Astrophysics and Stellar Astronomy

Swihart, Thomas L.; Noerdlinger, Peter D.

doi:10.1119/1.1986265

Cited by 5 publications

(14 citation statements)

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“…This result explains the statements in some older texts (Novotny 1973;Swihart 1968) attributed to unpublished calculations of the Arthur. N. Cox, the author of Cox (2000).…”

Section: A Practical Upper Limit To the Temperaturesupporting

confidence: 72%

“…At the opposite extreme, Hubeny & Mihalas (2014) gives an extensive summary of the theory described by Hummer & Mihalas (1988), but it provides no numerical values, while Griem (2005) presents an even more formal discussion from the laboratory plasma perspective. Some books (Swihart 1968;Novotny 1973;Allen 1973;Cox 2000) give tables of values for selected species over specific temperatures, but there is no detail of the procedures used nor guidance for the density or temperature dependence. Novotny (1973) and Swihart (1968), give U and comment that the reported values are valid regardless of temperature if the ion is abundant.…”

mentioning

confidence: 99%

“…Some books (Swihart 1968;Novotny 1973;Allen 1973;Cox 2000) give tables of values for selected species over specific temperatures, but there is no detail of the procedures used nor guidance for the density or temperature dependence. Novotny (1973) and Swihart (1968), give U and comment that the reported values are valid regardless of temperature if the ion is abundant. de Galan et al (1968); Sauval & Tatum (1984) give polynomial expressions for the partition function versus temperature, which give approximate estimates of U , but do not describe the density dependence.…”

mentioning

confidence: 99%

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A Practical Guide to the Partition Function of Atoms and Ions

Alimohamadi,

Ferland

2022

Preprint

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The partition function, U , the number of available states in an atom or molecules, is crucial for understanding the physical state of any astrophysical system in thermodynamic equilibrium. There are surprisingly few useful discussions of the partition function's numerical value. Textbooks often define U ; some give tables of representative values, while others do a deep dive into the theory of a dense plasma. Most say that it depends on temperature, atomic structure, density, and that it diverges, that is, it goes to infinity, at high temperatures, but few give practical examples. We aim to rectify this. We show that there are two limits, 1 & 2 electron (or closed-shell) systems like H or He, and species with a complicated electronic structure like C, N, O, and Fe.The high-temperature divergence does not occur for 1 & 2 electron systems in practical situations since, at high temperatures, species are collisionally ionized to higher ionization stages and are not abundant. The partition function is then close to the statistical weight of the ground state. There is no such simplification for many-electron species. U is temperature-sensitive across the range of temperatures where an ion is abundant but remains finite at even the highest practical temperatures. The actual value depends on highly uncertain truncation theories in high-density plasmas. We show that there are various theories for continuum lowering but that they are not in good agreement. This remains a long-standing unsolved problem.

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“…This result explains the statements in some older texts (Novotny 1973;Swihart 1968) attributed to unpublished calculations of the Arthur. N. Cox, the author of Cox (2000).…”

Section: A Practical Upper Limit To the Temperaturesupporting

confidence: 72%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Practical Guide to the Partition Function of Atoms and Ions

Alimohamadi,

Ferland

2022

Preprint

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“…The intensity (energy flux per unit wavelength per unit solid angle) of radiation 1(θ) passing through a medium at an angle 6 with the vertical can be calculated from the equation of radiative transfer given by: where T is the optical depth of the medium, ε is the emittance, u0 is the single scattering albedo, B is the Rayleigh-Jeans approximation to the Planck function, μ = cos θ, P(θ) is the scattering phase function, and T' and 8' are the integration variables for optical depth and scattering angle respectively (cf. Swihart 1971).…”

Section: Radiative Transfer Modelmentioning

confidence: 99%

Radiative Transfer Modeling of Microwave Emission and Dependence on Firn Properties

Comiso¹,

Zwally²,

Saba³

1982

Ann. Glaciol.

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The microwave emission from a model polar firn was calculated using a numerical solution of the radiative transfer equation that included angledependent Rayleigh scattering. The depth-dependent parameters in the equation were physical temperature and the coefficients of scattering and absorption. The coefficients were based on Rayleigh scattering from the snow grains. The bulk emissivity and the seasonal dependence of brightness temperature were calculated for seven locations at which grain sizes were measured as a function of depth. When the absorption and scattering coefficients are adjusted, the modeled emissivities agree with observed emissivities at these locations. The modeled seasonal dependence of brightness temperatures also compares well with values obtained at 1.55 cm wavelength by the Nimbus-5 satellite. Good agreement with data did not occur when the imaginary part of the index of refraction (and, hence, the absorption coefficient) had a significant temperature dependence between 210 and 250 K. INTRODUCTIONThe unexpectedly low microwave brightness temperatures of the Greenland and Antarctic ice sheets, as measured by the electrically scanning microwave radiometer (ESMR) on Nimbus-5, have been the subject of several theoretical investigations in recent years. Chang and others (1976) showed that, for a uniform snow medium, volume scattering is a dominant factor affecting the microwave emission and that the source of radiation can emanate from several meters within the medium. Subsequently, taking into account variations of snow properties with depth, Zwally (1977) used an analytic approximation to solve the microwave emissivity of the polar firn. The approximation overestimated the effect of radiative scattering in the medium because contributions due to multiple scattering were neglected. The scattering coefficient y s had to be multiplied by an empirical parameter f = 0.12 to obtain agreement between calculated emissivities, e(model) and observed emissivities e(obs). Using a different numerical technique for solving the radiative transfer equation from the one used in this paper, Chang and others (1980) showed good quantitative agreement with their quoted bright-

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“…On the opposite extreme, Hubeny & Mihalas (2014) give an extensive summary of the theory described by Hummer & Mihalas (1988), but it provides no numerical values, while Griem (2005) presents an even more formal discussion from the laboratory plasma perspective. Some books (Swihart 1968;Allen 1973;Novotny 1973;Cox 2000) give tables of values for selected species over specific temperatures, but there is no detail of the procedures used nor guidance for the density or temperature dependence. Novotny (1973) and Swihart (1968) give U and comment that the reported values are valid regardless of temperature if the ion is abundant.…”

Section: Introductionmentioning

confidence: 99%

A Practical Guide to the Partition Function of Atoms and Ions

Alimohamadi¹,

Ferland²

2022

PASP

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The partition function, U, the number of available states in an atom or molecules, is crucial for understanding the physical state of any astrophysical system in thermodynamic equilibrium. There are surprisingly few useful discussions of the partition function’s numerical value. Textbooks often define U; some give tables of representative values, while others do a deep dive into the theory of dense plasma. Most say that it depends on temperature, atomic structure, density, and that it diverges, that is, it goes to infinity, at high temperatures, but few give practical examples. We aim to rectify this. We show that there are two limits, one- and two-electron (or closed-shell) systems like H or He, and species with a complicated electronic structure like C, N, O, and Fe. The high-temperature divergence does not occur for one- and two-electron systems in practical situations because, at high temperatures, species are collisionally ionized to higher-ionization stages and are not abundant. The partition function is then close to the statistical weight of the ground state. There is no such simplification for many-electron species. U is temperature sensitive across the range of temperatures where an ion is abundant but remains finite at even the highest practical temperatures. The actual value depends on highly uncertain truncation theories in high-density plasmas. We show that there are various theories for continuum lowering but that they are not in good agreement. This remains a long-standing unsolved problem.

show abstract

Astrophysics and Stellar Astronomy

Cited by 5 publications

References 0 publications

A Practical Guide to the Partition Function of Atoms and Ions

A Practical Guide to the Partition Function of Atoms and Ions

Radiative Transfer Modeling of Microwave Emission and Dependence on Firn Properties

A Practical Guide to the Partition Function of Atoms and Ions

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