Abstract
Limits and approximations can serve as useful checks on calculated opacities. Scaling is important to estimate opacities for elements for which tables have not been calculated. Interpolation is indispensable to obtain opacities at density and temperature values for which they have not been tabulated. In the following, opacities, κ, are in units of m2∕kg.
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- 1.
For relativistic and collective effects, see Sect. 6.3.
- 2.
For application to H2O vapor, Penner (1959) used \(B_{\mathrm{e},i} \approx \sqrt{BC} = 1160\,\mathrm{{m}}^{-1}\). Penner and Olfe (1968) justified application of the model by the empirical observation that “adequate pressure broadening (i.e., so that rotational fine structure lines overlap) is achieved, at a few atmospheres at room temperature in dilute water vapor and that the widths for water – water collisions are 5 to 10 times larger than the widths for H2O – N2 or H2O – O2 collisions.” According to Penner and Varanasi (1965): “The relatively weak dependence of the theoretical value of A on B e, i is, in large measure, responsible for the success of the highly simplified procedure.”
- 3.
For small K-values ( < 0.05), I(K) ≈ 0. 5 K (within 2 %). For intermediate and large values of K, \(I(K) \approx 1.11{[2.303\log 1.21\,K]}^{1/2}\) (Penner and Olfe 1968).
- 4.
Values for α i for H2O vapor for four spectral regions (1.38, 1.87, 2.7, and 6. 3 μm) at T = 473, 673, 873, and 1,000 K with the T-dependence of the integrated radiances are given by Eq. (11.10) derived from measurements of Goldstein (1964), were compared with values calculated using the upper limit of the inequalities. They found that the ratio \(\alpha _{i,\mathrm{calc.}}/\alpha _{i,\mathrm{meas.}}\) for optical depths 0.03, 0.1, and 0.3 ranged from 0.82 to 1.1, with the larger discrepancies occurring for the 6. 3 μm band.
- 5.
The last term in brackets is a fit to the free–free Gaunt factor at u ≈ 7.
- 6.
See also Zel’dovich and Raĭzer (1966).
- 7.
It was found that replacing the coefficient 0.001 by 0.01 or 0.0001 would lead to substantially the same results [for calculated band emissivities for CO fundamental and first overtone, \(0.1\,\mathrm{MPa} \leq P \leq 5\,\mathrm{MPa}\), \(300 \leq T \leq 3,000\,\mathrm{K}\)]. The data show that a 50-fold increase in optical density changes the effective bandwidth by about 30 % for the fundamental and by about 10 % for the overtone band.
- 8.
See also approximate band models described in Sect 7.3.3.
- 9.
Numbers in brackets indicate the chapters in which the references are cited.
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Huebner, W.F., Barfield, W.D. (2014). Limits, Approximations, Scaling, and Interpolations. In: Opacity. Astrophysics and Space Science Library, vol 402. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8797-5_11
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