Abstract

Results are given of the spectrophotometric measurements of the earth’s atmosphere twilight aureole as first performed from the Soyuz-5 spacecraft. The analysis is made of the experimental findings depending on the wavelength, the perigee height of the sight ray above the earth’s surface, the sunset angle, etc. The comparison is carried out of the vertical profiles for the twilight aureole monochromatic brightness with the results of the corresponding theoretical calculations for the aerosol Elterman model (1968). The color diagram and pictures of the twilight aureole are built using theoretical values of the brightness aureole for different terrestrial atmospheric models and employing the experimental data and the results of visual observations from the Soyuz-5 spacecraft.

© 1971 Optical Society of America

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  1. K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).
  2. G. V. Rosenberg, V. V. Nikolaeva-Tereshkova, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 4 (1965).
  3. W. M. Shirra, Air-Ground Communications of the MA-8 Flight, Results of the Third U.S. Manned Orbital Spacecraft, NASA Publ. SP-12 (1962).
  4. L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).
  5. A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).
  6. C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).
  7. O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 1 (1969).
  8. O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 8 (1969).
  9. L. Elterman, “UV, Visible, and IR Attenuation for Altitudes to 50 km,” 1968 AFCRL-68-0153, Environmental Res. Paper 285 (1968).
  10. G. I. Marchuk, G. A. Mlikhailov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).
  11. O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).
  12. K. S. Shifrin, E. A. Chayanova, “Phase Function for the Junge Distribution and the Junge Type,” Proc. Main Geophysical Observatory 170 (1965).
  13. M. M. Gurevich, Color and Its Measurement (Izd. Akad. Nauk SSSR, Moscow).
  14. L. Elterman, “An Atlas of Aerosol Attenuation and Extinction Profiles for the Troposphere and Stratosphere,” Rept. AFCRL 66-828 (AFCRL, Bedford, Mass., 1966).

1969 (2)

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 1 (1969).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 8 (1969).

1967 (3)

G. I. Marchuk, G. A. Mlikhailov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

1966 (1)

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

1965 (3)

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

G. V. Rosenberg, V. V. Nikolaeva-Tereshkova, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 4 (1965).

K. S. Shifrin, E. A. Chayanova, “Phase Function for the Junge Distribution and the Junge Type,” Proc. Main Geophysical Observatory 170 (1965).

Chayanova, E. A.

K. S. Shifrin, E. A. Chayanova, “Phase Function for the Junge Distribution and the Junge Type,” Proc. Main Geophysical Observatory 170 (1965).

Dave, I. V.

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

Driving, A. Ya.

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

Dunkelman, L.

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

Elterman, L.

L. Elterman, “UV, Visible, and IR Attenuation for Altitudes to 50 km,” 1968 AFCRL-68-0153, Environmental Res. Paper 285 (1968).

L. Elterman, “An Atlas of Aerosol Attenuation and Extinction Profiles for the Troposphere and Stratosphere,” Rept. AFCRL 66-828 (AFCRL, Bedford, Mass., 1966).

Evans, D. C.

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

Feoktistov, K. P.

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Gill, I. R.

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

Gurevich, M. M.

M. M. Gurevich, Color and Its Measurement (Izd. Akad. Nauk SSSR, Moscow).

Marchuk, G. I.

G. I. Marchuk, G. A. Mlikhailov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

Mateer, C. L.

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

McDivitt, J. A.

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

Mikhailin, I. M.

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

Mlikhailov, G. A.

G. I. Marchuk, G. A. Mlikhailov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

Nikolaeva-Tereshkova, V. V.

G. V. Rosenberg, V. V. Nikolaeva-Tereshkova, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 4 (1965).

Roach, F. E.

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

Rosenberg, G. V.

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

G. V. Rosenberg, V. V. Nikolaeva-Tereshkova, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 4 (1965).

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Sandromirsky, A. B.

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Sergeevich, V. N.

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Shifrin, K. S.

K. S. Shifrin, E. A. Chayanova, “Phase Function for the Junge Distribution and the Junge Type,” Proc. Main Geophysical Observatory 170 (1965).

Shirra, W. M.

W. M. Shirra, Air-Ground Communications of the MA-8 Flight, Results of the Third U.S. Manned Orbital Spacecraft, NASA Publ. SP-12 (1962).

Smokty, O. I.

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 1 (1969).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 8 (1969).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

Sonechkin, D. A.

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Trifonov, G. I.

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

White, E. H.

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1)

C. L. Mateer, I. V. Dave, L. Dunkelman, D. C. Evans, Evidence of an Upper Stratospheric Dust Layer in a Satellite Twilight Color Photograph, NCAR ms. 465 (1967).

Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana (1)

G. V. Rosenberg, V. V. Nikolaeva-Tereshkova, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 4 (1965).

Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana (5)

A. Ya. Driving, I. M. Mikhailin, G. V. Rosenberg, A. B. Sandromirsky, G. I. Trifonov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 2, 6 (1966).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 1 (1969).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 5, 8 (1969).

G. I. Marchuk, G. A. Mlikhailov, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

O. I. Smokty, Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana 3, 4 (1967).

Proc. Main Geophysical Observatory (1)

K. S. Shifrin, E. A. Chayanova, “Phase Function for the Junge Distribution and the Junge Type,” Proc. Main Geophysical Observatory 170 (1965).

Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1)

K. P. Feoktistov, G. V. Rosenberg, A. B. Sandromirsky, V. N. Sergeevich, D. A. Sonechkin, “Some Results of Optical Observations from the Vostok Spacecraft,” Sb. Issl. kosm. prostr. Moscow Izd. Nauka (1965).

Other (5)

W. M. Shirra, Air-Ground Communications of the MA-8 Flight, Results of the Third U.S. Manned Orbital Spacecraft, NASA Publ. SP-12 (1962).

L. Dunkelman, I. R. Gill, J. A. McDivitt, F. E. Roach, E. H. White, Geoastronomical Observations, Manned Space Flight Experiments Symposium Gemini 3 and 4, NASA Publ. (1965).

L. Elterman, “UV, Visible, and IR Attenuation for Altitudes to 50 km,” 1968 AFCRL-68-0153, Environmental Res. Paper 285 (1968).

M. M. Gurevich, Color and Its Measurement (Izd. Akad. Nauk SSSR, Moscow).

L. Elterman, “An Atlas of Aerosol Attenuation and Extinction Profiles for the Troposphere and Stratosphere,” Rept. AFCRL 66-828 (AFCRL, Bedford, Mass., 1966).

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Figures (17)

Fig. 1
Fig. 1

Optical scheme of the hand spectrograph: (1) entrance objective of the spectrograph; (2) entrance objective of the photo-fixation channel; (3) shutter of the photo-fixation channel; (4,5,7,8) flat mirrors; (6) photo-film; (9) collimating objective; (10) diffraction grating; (11) entrance slit of spectrograph; and (12) shutter of the spectrograph channel.

Fig. 2
Fig. 2

Color photo of the twilight aureole of different models of the earth atmosphere: (a) molecular atmosphere; (b) molecular atmosphere with the presence of aerosol; (c) molecular atmosphere with the presence of aerosol and ozone. [These three curves—primary, scattering, Elterman model (1968); δ = 0°, φ = 8°, altitude of the spacecraft h = 252.3 km]; (d) visual data, Soyuz-5, 2nd orbit, δ = 3–10°, φ = 0°, altitude of the spacecraft h ≃ 233 km; (e) experimental data Soyuz-5, 6th orbit, δ ≃ 0°, φ = 8°, altitude of the spacecraft h = 252.3 km.

Fig. 3
Fig. 3

Photograph and spectrum of the earth atmosphere aureole (Frame 6).

Fig. 4
Fig. 4

Photograph and spectrum of the earth atmosphere aureole (Frame 7).

Fig. 5
Fig. 5

Spectral variation in the twilight aureole brightness of the earth atmosphere for different altitudes: — theoretical results; -×-×- experimental results (Soyuz-5).

Fig. 6
Fig. 6

Monochromatic brightness curves for the earth atmosphere twilight aureole. Frame 7 (Soyuz-5). In this figure and all subsequent figures, the units for Iλ are W sr−1 m−2 nm−1.

Fig. 7
Fig. 7

Sight geometry of the twilight atmosphere from the spacecraft.

Fig. 8
Fig. 8

Monochromatic brightness curves for the twilight aureole of the earth molecular atmosphere. Without the consideration of ozone absorption, the Elterman model, 1968, δ = 0°, φ = 8°, h = 252.3 km. In Figs. 8, 9 and 10 the wavelength sequence of the curves is: 400, 450, 500, 550, 600, 700, and 800 nm.

Fig. 9
Fig. 9

Monochromatic brightness curves for the twilight aureole of the molecular earth atmosphere in the presence of nonabsorbing particles and ozone (the 1968 Elterman model, δ = 0°, φ = 8°, h = 252.3 km).

Fig. 10
Fig. 10

Monochromatic brightness curves for the twilight aureole, of the molecular earth atmosphere with the presence of aerosol particles and ozone (the 1968 Elterman model, δ = 0°, φ = 8°, h = 22.3 km).

Fig. 11
Fig. 11

Comparison of theoretical and experimental values for the monochromatic brightness of the earth atmosphere twilight aureole: — molecular atmosphere with the presence of ozone and aerosol; -×-×- molecular atmosphere; -○-○- experimental data (Soyuz-5).

Fig. 12
Fig. 12

Comparison of theoretical and experimental results at 500 nm: — Elterman model (1968); -×-×- molecular atmosphere; -○-○- experimental data (Soyuz-5).

Fig. 13
Fig. 13

Comparison of theoretical and experimental results at 550 nm: — Elterman model (1968); -×-×- molecular atmosphere; -○-○- experimental data (Soyuz-5).

Fig. 14
Fig. 14

Comparison of theoretical and experimental results at 600 nm: — Elterman model (1968); -×-×- molecular atmosphere; -○-○- experimental data (Soyuz-5).

Fig. 15
Fig. 15

Comparison of theoretical and experimental results at 650 nm: — Elterman model (1968); -×-×- molecular atmosphere; -○-○- experimental data (Soyuz-5).

Fig. 16
Fig. 16

Monochromatic brightness curves for the earth atmosphere twilight aureole: - - - data of Mateer et al.6; — data of this paper.

Fig. 17
Fig. 17

Color diagram for the twilight aureole of the real earth atmosphere (angle for the sun setting beyond the horizon is δ = 0°, the orbital spacecraft altitude is h = 252.3 km, azimuth is φ = 8°): -●- molecular atmosphere; -○- molecular atmosphere with the presence of aerosol; -×- molecular atmosphere with the presence of aerosol and ozone [these three curves—primary, scattering, Elterman model (1968)]; -□- experimental data (Soyuz-5, 6th orbit, overcast sky).

Tables (1)

Tables Icon

Table I Color Coordinates X and Y of the Twilight Aureole for Different Models of the Earth’s Atmosphere (Primary Scattering, the Angle of the Sunset δ = 00, Orbital Altitude of the Spacecraft h = 252.3 km, Azimuth φ = 8°)

Equations (19)

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P = 2 [ I 1 ( I 1 - I 2 ) + I 2 ( I 2 - I 3 ) + I 3 ( I 3 - I 1 ) ] 1 2 I 1 + I 2 + I 3 ,
cos θ I λ r - sin θ r I λ θ + sin θ cos φ r I λ ψ - cot ψ sin θ sin φ r × I λ φ = - α λ ( r ) I λ ( r , θ , ψ , φ ) + B λ ( r , θ , ψ , φ ) ,
B λ ( r , θ , ψ , φ ) = a λ ( r ) 4 π Ω I λ ( r , ψ , θ , φ ) x λ ( r , γ ) d ω + a λ ( r ) S λ 4 x λ ( r 1 γ 1 ) exp [ - T λ ( r , ψ ) ] ,
a λ ( r ) = σ mol ( r , λ ) + aer ( r , λ ) σ mol ( r , λ ) + σ aer ( λ , r ) + K ( λ , r ) ,
a λ ( r ) = a mol ( r , λ ) + a aer ( r , λ ) ,
a mol ( r , λ ) = σ mol ( r , λ ) σ mol ( r , λ ) + σ aer ( r , λ ) + K ( λ , r ) , a aer ( r , λ ) = σ aer ( r , λ ) σ aer ( r , λ ) + σ mol ( r , λ ) + K ( λ , r ) ,
x ( r , γ , λ ) = σ mol ( r , λ ) x mol ( r , γ , λ ) σ mol ( r , λ ) + σ aer ( r , λ ) + σ aer ( r , λ ) x aer ( r , γ , λ ) σ aer ( r , λ ) + σ mol ( r , λ ) ,
1 4 π Ω x mol ( γ , r , λ ) d ω = 1 4 π Ω x aer ( γ , r , λ ) d ω = 1.
I λ ( r , ψ , θ , φ ) = 0 at θ > π / 2
I λ ( r , ψ , θ , φ ) = I *             at             θ < π / 2 ,
I λ ( r , θ , ψ , φ ) = R 0 r B λ ( r , ψ , θ , φ ) exp [ - r r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 + A λ π 0 2 π d φ 0 π / 2 I λ ( R 0 , ψ , π - θ , φ ) sin θ sin θ d θ exp [ - R 0 r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] ,             at             0 θ θ 0 ,
I λ ( r , θ , ψ , φ ) = r 0 R 1 B λ ( r , θ , ψ , φ ) exp [ - r 0 r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 - r 0 r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] × α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 + r 0 r B λ ( r , θ , ψ , φ ) exp [ - r r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2             at             θ 0 θ π / 2 ,
I λ ( r , θ , ψ , φ ) = r R 1 B λ ( r , θ , ψ , φ ) exp [ - r r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2             at             π / 2 θ < π - θ 0 .
B λ ( r , θ , ψ , φ ) 0 ,
B λ ( r , θ , ψ , φ ) = S λ 4 [ a mol ( r , λ ) x ( γ 4 , λ ) + a aer ( r , λ ) x aer ( γ 1 , λ ) ] × exp [ - T λ ( r , ψ ) ] .
I λ ( 1 ) ( r , θ , ψ , φ ) = S λ 4 r c R t [ σ mol ( r , λ ) x mol ( γ 1 , λ ) + σ aer ( r 1 λ ) x aer ( γ 1 , λ ) ] exp [ - T λ ( r 1 ψ ) - r 0 r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 - r 0 r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] × r d r ( r 2 - r 0 2 ) 1 2 + S λ 4 r 0 r c σ mol ( r 1 λ ) x mol ( γ 1 , λ ) + σ aer ( r 1 λ ) x aer ( γ 1 , λ ) ] exp [ - T λ ( r , 1 ψ ) - r r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] r d r ( r 2 - r 0 2 ) 1 2             at             θ 0 θ < π / 2 .
I λ ( 1 ) ( r , θ , ψ , φ ) = S λ 4 r c R 1 [ σ mol ( r , λ ) x mol ( γ 1 , λ ) + σ aer ( r , λ ) x aer ( γ 1 λ ) ] × exp [ - T λ ( r , ψ ) - r r α λ ( r ) r d r ( r 2 - r 0 2 ) 1 2 ] r d r ( r 2 - r 0 2 ) 1 2             at             π 2 θ < π - θ 0 ,
X = 0 λ I ( λ ) x ¯ ( λ ) d λ Δ λ I ( λ ) [ x ¯ ( λ ) + y ¯ ( λ ) + z ¯ ( λ ) ] d λ ,
Y = Δ λ I ( λ ) y ¯ ( λ ) d λ Δ λ I ( λ ) [ x ¯ ( λ ) + y ¯ ( λ ) + z ¯ ( λ ) ] d λ

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