problem can also be applied directly to problems with a non-grey opacity involving many frequencies simultaneously. Title: The Formation of Spectral Lines Authors: Hummer, D. G.; Rybicki, G. Bibcode: 1971ARA&A...9..237H Altcode: No abstract at ADS Title: Radiative transfer in spherically symmetric systems. The conservative grey case Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1971MNRAS.152....1H Altcode: A practical computational method is presented for the solution of radiative transfer problems in spherically symmetric systems. This procedure involves iteration on the `Eddington factor ` f = K/J and is designed to handle the outward peaking of the radiation field in extended spherical systems. Extensive numerical results are obtained and discussed for systems in which `cp = o< , for n = 3/2, 2 and 3. Title: Electron Impact Excitation of Positive Ions Authors: Burgess, A.; Hummer, D. G.; Tully, J. A. Bibcode: 1970RSPTA.266..225B Altcode: Non-relativistic Coulomb-Born-Oppenheimer reactance matrices and cross-sections are given for all transitions between the 1s, 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. Title: Joint Institute for Laboratory Astrophysics of the National Bureau of Standards and the University of Colorado, Boulder, Colorado. Report 1968-1969. Authors: Hummer, D. G. Bibcode: 1970BAAS....2...59H Altcode: No abstract at ADS Title: Non-coherent scattering - VII. Frequencydependent thermalization lengths and scattering with continuous absorption Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1970MNRAS.150..419H Altcode: Our previous definition of the thermalization length is generalized to cases in which the initial photon frequency is prescribed. This permits treatment of cases in which the emissivity does not have a unique frequency dependence. This definition is then applied to the case of a line formed by scattering in the presence of continuous opacity. In this case the emissivity in the line has a unique frequency dependence but the total emissivity does not. Equations for the distribution of thermalization distances are derived both by a diagrammatic technique and by use of resolvents. Median thermalization lengths are defined in terms of these distributions. Extensive numerical results are reported and the utility of this approach is discussed. Title: Model atmospheres for the central stars of planetary nebulae. Authors: Hummer, D. G.; Mihalas, D. Bibcode: 1970MNRAS.147..339H Altcode: Approximately 70 model atmospheres for the central stars of planetary nebulae have been computed under the assumptions of hydrostatic, radiative and local thermodynamic equilibrium and of stratification. These models have effective temperatures and surface gravities in the range and log . The atmospheres have been taken to consist of hydrogen, helium, oxygen, nitrogen, carbon and neon, and the opacity included contributions from both ground and excited states of each ion. The transfer equation is solved using Feautrier's method and the temperature corrections are calculated by means of the Krook-Avrett procedure. Particular attention is given to the effects of gravity and chemical composition on the surface fluxes. The photon fluxes in the H I, He I and He ii continua and the stellar flux at H have been tabulated for use in the determination of Zanstra temperatures. Title: Spectral Line Formation by Noncoherent Scattering with a Dipole Phase Function Authors: Hummer, D. G. Bibcode: 1970ApL.....5....1H Altcode: No abstract at ADS Title: Part C Chromospheres and Coronae of Stars Authors: Pecker, J. C.; Hummer, D. G. Bibcode: 1970sfss.coll..239P Altcode: 1970IAUCo...2..239P No abstract at ADS Title: Non-coherent scattering-VI. Solutions of the transfer problem with a frequency-dependent source function Authors: Hummer, D. G. Bibcode: 1969MNRAS.145...95H Altcode: A generalized discrete-ordinate method is used to obtain accurate numerical solutions of the line transfer problem in which the scattering is described by a redistribution function. Extensive results are obtained and discussed for the cases of pure Doppler broadening and of Doppler and natural broadening combined. It is shown that, in the latter case, the intensity of radiation emerging from a semi-infinite isothermal atmosphere approaches that for coherent scattering in the line wings instead of approaching the value of the Planck function. Title: The normalized on-the-spot approximation for line transfer problems. Authors: van Blerkom, D.; Hummer, D. G. Bibcode: 1969JQSRT...9.1567V Altcode: No abstract at ADS Title: Non-coherent scattering-V. Thermalization distances and their distribution function Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1969MNRAS.144..313R Altcode: The distribution function for thermalization distances is derived for an infinite atmosphere with a plane source. Precise definitions of the thermalization length are discussed from the point of view of representing the distribution by a single characteristic length; of these a definition in terms of the median of the distribution seems to be most useful. The distribution of longest flights is derived and shown to provide a good approximation to the distribution of thermalization lengths at large distances from the source. Extensive numerical illustrations are provided. Title: Line Formation in Differentially Moving Media with Temperature Gradients Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1968rla..conf..213H Altcode: No abstract at ADS Title: A Non-Lte Theory of Overlapping Lines Near the Series Limit Authors: van Blerkom, D.; Hummer, D. G. Bibcode: 1968ApJ...154..741V Altcode: The effects of overlapping on the formation of spectral lines near the series limit are investigated through the introduction of a band model Numerical solutions are obtained which show that the source function can be increased by a factor of 2 for e = 0 1 and by much larger amounts for smaller A cri- terion is presented to check when overlapping is likely to be important Title: Redshifted Line Profiles from Differentially Expanding Atmospheres Authors: Hummer, D. G.; Rybicki, G. B. Bibcode: 1968ApJ...153L.107H Altcode: Accurate numerical solutions of the non-LTE radiative-transfer problem with non-coherent scatter- ing have been obtained using a generalization of the Riccati method of Rybicki and Hummer (1967). Differential expansion in the direction of the observer is found to produce significantly redshifted line profiles Title: Radiative Transfer Processes in Planetary Nebulae Authors: Hummer, D. G. Bibcode: 1968IAUS...34..166H Altcode: No abstract at ADS Title: Summary-introduction to radiative transfer problems in stellar atmospheres Authors: Hummer, D. G. Bibcode: 1968JQSRT...8..193H Altcode: A central problem in the theory of stellar atmospheres is the determination of the radiation field and distribution of atomic states throughout the atmosphere, given the temperature-pressure structure of the atmosphere, and the chemical composition and the dynamical state of the gas. This problem has a diagnostic aspect, since it can be used to obtain the external radiation field for the purpose of comparing stellar models with observations. There is also a constructive aspect, since the computation of a self-consistent model, without the LTE assumptions, must necessarily involve the problem, usually as one step in an iterative process. Here attention is limited to the radiation field in resonance lines, in which the problem is especially simple. From the considerable body of numerical solutions now available, it is possible to discern some rather general features of these problems, which can be conveniently discussed in terms of processes coupling the radiation field to the electron gas. A concept which has proven useful in understanding these solutions is that of the so-called thermalization length, defined as the optical distance between the point where a photon is created at the expense of electron kinetic energy and the point where it is converted back into kinetic energy. For example, it is seen that a rather severe limit is placed on the resolution with which any observation of the radiation field can determine the temperature-pressure structures of an atmosphere. One technical difficulty encountered in this work is that of solving the combined transfer and statistical equilibrium equations for an atmosphere with optical properties depending on depth. Mathematically, the problem is the numerical solution of a set of coupled linear differential equations with non-constant coefficient, with two-point boundary conditions. Some recent work on this problem by Dr. G. B. Rybicki and the author will be discussed, in which the linear equations are converted into a set of coupled non-linear equations with a one-point boundary condition by means of a transformation due to Rybicki and Usher. Title: Non-coherent scattering-III. The effect of continuous absorption on the formation of spectral lines Authors: Hummer, D. G. Bibcode: 1968MNRAS.138...73H Altcode: The formation of spectral lines by completely redistributed non-coherent scattering is systematically investigated for situations in which photo-ionization or extinction by dust grains occurs at the line frequencies. Particular attention is given to cases for which the source function associated with the continuous opadty differs from the Planck function at the local electron temp eratare. Very large effects are found to occur, even when the ratio of continuous opacity to line opacity is very small. Extensive numerical results are presented, and the relevant scaling laws and the generalized thermalization length are discussed. Title: Non-coherent scattering-IV. Doppler redistribution functions in moving atmospheres Authors: Hummer, D. G. Bibcode: 1968MNRAS.141..479H Altcode: Redistribution functions and their directional averages are derived for scattering by an assembly of atoms possessing a Maxwellian distribution in a frame moving with respect to the observer. Various assumptions concerning the form of the absorption coefficient and the coherence properties in the atom's rest frame are considered. The exact redistribution function for pure Doppler broadening is expanded in Legendre polynomials to facilitate evaluation of the source function. From the expansion it is clear that the use of directionaveraged redistribution functions will be significantly less accurate for moving than for static media. Title: Spectral Line Formation in Variable-Property Media: the Riccati Method Authors: Rybicki, G. B.; Hummer, D. G. Bibcode: 1967ApJ...150..607R Altcode: A numerical method, based on a generalized Riccati transformation of Rybicki and Usher, is presented for the integration of the radiative transfer equation for spectral line formation by non-coherent scattering in inbomogeneous plane-parallel media. A model atom with two discrete levels is assumed. An asymptotic theory is developed which permits the application of this method to problems involving semi-infinite media. Numerical results obtained by the Riccati method are used to discuss the effects of spatial variations in the Doppler width on lines formed by pure Doppler broadening. Title: Line Formation with Non-Coherent Electron Scattering in O and B Stars Authors: Hummer, D. G.; Mihalas, D. Bibcode: 1967ApJ...150L..57H Altcode: No abstract at ADS Title: The ionization structure of planetary nebulae-VI. The Lyman continuum problem Authors: van Blerkom, D.; Hummer, D. G. Bibcode: 1967MNRAS.137..353V Altcode: Because of the recent evidence that a planetary nebula does not completely absorb Lyman continuum radiation for a substantial part of its life, we have used a generalized discrete-ordinate method to obtain accurate numerical solutions to the ionization balance problem in pure hydrogen, plane-parallel model nebulae of various optical thicknesses. The effects of incomplete absorption, interior boundary conditions and non-Plancidan stellar fluxes are examined. We find that the density of neutral hydrogen in the nebula increases by less than a factor of two as the optical thickness varies from to Co, and is insensitive to large deviations in the stellar flux from the Planckian distribution, if the total number of Lyman photons is constant. The radius of the Strbmgren sphere differs by about 5 per cent for the two boundary conditions considered. The accuracies of two simple approximations are assessed, and conditions for their validity are obtained. Title: Computational Methods for Non-LTE Line_transfer Problems Authors: Hummer, D. G.; Rybicki, G. Bibcode: 1967MComP...7...53H Altcode: No abstract at ADS Title: Thermalization Lengths and the Homogeneous Transfer Equation in Line Formation Authors: Hummer, D. G.; Stewart, J. C. Bibcode: 1966ApJ...146..290H Altcode: No abstract at ADS Title: General Noncoherent Scattering Authors: Hummer, D. G. Bibcode: 1965SAOSR.174..143H Altcode: No abstract at ADS Title: The Emission Coefficient Authors: Hummer, D. G. Bibcode: 1965SAOSR.174...13H Altcode: No abstract at ADS Title: Non-coherent scattering, II: Line formation with a frequency independent source function Authors: Avrett, E. H.; Hummer, D. G. Bibcode: 1965MNRAS.130..295A Altcode: The frequency-independent source function for a two-level atom in a self-excited atmosphere with no continuous absorption is encountered, for example, in theories of line formation in the solar corona and outer chromosphere. We present accurate numerical solutions to this transfer problem in finite and semi-infinite atmospheres for physically interesting values of the relevant parameters and discuss the mathematical and physical features of the source functions we obtain. We do not here apply our results to any particular astrophysical problem, but rather concentrate on understanding the physics of the transfer problem. Particular attention is given to the effects of collision broadening. Finally, the assumption of a frequency-independent source function for Doppler broadening is examined for self-consistency. Title: The Voigt function: An eight-significant-figure table and generating procedure Authors: Hummer, D. G. Bibcode: 1965MmRAS..70....1H Altcode: No abstract at ADS Title: The Mean Number of Scatterings by a Resonance-Line Photon. Authors: Hummer, D. G. Bibcode: 1964ApJ...140..276H Altcode: A rigorous upper bound to the mean number of scatterings (N) experienced by a resonance-line photon is obtained very simply. For completely non-coherent scattering this bound provides a good estimate of (N) and explicit expressions are obtained for the cases of Doppler, Voigt, and Lorentz absorption coefficients. These results are contrasted with those of Osterbrock (1962) and are interpreted in terms of the concept of effective band width. Title: The ionization structure of planetary nebulae, III Authors: Hummer, D. G.; Seaton, M. J. Bibcode: 1964MNRAS.127..217H Altcode: In planetary nebulae the helium to hydrogen abundance ratio, by numbers of atoms, is Y=o I to C 2 the abundance of heavier elements, by numbers of atoms relative to hydrogen, is Z IC- . The heavier elements may be neglected when considering the ionization of hydrogen and helium, but are of importance in considering the thermal balance. Three main regions are considered In region (i), that is, the innermost region, the most abundant ions are He+2 and H+. The He+2 - He+ ionization equilibrium is calculated by the method of Paper I. Recombinations of He+2 to He+ produce sufficient quanta to keep the hydrogen ionized. With Z =0, the electron temperature in region (i) would be very high (T8 >7.5 x io4 K with no collisional excitation), but with Z io an efficient cooling process is shown to be excitation of allowed transitions in lithium-like ions such as C+3, giving Te 2 X i04 K. In region (ii) the most abundant ions are He+ and H+. The He+-He0 and H+-H0 ionization equilibria are calculated on solving coupled equations. At lower star temperatures (Ts <5 X I0 K for Y= ), the inner part of region (ii) contains He+ and H+ and the outer part contains He0 and H+. At higher star temperatures the transitions from He + to He0 and from H+ to H0 occur at approximately the same point. In region (iii) none of the gas is ionized. This region exists only if the total amount of gas is sufficient to produce absorption of all ionizing stellar quanta. Title: The ionization structure of planetary nebulae, II. Collisional cooling of pure hydrogen nebulae Authors: Hummer, D. G. Bibcode: 1963MNRAS.125..461H Altcode: Collisional excitation and ionization of hydrogen are mechanisms which lead to the removal of energy from planetary nebulae. The ionization and thermal balance equations including these effects are solved for a pure hydrogen nebula to determine the electron temperature in the nebula for a wide range of star temperatures. For reasonable stellar fluxes the electron temperature considered as a function of star temperature assumes a maximum value of 2>c I0 0K. Collisional excitation greatly enhances the line emission from low-lying states of hydrogen; in particular, it may be important to allow for collisional excitation of Ly o' when one obtains star temperatures from estimates of the total energy radiated by the nebula. The role of hydrogen collisional cooling in real nebulae is discussed briefly. Accurate and convenient expressions are obtained for the collisional excitation and ionization rates for hydrogen. Title: Atomic collision and radiative transfer phenomena in planetary nebulae Authors: Hummer, David Graybill Bibcode: 1963PhDT........65H Altcode: No abstract at ADS Title: The ionization structure of planetary nebulae, I. Pure hydrogen nebulae Authors: Hummer, D. G.; Seaton, M. J. Bibcode: 1963MNRAS.125..437H Altcode: The problem is formulated in terms of the equations of radiative transfer, of ionization equilibrium and of the thermal balance. The intensity of ionizing radiation is I = lyS + 1yd where lyS is the attenuated intensity of stellar radiation and lyd the intensity of diffuse radiation produced in the nebula. In Approximation I it is assumed that there is no transfer problem for the diffuse radiation, emission and absorption occurning at the same place. The problem is then solved using a method due to Zanstra and de Jong. Assuming the star to radiate as a black body, numerical results are obtained for a wide range of star temperatures. In Approximation II the transfer equation for 1yd is solved using the source function from Approximation I. For an isothermal plane parallel model it is found that the mean intensities Jy as given by Approximations I and II, never differ by more than a few per cent. Title: Non-coherent scattering: I. The redistribution function with Doppler broadening Authors: Hummer, D. G. Bibcode: 1962MNRAS.125...21H Altcode: The redistribution in frequency of radiation scattered from moving atoms is examined in some generality, allowing for the different types of scattering which occur in the atom's rest frame under different circumstances. Some general formulae are obtained and a number of explicit results are given. Finally some attention is devoted to the properties of the redistribution functions and to the methods which may be used for computing them. Title: The Ultra-Violet Radiation of the Central Stars of Planetary Nebulae Authors: Hummer, D. G.; Seaton, M. J. Bibcode: 1961LIACo..10..539H Altcode: 1961MSRSL...4..538H; 1961LIACo..10..538H No abstract at ADS