Abstract

We calculate the radiation torque exerted by a monochromatic plane wave, either unpolarized or linearly polarized, on aggregates of spheres and investigate the stability of the resulting rotational motion. In fact, neglecting any braking momenta we calculate the component of the electromagnetic torque orthogonal to the principal axis of maximum moment of inertia through the center of mass (transverse torque), as a function of the direction of propagation of the incident field. The aggregates we study are composed of homogeneous spheres, possibly of different materials. The electromagnetic torque is calculated through the transition matrix approach along the lines of the theory reported in our recent paper [F. Borghese, P. Denti, R. Saija and M. A. Iatì, Opt. Express 14, 9508 (2006)]. When the transverse component of the electromagnetic torque is small or vanishes the rotational motion driven by the component along the principal axis of inertia may be nearly stable.

© 2007 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd ed., (Wiley, New York, 1975).
  2. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
    [CrossRef]
  3. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
    [CrossRef]
  4. P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclassical Opt. 4, S78-S81 (2002).
    [CrossRef]
  5. E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979).
    [CrossRef]
  6. A. Lazarian, "Physics of grain alignment," in Cosmic evolution and galaxy formation, AIP Conference series 3, (1999).
  7. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
    [CrossRef]
  8. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
    [CrossRef]
  9. E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
  10. B. T. Draine and P. J. Flatau, "Discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
    [CrossRef]
  11. P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).
  12. P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).
    [CrossRef]
  13. F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Trasfer 89, 3-9 (2004).
    [CrossRef]
  14. F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
    [CrossRef]
  15. F. Borghese, P. Denti, R. Saija and M. A. Iatì, "Radiation torque on nonspherical particles in the transition matrix formalism," Opt. Express 14, 9508-9521 (2006).
    [CrossRef] [PubMed]
  16. F. Borghese, P. Denti and R. Saija, Scattering from Model Nonspherical Particles, 2nd ed., (Springer, Berlin, 2007).
  17. R. Saija, M. A. Iatì, P. Denti, F. Borghese, A. Giusto and O. I. Sindoni, "Efficient light-scattering calculations for aggregates of large spheres," Appl. Opt. 42, 2785-2793 (2003).
    [CrossRef] [PubMed]
  18. R. Saija, M. A. Iatì, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie Theory: The transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
    [CrossRef]
  19. Y.-L. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995).
    [CrossRef] [PubMed]
  20. W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Eletromagnetic Waves (IEEE, Piscataway, N. J., 1990).
  21. F. Borghese, P. Denti, R. Saija and M. A. Iatì, "Radiation torque on nonspherical particles in the transition matrix formalism: erratum," Opt. Express 15, 6946 (2007).
    [CrossRef] [PubMed]
  22. The Erratum to Ref. [16] can be found at http://dfmtfa.unime.it/profs/borghese/ferdinandoborghese.html.
  23. M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
    [CrossRef]
  24. E. M. Rose, Elementary Theory of Angular Momentum, (Wiley, New York, 1956).
  25. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., (Addison-Wesley, Reading, Mass., 2002).
  26. B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984).
    [CrossRef]
  27. P. H. Berning, G. Hass and P. R. Madden, "Reflectance-increasing coatings for the vacuum ultraviolet and their applications," J. Opt. Soc. Am. 50, 586-597 (1960).
    [CrossRef]
  28. U. Kreibig, "Electronic properties of small silver particles: the optical constants and their temperature dependence," J. Phys. F: Metal Phys. 4, 999-1014 (1974).
    [CrossRef]
  29. G. Wurm and M. Schneiter, "Coagulation as a unifying element for interstellar polarization," Astrophys. J. 567, 370-375 (2002).
    [CrossRef]

2007 (1)

2006 (1)

2004 (2)

F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Trasfer 89, 3-9 (2004).
[CrossRef]

F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
[CrossRef]

2003 (1)

2002 (2)

P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclassical Opt. 4, S78-S81 (2002).
[CrossRef]

G. Wurm and M. Schneiter, "Coagulation as a unifying element for interstellar polarization," Astrophys. J. 567, 370-375 (2002).
[CrossRef]

2001 (2)

M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

R. Saija, M. A. Iatì, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie Theory: The transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

1998 (2)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

1997 (1)

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

1996 (1)

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

1995 (1)

1994 (1)

1984 (2)

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).
[CrossRef]

B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984).
[CrossRef]

1979 (1)

E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979).
[CrossRef]

1974 (1)

U. Kreibig, "Electronic properties of small silver particles: the optical constants and their temperature dependence," J. Phys. F: Metal Phys. 4, 999-1014 (1974).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

1971 (1)

P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).

1960 (1)

Aiello, S.

R. Saija, M. A. Iatì, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie Theory: The transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

Berning, P. H.

Borghese, F.

Cecchi-Pestellini, C.

R. Saija, M. A. Iatì, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie Theory: The transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

Crichton, J. H.

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).
[CrossRef]

Denti, P.

Draine, B. T.

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

B. T. Draine and P. J. Flatau, "Discrete dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11, 1491-1499 (1994).
[CrossRef]

B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984).
[CrossRef]

Flatau, P. J.

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

Galajda, P.

P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclassical Opt. 4, S78-S81 (2002).
[CrossRef]

García de Abajo, F. J.

F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Trasfer 89, 3-9 (2004).
[CrossRef]

F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
[CrossRef]

Giusto, A.

Hass, G.

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

Iatì, M. A.

Kreibig, U.

U. Kreibig, "Electronic properties of small silver particles: the optical constants and their temperature dependence," J. Phys. F: Metal Phys. 4, 999-1014 (1974).
[CrossRef]

Lee, H. M.

B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984).
[CrossRef]

Madden, P. R.

Marston, P. L.

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

Ormos, P.

P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclassical Opt. 4, S78-S81 (2002).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

Saija, R.

Schneiter, M.

G. Wurm and M. Schneiter, "Coagulation as a unifying element for interstellar polarization," Astrophys. J. 567, 370-375 (2002).
[CrossRef]

Sindoni, O. I.

Waterman, P. C.

P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).

Weingartner, J. C.

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

Wurm, G.

G. Wurm and M. Schneiter, "Coagulation as a unifying element for interstellar polarization," Astrophys. J. 567, 370-375 (2002).
[CrossRef]

Xu, Y.-L.

Appl. Opt. (2)

Astrophys. J. (7)

B. T. Draine and H. M. Lee, "Optical properties of interstellar graphite and silicate grains," Astrophys. J. 285, 89-108 (1984).
[CrossRef]

G. Wurm and M. Schneiter, "Coagulation as a unifying element for interstellar polarization," Astrophys. J. 567, 370-375 (2002).
[CrossRef]

R. Saija, M. A. Iatì, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie Theory: The transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

E. M. Purcell, "Suprathermal rotation of interstellar grains," Astrophys. J. 231, 404-416 (1979).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

P. Galajda and P. Ormos, "Rotation of microscopic propellers in laser tweezers," J. Opt. B: Quantum Semiclassical Opt. 4, S78-S81 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. F: Metal Phys. (1)

U. Kreibig, "Electronic properties of small silver particles: the optical constants and their temperature dependence," J. Phys. F: Metal Phys. 4, 999-1014 (1974).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transf. 70, 811-816 (2001).
[CrossRef]

J. Quant. Spectrosc. Radiat. Trasfer (1)

F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Trasfer 89, 3-9 (2004).
[CrossRef]

Nature (2)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348-349 (1998).
[CrossRef]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Erratum: Optical alignment and spinning of laser-trapped microscopic particles," Nature 395, 621 (1998).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).
[CrossRef]

Phys. Rev. B (1)

F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).

Other (7)

J. D. Jackson, Classical Electrodynamics, 2nd ed., (Wiley, New York, 1975).

A. Lazarian, "Physics of grain alignment," in Cosmic evolution and galaxy formation, AIP Conference series 3, (1999).

F. Borghese, P. Denti and R. Saija, Scattering from Model Nonspherical Particles, 2nd ed., (Springer, Berlin, 2007).

The Erratum to Ref. [16] can be found at http://dfmtfa.unime.it/profs/borghese/ferdinandoborghese.html.

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Eletromagnetic Waves (IEEE, Piscataway, N. J., 1990).

E. M. Rose, Elementary Theory of Angular Momentum, (Wiley, New York, 1956).

H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., (Addison-Wesley, Reading, Mass., 2002).

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Figures (8)

Fig. 1.
Fig. 1.

T as a function of the angles of incidence ϑ I and φ I for cluster A of four spheres of astronomical silicates (gray) and one sphere of aluminum (blue) whose geometry is shown in (a). The light is unpolarized in (b), and linearly polarized with η = 1, 2 in (c) and (d), respectively.

Fig. 2.
Fig. 2.

Same of Fig. 1 for cluster B of four spheres of amorphous carbon (gray) and one sphere of aluminum (blue) whose geometry is shown in (a).

Fig. 3.
Fig. 3.

Same of Fig. 1 for cluster C of four spheres of amorphous carbon (gray) and one sphere of silver (blue) whose geometry is shown in (a).

Fig. 4.
Fig. 4.

Same of Fig. 1 for cluster D of four spheres of amorphous carbon (gray) and one sphere of silver (blue) whose geometry is shown in (a). The radius of the silver sphere is 28nm so that its mass is equal to the mass of a sphere of carbon.

Fig. 5.
Fig. 5.

Axial averages 〈T⊥〉 and 〈T 〉 as a function of θ I for cluster A.

Fig. 6.
Fig. 6.

Same as Fig. 5 for cluster B.

Fig. 7.
Fig. 7.

Same as Fig. 5 for cluster C.

Fig. 8.
Fig. 8.

Same as Fig. 5 for cluster D.

Tables (3)

Tables Icon

Table 1. Densities in g/cm3 and dielectric constants at λ = 380nm of the materials that compose the clusters

Tables Icon

Table 2. Composition of the clusters and radii of the component spheres in nm

Tables Icon

Table 3. Coordinates of the centers of the spheres (in nm) with respect to the frame of reference with origin at the center of mass. The principal moments of inertia are also given in g∙cm2 × 1028

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

Γ rad = n ˆ · Т M × r d S ,
Т M = 1 8 π Re [ n 2 E E * + B B * 1 2 ( n 2 E · E * + B · B * ) ]
E = E I + E S , B = B I + B S ,
Γ rad = r 3 Ω r ˆ · Т M × r ˆ .
E S = exp ( ikr ) r f ( k ˆ S , k ˆ I )
u ˆ 1 × u ˆ 2 = k ˆ I ,
E I = η E 0 η u ˆ η exp ( i k · r ) = η E 0 η plm J l m ( p ) r k W l m ( p ) ,
E S = η E 0 η plm H l m ( p ) ( r , k ) A η l m ( p ) ,
J l m ( 1 ) ( r , k ) = j l ( k r ) X lm ( r ˆ ) , J l m ( 2 ) ( r , k ) = 1 k × J l m ( 1 ) ( r , k ) ,
A η l m ( p ) = p l m 𝓢 l m l ′m ( p p ) W I η l m ( p ) .
Γ Rad = Re ( Γ )
Γ Rad x = Re [ 1 2 ( Γ 1 Γ 1 ) ] , Γ Rad y = Re [ i 2 ( Γ 1 + Γ 1 ) ] , Γ Rad z = Re ( Γ 0 ) ,
Γ μ = η η ¯ I I η ¯ η Γ μ ; η ¯ η ,
Γ μ ; η ¯ η = Γ μ ; η ¯ η ( ext ) Γ μ ; η ¯ η ( sca ) ,
Γ μ ; η ¯ η ( ext ) = c Γ plm s μ ; l m W l , m μ ( p ) A η ¯ l m ( p ) * ,
s 1 ; l m = ( l m ) ( l + 1 + m ) 2 , s 0 ; l m = m , s 1 ; l m = ( l + m ) ( l + 1 m ) 2 .
ω T = k B T I ,
Γ μ ; η ¯ η ( ext ) = c Γ plm s μ ; l m p ¯ l ¯ W I η l , m μ ( p ) 𝓢 ¯ lm l ¯ m ( p p ¯ ) * W I η ¯ l ¯ m ( p ¯ ) * ,
Γ μ ; η ¯ η ( sca ) = c Γ plm s μ ; l m p ¯ p ¯ l ¯ l ¯ m 𝓢 ¯ l , m μ , l ¯ , m μ ( p p ¯ ) W I η l ¯ , m μ ( p ¯ ) 𝓢 ¯ l m l ¯ m ( p p ¯ ) * W I η l ¯ m ( p ) * ,
T η = 8 π k I I σ T η Γ Rad η ,

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