Abstract

The Kolmogorov theory is used to illustrate that the mean-square wave-front error E2, which results when the wave-front distortion associated with an artificial guide-star reference is used to compensate a telescope aperture of diameter D, for imaging an object at infinity is given by E2 = (D/d0)5/3. The quantity d0 is a measure of the effective diameter of the compensated imaging system (i.e., a telescope with a diameter equal to d0 will have 1 rad of rms wave-front error) and is expressed as an integral over the Cn2 profile. The Cn2-weighting function is expressed in terms of hypergeometric functions whose series representation converges very rapidly. (Typically only a few terms are required). As a result d0 can be evaluated quite quickly on a microcomputer or a scientific calculator. In this study the quantity d0 is evaluated for six Cn2 profiles of interest, illustrating the importance of including the altitude weighting in the theoretical formulation of d0. In addition, this study illustrates the importance of removing the piston and the tilt from the wave-front distortion when assessing the performance of an imaging system.

© 1994 Optical Society of America

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References

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  1. G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).
  2. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
    [CrossRef]
  3. D. L. Fried, J. L. Belsher, “Analysis of artificial guide-star (AGS) adaptive-optics system (AOS) performance for astronomical imaging,” (Optical Sciences Company, Placentia, Calif., 1991).
  4. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [CrossRef]
  5. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]
  6. V. I. Tatarski, Wave Propagation through a Turbulent Medium (McGraw-Hill, New York, 1961).
  7. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 15.1.20.
  8. Ref. 7, Eq. 22.9.3.
  9. Ref. 7, Eq. 22.3.12.
  10. Ref. 7, Eq. 15.1.1.
  11. Ref. 7, Eq. 15.1.20.

1991 (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

1989 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 15.1.20.

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Belsher, J. L.

D. L. Fried, J. L. Belsher, “Analysis of artificial guide-star (AGS) adaptive-optics system (AOS) performance for astronomical imaging,” (Optical Sciences Company, Placentia, Calif., 1991).

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

D. L. Fried, J. L. Belsher, “Analysis of artificial guide-star (AGS) adaptive-optics system (AOS) performance for astronomical imaging,” (Optical Sciences Company, Placentia, Calif., 1991).

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 15.1.20.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation through a Turbulent Medium (McGraw-Hill, New York, 1961).

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

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

Nature (London) (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Proc. IEEE (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Other (8)

G. A. Tyler, “Rapid evaluation of d0,” (Optical Sciences Company, Placentia, Calif., 1984).

D. L. Fried, J. L. Belsher, “Analysis of artificial guide-star (AGS) adaptive-optics system (AOS) performance for astronomical imaging,” (Optical Sciences Company, Placentia, Calif., 1991).

V. I. Tatarski, Wave Propagation through a Turbulent Medium (McGraw-Hill, New York, 1961).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. 15.1.20.

Ref. 7, Eq. 22.9.3.

Ref. 7, Eq. 22.3.12.

Ref. 7, Eq. 15.1.1.

Ref. 7, Eq. 15.1.20.

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Figures (3)

Fig. 1
Fig. 1

Function Fn(h/H). The break at h/H = 1 occurs because of the contribution of the turbulence above the backscatter altitude H. The dashed curve is for the case with the tilt included (u = 0), and the solid curve is for the case with the tilt removed (u = 1). The piston is removed regardless of the value of u.

Fig. 2
Fig. 2

Turbulence models studied in this research. (a) The SLC-Day model (top curve) and the SLC-Night model (bottom curve). From top to bottom the three solid curves in (b) illustrate the turbulence models HV70, HV54, and HV27, respectively. The dotted (lowest) curve pertains to HV5/7.

Fig. 3
Fig. 3

Altitude dependence of d0 for various turbulence models. (a), (b), (c), (d), (e), (f) The results associated with the turbulence models SLC-Day, SLC-Night, HV5/7, HV27, HV54, and HV70, respectively. In each figure the curves correspond to propagation at zenith with an operating wavelength equal to 0.5 μm. The top two curves in each figure were obtained from Eq. (60), where the upper curve is for the tilt removed (u = 1) and the lower curve is for the tilt included (u = 0). The lowest curve in each figure is for the case with the piston and the tilt included and was obtained from Eq. (C23).

Tables (1)

Tables Icon

Table 1 Function Fn(h/H)

Equations (130)

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E 2 = d r W ( r / R ) [ ϕ ( r ) - ϕ s ( r ) - ϕ s ( r ) ] 2 / d r W ( r / R ) ,
W ( r ) = { 1 for r 1 0 otherwise .
r = ρ R
E 2 = 1 π d ρ W ( ρ ) [ ϕ ( ρ R ) - ϕ s ( ρ R ) ] 2 .
E 2 = 1 π d ρ W ( ρ ) ϕ 2 ( ρ R ) - 2 π d ρ W ( ρ ) ϕ ( ρ R ) ϕ s ( ρ R ) + 1 π d ρ W ( ρ ) ϕ s 2 ( ρ R ) .
E 2 = ( D / d 0 ) 5 / 3 ,
d 0 = D ( E 2 ) - 3 / 5 .
ϕ a ( r ) = ϕ T a ( r ) - A - 1 d r W ( r / R ) ϕ T a ( r ) - u a 64 π D 4 d r W ( r / R ) ϕ T a ( r ) r · r ,
A = d r W ( r / R ) ,
ϕ T a ( r ) = k 0 R a d z n [ r ( 1 - z / R a ) , z ] .
ϕ a ( r ) = k A - 1 0 R a d r d z W ( r / R ) × { n [ r ( 1 - z R a ) , z ] - n [ r ( 1 - z R s ) , z ] - 4 π u a A - 1 r · r n [ r ( 1 - z R a ) , z ] } ,
ϕ b ( r ) = k A - 1 0 R b d r d z W ( r / R ) × { n [ r ( 1 - z R b ) , z ] - n [ r ( 1 - z R b ) , z ] - 4 π n b A - 1 r · r n [ r ( 1 - z R b ) , z ] } .
ϕ a ( r 1 ) ϕ b ( r 2 ) = k 2 A - 2 0 R b 0 R b d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) × ( { n [ r 1 ( 1 - z 1 R a ) , z 1 ] - n [ r 1 ( 1 - z 1 R a ) , z 1 ] } × { n [ r 2 ( 1 - z 2 R b ) , z 2 ] - n [ r 2 ( 1 - z 2 R b ) , z 2 ] } - 4 π u b A - 1 r 2 · r 2 { n [ r 1 ( 1 - z 1 R a ) , z 1 ] - n [ r 1 ( 1 - z 1 R a ) , z 1 ] } n [ r 2 ( 1 - z 2 R b ) , z 2 ] - 4 π u a A - 1 r 1 · r 1 { n [ r 2 ( 1 - z 2 R b ) , z 2 ] - n [ r 2 ( 1 - z 2 R b ) , z 2 ] } n [ r 1 ( 1 - z 1 R a ) , z 1 ] + 16 π 2 u a u b A - 2 ( r 1 · r 1 ) ( r 2 · r 2 ) × n [ r 1 ( 1 - z 1 R a ) , z 1 ] n ( r 2 - z 2 R b , z 2 ) ) .
[ n ( r 1 , z 1 ) - n ( r 2 , z 1 ) ] n ( r , z ) = - 1 2 { [ n ( r 1 , z 1 ) - n ( r , z ) ] 2 - [ n ( r 2 , z 1 ) - n ( r , z ) ] 2 } .
[ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 = C n 2 ( z 1 + z 2 2 ) × ( r 1 - r 2 2 + z 1 - z 2 2 ) 1 / 3 .
z + = 1 2 ( z 1 + z 2 ) ,
z - = z 1 - z 2 .
ϕ a ( r 1 ) ϕ b ( r 2 ) = - 1 2 k 2 A - 2 0 R b d r 1 d r 2 d z + d z - W ( r 1 / R ) W ( r 2 / R ) C n 2 ( z + ) × ( [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 - z - 2 / 3 - [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 + z - 2 / 3 - [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 + z - 2 / 3 + [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 - z - 2 / 3 - 4 π u b A - 1 r 2 · r 2 { [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 - z - 2 / 3 - [ | r 1 ( 1 - z + + ½ z - R b ) - r 2 ( 1 - z + - ½ z - R a ) | 2 + z - 2 ] 1 / 3 + z - 2 / 3 } - 4 π u a A - 1 r 1 · r 1 { [ | r 2 ( 1 - z + - ½ z - R b ) - r 1 ( 1 - z + + ½ z - R a ) | 2 + z - 2 ] 1 / 3 - z - 2 / 3 - [ | r 2 ( 1 - z + - ½ z - R a ) - r 1 ( 1 - z + + ½ z - R b ) | 2 + z - 2 ] 1 / 3 + z - 2 / 3 } + 16 π 2 u a u b A - 2 ( r 1 · r 1 ) ( r 2 · r 2 ) × { [ | r 1 ( 1 - z + + ½ z - R a ) - r 2 ( 1 - z + - ½ z - R b ) | 2 + z - 2 ] 1 / 3 - z - 2 / 3 } ) .
d z [ ( A 2 + z 2 ) 1 / 3 - z 2 / 3 ] = 2 1 / 3 Γ 2 ( 1 / 6 ) 5 Γ ( 1 / 3 ) A 5 / 3 ,
2 1 / 3 Γ 2 ( 1 / 6 ) 5 Γ ( 1 / 3 ) 2.91 ,
ϕ a ( r 1 ) ϕ b ( r 2 ) = - 2.91 2 k 2 A - 2 0 R b d r 1 d r 2 d z W ( r 1 / R ) W ( r 2 / R ) C n 2 ( z ) × { r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 - r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 - r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 + r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 - 4 π u b A - 1 r 2 · r 2 [ r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 - r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 ] - 4 π u a A - 1 r 1 · r 1 [ r 2 ( 1 - z / R b ) - r 1 ( 1 - z / R a ) 5 / 3 - r 2 ( 1 - z / R b ) - r 1 ( 1 - z / R a ) 5 / 3 ] + 16 π 2 u a u b A - 2 ( r 1 · r 1 ) ( r 2 · r 2 ) r 1 ( 1 - z / R a ) - r 2 ( 1 - z / R b ) 5 / 3 } .
r 1 = R ρ 1 ,
r 2 = R ρ 2 ,
r 1 = R ρ 1 ,
r 2 = R ρ 2 .
r 0 = [ Γ 2 ( 1 / 6 ) sec ( ψ ) 2 2 / 3 5 Γ ( 1 / 3 ) [ ²⁴ / Γ ( 6 / 5 ) ] 5 / 6 ( 2 π λ ) 2 d h C n 2 ( h ) ] - 3 / 5 ,
Γ 2 ( 1 / 6 ) 2 2 / 3 5 Γ ( 1 / 3 ) [ ²⁴ / Γ ( 6 / 5 ) ] 5 / 6 2.91 6.88
z = h sec ( ψ ) ,
R a = H a sec ( ψ ) ,
R b = H b sec ( ψ ) ,
ϕ a ( ρ 1 R ) ϕ b ( ρ 2 R ) / ( D / r 0 ) 5 / 3 = - 6.88 π 2 2 8 / 3 [ d h C n 2 ( h ) ] - 1 0 H b d ρ 1 d ρ 2 d h W ( ρ 1 ) W ( ρ 2 ) × C n 2 ( h ) ( 1 - h / H a ) 5 / 3 [ ρ 1 - α ρ 2 5 / 3 - ρ 1 - α ρ 2 5 / 3 - ρ 1 - α ρ 2 5 / 3 + ρ 1 - α ρ 2 5 / 3 - 4 u b ρ 2 · ρ 2 ( ρ 1 - α ρ 2 5 / 3 - ρ 1 - α ρ 2 5 / 3 ) - 4 u a ρ 1 · ρ 1 ( ρ 1 - α ρ 2 5 / 3 - ρ 1 - α ρ 2 5 / 3 ) + 16 u a u b ( ρ 1 · ρ 1 ) ( ρ 2 · ρ 2 ) ρ 1 - α ρ 2 5 / 3 ] ,
α = ( 1 - h / H b ) / ( 1 - h / H a ) .
1 π d ρ W ( ρ ) ϕ a ( ρ R ) ϕ b ( ρ R ) / ( D / r 0 ) 5 / 3 = - 6.88 2 8 / 3 [ d h C n 2 ( h ) ] - 1 0 H b d h C n 2 ( h ) ( 1 - h / H a ) 5 / 3 × 1 π 3 d ρ d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) W ( ρ ) × [ ρ - α ρ 5 / 3 - ρ - α ρ 2 5 / 3 - ρ 1 - α ρ + ρ 1 - α ρ 2 5 / 3 - 4 u b ρ 2 · ρ ( ρ - α ρ 2 5 / 3 - ρ 1 - α ρ 2 5 / 3 ) - 4 u a ρ 1 · ρ ( ρ 1 - α ρ 5 / 3 - ρ 1 - α ρ 2 5 / 3 ) + 16 u a u b ( ρ 1 · ρ ) ( ρ 2 · ρ ) ρ 1 - α ρ 2 5 / 3 ] ,
( ρ 1 · ρ ) ( ρ 2 · ρ ) = 1 2 ( ρ · ρ ) ( ρ 1 · ρ 2 ) ,
1 π d ρ W ( ρ ) ϕ a ( ρ R ) ϕ b ( ρ R ) / ( D / r 0 ) 5 / 3 = 6.88 2 8 / 3 [ d h C n 2 ( h ) ] - 1 0 H b d h C n 2 ( h ) ( 1 - h / H a ) 5 / 3 × { - ( 1 - α ) 5 / 3 1 π d ρ W ( ρ ) ρ 5 / 3 + 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 + [ 4 ( u a + u b ) - 8 ( u a u b ) 1 π d ρ W ( ρ ) ρ · ρ ] × 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 ρ 1 · ρ 2 } .
1 π d ρ W ( ρ ) ρ 5 / 3 = 6 11 ,
1 π d ρ W ( ρ ) ρ · ρ = 1 2 ,
1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 = 6 11 F 2 1 ( - 11 6 , - 5 6 ; 2 ; α 2 ) ,
1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 ρ 1 · ρ 2 = 5 22 α 2 F 1 ( - 11 6 , 1 6 ; 3 ; α 2 ) .
1 π d ρ W ( ρ ) ϕ a ( ρ R ) ϕ b ( ρ R ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 5 ) ] 5 / 6 [ d h C n 2 ( h ) ] - 1 0 H b d h C n 2 ( h ) ( 1 - h / H a ) 5 / 3 × [ 6 11 F 2 1 ( - 11 6 , - 5 6 ; 2 ; α 2 ) - 6 11 ( 1 - α ) 5 / 3 - 10 11 ( u a + u b - u a u b ) α 2 F 1 ( - 11 6 , 1 6 ; 3 ; α 2 ) ] .
6.88 = 2 [ 24 5 Γ ( 6 / 5 ) ] 5 / 6 .
Γ ( 11 / 5 ) = ( 6 / 5 ) Γ ( 6 / 5 ) ,
u = u a + u b - u a u b .
H a = ,
H b =
α = 1.
1 π d ρ W ( ρ ) ϕ 2 ( ρ R ) / ( D / r 0 ) 5 / 3 = [ Γ ( 11 / 15 ) ] 5 / 6 [ 6 11 F 2 1 ( - 11 6 , - 5 6 ; 2 ; 1 ) - 10 11 u F 2 1 ( - 11 6 , 1 6 ; 3 ; 1 ) ] .
F 2 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) .
1 π d ρ W ( ρ ) ϕ 2 ( ρ R ) / ( D / r 0 ) 5 / 3 = 1.0324216 - u ( 0.8977579 ) .
1 π d ρ W ( ρ ) ϕ 2 ( ρ R ) / ( D / r 0 ) 5 / 3 = { 1.0324216 ( tilt included ) 0.1346636 ( tilt removed ) .
H a = ,
H b = H ,
α = 1 - h / H .
- 2 π d ρ W ( ρ ) ϕ ( ρ R ) ϕ s ( ρ R ) / ( D / r 0 ) 5 / 3 = - 2 [ Γ ( 11 / 5 ) ] 5 / 6 [ d h C n 2 ( h ) ] - 1 0 H d h C n 2 ( h ) × { 6 11 F 2 1 [ - 11 6 , - 5 6 ; 2 ; ( 1 - h / H ) 2 ] - 6 11 ( h / H ) 5 / 3 - u 10 11 ( 1 - h / H ) F 2 1 [ - 11 6 , 1 6 ; 3 ; ( 1 - h / H ) 2 ] } .
H a = H ,
H b = H ,
α = 1.
1 π d ρ W ( ρ ) ϕ s 2 ( ρ R ) / ( D / r 0 ) 5 / 3 = [ 1.0324216 - u ( 0.8977579 ) ] [ d h C n 2 ( h ) ] - 1 × 0 H d h C n 2 ( h ) ( 1 - h / H ) 5 / 3 .
E 2 / ( D / r 0 ) 5 / 3 = [ 1.0324216 - u ( 0.8977579 ) ] × { 1 + [ d h C n 2 ( h ) ] - 1 0 H d h C n 2 ( h ) × ( 1 - h / H ) 5 / 3 } - 2 [ Γ ( 11 / 5 ) ] 5 / 6 × [ d h C n 2 ( h ) ] - 1 0 H d h C n 2 ( h ) × { 6 11 F 2 1 [ - 11 6 , - 5 6 ; 2 ; ( 1 - h / H ) 5 / 3 ] - 6 11 ( h / H ) 5 / 3 - u 10 11 ( 1 - h / H ) × F 2 1 [ - 11 6 , 1 6 ; 3 ; ( l - h / H ) 2 ] } .
d 0 = r 0 ( [ 1.0324216 - u ( 0.8977579 ) ] { 1 + [ d h C n 2 ( h ) ] - 1 × 0 H d h C n 2 ( h ) ( 1 - h / H ) 5 / 3 } - 2 [ Γ ( 11 / 5 ) ] 5 / 6 [ d h C n 2 ( h ) ] - 1 0 H d h C n 2 ( h ) × { 6 11 F 2 1 [ - 11 6 , - 5 6 ; 2 ; ( 1 - h / H ) 5 / 3 ] - 6 11 ( h / H ) 5 / 3 - u 10 11 ( 1 - h / H ) × F 2 1 [ - 11 6 , 1 6 ; 3 ; ( l - h / H ) 2 ] } ) - 3 / 5 .
d 0 = λ 6 / 5 cos 3 / 5 ( ψ ) [ d h C n 2 ( h ) F ( h / H ) ] - 3 / 5 ,
F ( h / H ) = 16.71371210 ( ( 1.032421640 - u 0.8977579487 ) × [ 1 + ( 1 - h / H ) 5 / 3 ] - 2.168285442 × { 6 11 F 2 1 [ - 11 6 , - 5 6 ; 2 ; ( 1 - h H ) 2 ] - 6 11 ( h / H ) 5 / 3 - u 10 11 ( 1 - h / H ) × F 2 1 [ - 11 6 , 1 6 ; 3 ; ( 1 - h H ) 2 ] } ) for h < H ,
F ( h / H ) = 16.71371210 ( 1.032421640 - u 0.8977579487 ) for h > H ,
d 0 = 2.8772318 ϑ 0 R ,
I 1 ( α ) = 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3
ρ = α ρ 2 ,
r = ρ 1 ,
I 1 ( α ) = 1 π α 2 d ρ W ( ρ / α ) 1 π d r W ( r ) r - ρ 5 / 3 ,
I 1 ( α ) = 1 π α 2 d ρ W ( ρ / α ) 1 π 0 1 r d r 0 2 / π d ϕ × [ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 .
n = 0 C n α ( x ) z n = ( 1 - 2 x z + z 2 ) - α             valid for z < 1 ,             a 0 ,
C n α cos ( ϕ ) = m = 0 n Γ ( a + m ) Γ ( α + n - m ) m ! ( n - m ) ! [ Γ ( α ) ] 2 × cos [ ( n - 2 m ) ϕ ] .
[ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 = { r 5 / 3 [ 1 - 2 ρ r cos ( ϕ ) + ρ 2 r 2 ] 5 / 6 for r > ρ ρ 5 / 3 [ 1 - 2 r ρ cos ( ϕ ) + r 2 ρ 2 ] 5 / 6 for r < ρ .
[ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 = { r 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] 2 cos [ ( n - 2 m ) ϕ ] ( ρ / r ) n r > ρ ρ 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] 2 cos [ ( n - 2 m ) ϕ ] ( r / ρ ) n r < ρ .
0 2 π d ϕ [ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 = { 2 π r 5 / 3 n = 0 a n ( ρ / r ) 2 n r > ρ 2 π ρ 5 / 3 n = 0 a n ( r / ρ ) 2 n r < ρ ,
a n = [ Γ ( - + n ) n ! Γ ( - ) ] 2 .
I 1 ( α ) = 2 π α 2 d ρ W ( ρ / α ) × n = 0 a n ( ρ 5 / 3 - 2 n 0 ρ d r r 2 n + 1 + ρ 2 n ρ 1 d r r 8 / 3 - 2 n ) .
I 1 ( α ) = 4 α 2 0 α ρ d ρ × n = 0 a n ( ρ 11 / 3 2 n + 2 + ρ 2 n 11 / 3 - 2 n - ρ 11 / 3 11 / 3 - 2 n ) .
S 1 = n = 0 a n 1 2 n + 2 ,
S 2 = n = 0 a n 1 ¹¹ / - 2 n .
S 1 = n = 0 [ Γ ( - + n ) n ! Γ ( - ) ] 2 1 2 n + 2 .
S 1 = 1 2 Γ ( 2 ) Γ ( - ) Γ ( - ) n = 0 Γ ( - + n ) Γ ( - + n ) Γ ( 2 + n ) 1 n ! .
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) z n n ! ,
S 1 = 1 2 F 2 1 ( - 5 6 , - 5 6 ; 2 ; 1 ) .
F 2 1 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) .
S 1 = 1 2 Γ ( 11 / 3 ) [ Γ ( 17 / 6 ) ] 2 ,
Γ ( 2 ) = 1.
S 2 = n = 0 [ Γ ( - + n ) n ! Γ ( - ) ] 2 1 11 / 3 - 2 n
S 2 = 3 11 Γ ( 1 ) Γ ( - ) Γ ( - ¹¹ / ) n = 0 Γ ( - + n ) Γ ( - ¹¹ / + n ) Γ ( 1 + n ) 1 n ! .
Γ ( z + 1 ) = z Γ ( z ) .
S 2 = 3 11 F 2 1 ( - 5 6 , - 11 6 ; 1 ; 1 ) ,
S 2 = 3 11 Γ ( 1 ) Γ ( 11 / 3 ) Γ ( ¹⁷ / ) Γ ( ¹¹ / ) .
S 2 = 1 2 Γ ( 11 / 3 ) [ Γ ( 17 / 6 ) ] 2 .
n = 0 a n 1 2 n + 2 = n = 0 a n 1 11 / 3 - 2 n .
I 1 ( α ) = 4 α 2 n = 0 a n 1 ¹¹ / - 2 n 0 α d ρ ρ 2 n + 1 .
I 1 ( α ) = 4 α 2 n = 0 [ Γ ( - + n ) n ! Γ ( - ) ] 2 1 11 / 3 - 2 n α 2 n + 2 2 n + 2 .
I 1 ( α ) = 6 11 Γ ( 2 ) Γ ( - ¹¹ / ) Γ ( - ) n = 0 Γ ( - ¹¹ / + n ) Γ ( - + n ) Γ ( 2 + n ) α 2 n n ! .
1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 = 6 11 F 2 1 ( - 11 6 , - 5 6 ; 2 ; α 2 ) .
I 2 ( α ) = 1 π 2 d ρ 1 d ρ 2 W ( ρ 1 ) W ( ρ 2 ) ρ 1 - α ρ 2 5 / 3 ρ 1 · ρ 2
ρ = α ρ 2 ,
r = ρ 1 .
I 2 ( α ) = 1 π α 3 d ρ W ( ρ / α ) 1 π ρ 0 1 r 2 d r 0 2 π d ϕ × cos ( ϕ ) [ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 .
r · ρ = r ρ cos ( ϕ ) .
[ r 2 - 2 r ρ + ρ 2 ] 5 / 6 = { r 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] 2 cos [ ( n - 2 m ) ϕ ] ( ρ / r ) n r > ρ ρ 5 / 3 n = 0 m = 0 n Γ ( - + m ) Γ ( - + n - m ) m ! ( n - m ) ! [ Γ ( - ) ] cos [ ( n - 2 m ) ϕ ] ( r / ρ ) n r < ρ .
0 2 π d ϕ cos ( k ϕ ) cos ( l ϕ ) = π δ k l ,             k 0 ,
n - 2 m = ± 1.
n = 2 p + 1 ,
m = p , p + 1 ,
p = 0 , 1 , 2 , .
0 2 π d ϕ cos ( ϕ ) [ r 2 - 2 r ρ cos ( ϕ ) + ρ 2 ] 5 / 6 = { 2 π r 5 / 3 p = 0 b p ( ρ / r ) 2 p + 1 r > ρ 2 π ρ 5 / 3 p = 0 b p ( r / ρ ) 2 p + 1 r < ρ ,
b p = Γ ( - + p ) Γ ( + p ) p ! ( p + 1 ) ! [ Γ ( - ) ] 2 .
I 2 ( α ) = 2 π α 3 d ρ W ( ρ / α ) ρ × p = 0 b p ( ρ 2 / 3 - 2 p 0 ρ d r r 2 p + 3 + ρ 2 p + 1 p 1 d r r 5 / 3 - 2 p ) .
I 2 ( α ) = - 5 22 α 2 F 1 ( - 11 6 , 1 6 ; 3 ; α 2 ) .
ϕ ( r ) = k d z n ( r , z ) ,
ϕ s ( r ) = k d z n [ r ( 1 - z R ) , z ] .
[ ϕ ( r ) - ϕ s ( r ) ] 2 = k 2 d z 1 d z 2 { n ( r , z 1 ) - n [ r ( 1 - z 1 R ) , z 1 ] } × { n ( r , z 2 ) - n [ r ( 1 - z 2 R ) , z 2 ] } .
[ ϕ ( r ) - ϕ s ( r ) ] 2 = - 1 2 k 2 d z 1 d z 2 ( [ n ( r , z 1 ) - n ( r , z 2 ) ] 2 - { n ( r , z 1 ) - n [ r ( 1 - z 2 R ) , z 2 ] } 2 - { n [ r ( 1 - z 1 R ) , z 1 ] - n ( r , z 2 ) } 2 + { n [ r ( 1 - z 1 R ) , z 1 ] - n [ r ( 1 - z 2 r ) , z 2 ] } 2 ) .
[ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 = C n 2 ( z 1 + z 2 2 ) [ r 1 - r 2 2 + z 1 - z 2 2 ] 1 / 3 .
d z [ ( A 2 + z 2 ) 1 / 3 - z 2 / 3 ] = 2 1 / 3 5 Γ 2 ( ) Γ ( ) A 5 / 3 .
[ ϕ ( r ) - ϕ s ( r ) ] 2 = 2 1 / 3 Γ 2 ( ) 5 Γ ( ) k 2 d z C n 2 ( z ) | r z R | 5 / 3 .
ϑ 0 = [ 2 1 / 3 Γ 2 ( ) 5 Γ ( ) k 2 d z C n 2 ( z ) z 5 / 3 ] - 3 / 5 .
[ ϕ ( r ) - ϕ s ( r ) ] 2 = | r ϑ 0 R | 5 / 3 .
E 2 = 1 π d ρ W ( ρ ) [ ϕ ( ρ R ) - ϕ s ( ρ R ) ] 2 .
1 π d ρ W ( ρ ) ρ 5 / 3 = 6 11 .
E 2 = 3 11 2 - 2 / 3 ( D ϑ 0 R ) 5 / 3 .
E 2 = ( D / d 0 ) 5 / 3 ,
d 0 = 2.8772318 ϑ 0 R .
z = h sec ( ψ ) ,
R = H sec ( ψ ) ,
d 0 = λ 6 / 5 cos 3 / 5 ( ψ ) [ d h C n 2 ( h ) F ( h / H ) ] - 3 / 5 ,
F ( h / H ) = 19.76732652 ( h / H ) 5 / 3 .

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