Abstract

For thermal emission from particles the conventional size parameter X = 2πa/λ does not distinguish between small and large particles. We show that the opacity parameter Ω = 4πka/λ = 2kX is a more accurate means of demarcating the two emission regimes. Ω is approximately equal to the particle’s mean optical depth, and it can be derived from both scattering theory and geometrical optics.

© 1999 Optical Society of America

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Errata

David K. Lynch and Stephan Mazuk, "Size parameter for thermally emitting particles: erratum," Appl. Opt. 38, 7467-7467 (1999)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-38-36-7467

References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. Irradiance F from a particle is proportional to emissivity × Planck function = Qabs *Bλ(T). For a blackbody, Qabs is unity. When Qabs ≪ 1, F shows a spectral structure that indicates its optical constants, which in turn are determined by the chemical composition and crystalline state. When Qabs ≈ 1 as it does when the particle is large, the emission is broad and featureless with only a single maximum whose width is about the same as that of the Planck function. Although there may be minor deviations of Qabs from unity that disqualifies it from being a true blackbody, the radiation is nonetheless roughly Planckian and entirely different from when Qabs ≪ 1.
  4. T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

1995

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

Begemann, B.

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Dorschner, J.

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

Henning, T.

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Mutschke, H.

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Astron. Astrophys. Suppl.

T. Henning, B. Begemann, H. Mutschke, J. Dorschner, “Optical properties of oxide dust grains,” Astron. Astrophys. Suppl. 112, 143–149 (1995).

Other

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Irradiance F from a particle is proportional to emissivity × Planck function = Qabs *Bλ(T). For a blackbody, Qabs is unity. When Qabs ≪ 1, F shows a spectral structure that indicates its optical constants, which in turn are determined by the chemical composition and crystalline state. When Qabs ≈ 1 as it does when the particle is large, the emission is broad and featureless with only a single maximum whose width is about the same as that of the Planck function. Although there may be minor deviations of Qabs from unity that disqualifies it from being a true blackbody, the radiation is nonetheless roughly Planckian and entirely different from when Qabs ≪ 1.

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Figures (4)

Fig. 1
Fig. 1

Emissivity Q abs calculated for various particle radii a as a function of wavelength for olivine.4 Note that as the particle becomes larger, the contrast in the spectral structure indicative of the optical constants begins to decrease. In the large-particle limit the contrast nearly vanishes and the Q abs spectrum becomes almost constant at unity.

Fig. 2
Fig. 2

Calculated emissivity Q abs using Mie theory as a function of particle radius a at λ = 10.0 µm in a range of k, the imaginary part of the index of refraction where the real part of the index n = 1.5. The loci of X = 1 (vertical line) and Ω = 1 (dashed curve) are also shown. For small particles, Q abs varies as λ-1 whereas for large particles Q abs is approximately constant and near unity. Particles to the right of X R = 1 are often termed large. From a scattering viewpoint this is correct, but their emissivity is still far below unity. Thus, from an emission standpoint, they are considered small. The Ω = 1 line follows the transition from small to large particles for every value of k.

Fig. 3
Fig. 3

Physical basis of Ω derived from a slab of matter with index N = n + ik, an absorption coefficient of 4πk/λ, and thickness a. Ignoring surface reflections, we determine the ratio of the incoming radiance to the transmitted radiance I/ I o by its absorption optical depth 4πka/λ = Ω, i.e., I/ I o = exp(-4πka/λ). The mean thickness (chord length) of a sphere is 4a/3 ≈ a, and therefore a sphere’s radius is a good approximation of its average thickness. The opacity parameter Ω is essentially the particle’s mean optical depth.

Fig. 4
Fig. 4

Emissivity Q abs plotted against the opacity parameter Ω. Particle radius a = Ωλ/4πk varies from 10-6 on the left to 104 on the right. The refractive index n was fixed at 1.5 + i1.0, and the wavelength λ was set at 10 µm. Ω was varied from 10-4 to 104, and then the radius was computed followed by Q abs. Permuting the fixed and variable parameters (index, wavelength, and radius) gave essentially the same results although the radius scale is shifted relative to Ω. When Ω ≪ 1, then Q abs ≪ 1 and varies as 1/Ω. When Ω ≫ 1, then Q abs ≈ 1 and remains constant with increasing Ω. Note that the break in the curve occurs near Ω ≈ 1, confirming that Ω is the desired indicator.

Equations (2)

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mX=2πNPa/λ=2πa/λn+ik=2πna/λ+i2πka/λ=XR+XI,
Ω=4πka/λ=2XR.

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