Abstract

Basic thermal perturbations are considered for a fused-silica primary mirror in an orbiting space telescope within 800 km of the earth’s surface. In such an orbit, the change of thermal environment with pointing direction leads almost inevitably to an axial temperature gradient dT/dz through the primary mirror, and to a consequent bending of the mirror and a change ΔF of focal length. If ΔF is constant across the mirror to within about ±3 f2λ, where f is the focal ratio and λ the wavelength, the rms deviation of the mirror surface from a paraboloid will not exceed λ/50, and diffraction-limited performance is possible. Nonuniformities of ΔF can result either from lateral changes of dT/dz across the face of the mirror or from axial changes of dT/dz, i.e., from nonlinear gradients. A numerical analysis of the radiation transfer in the telescope tube, when this is illuminated from one side, shows that even in the unfavorable case of a two-dimensional telescope the lateral variation of dT/dz can be kept sufficiently small if the telescope tube is substantially longer than the mirror diameter, and if the sunlit earth does not shine directly on the mirror at any time. Analysis of the thermal distortion near the edge of a flat disk subject to a nonlinear axial gradient and of a spherical cap subject to a radial gradient dT/dR indicates that the likely distortions are certainly within optical tolerance for a 1-m telescope and probably also for a 3-m telescope. Deviations from uniformity resulting from the mirror supports or from inhomogeneities in the mirror itself are not considered.

© 1967 Optical Society of America

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References

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  1. L. Berkner and H. Odishaw, Eds., Science in Space (McGraw-Hill Book Co., New York, 1961);Report of Iowa Space Study Group, 1963 (National Academy of Science);Report of Woods Hole Space Study Group, 1966 (National Academy of Science).
  2. L. Spitzer, Astron. J. 65, 242 (1960).
    [Crossref]
  3. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (John Wiley & Sons, Inc., New York, 1960).
  4. R. M. Scott, Appl. Opt. 1, 387 (1962).
    [Crossref]
  5. We are indebted to D. Markle of the Perkin-Elmer Corp. for this result.
  6. We are grateful to C. Smith and Mrs. H. Selberg at the Princeton Plasma Physics Laboratory for programming and executing these calculations.
  7. M. Picone, Math. Ann. 101, 701 (1929).
    [Crossref]
  8. F. Timoshenko and F. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill Book Co., New York, 1959).

1962 (1)

1960 (1)

L. Spitzer, Astron. J. 65, 242 (1960).
[Crossref]

1929 (1)

M. Picone, Math. Ann. 101, 701 (1929).
[Crossref]

Boley, B. A.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (John Wiley & Sons, Inc., New York, 1960).

Picone, M.

M. Picone, Math. Ann. 101, 701 (1929).
[Crossref]

Scott, R. M.

Spitzer, L.

L. Spitzer, Astron. J. 65, 242 (1960).
[Crossref]

Timoshenko, F.

F. Timoshenko and F. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill Book Co., New York, 1959).

Weiner, J. H.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (John Wiley & Sons, Inc., New York, 1960).

Woinowski-Krieger, F.

F. Timoshenko and F. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill Book Co., New York, 1959).

Appl. Opt. (1)

Astron. J. (1)

L. Spitzer, Astron. J. 65, 242 (1960).
[Crossref]

Math. Ann. (1)

M. Picone, Math. Ann. 101, 701 (1929).
[Crossref]

Other (5)

F. Timoshenko and F. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill Book Co., New York, 1959).

L. Berkner and H. Odishaw, Eds., Science in Space (McGraw-Hill Book Co., New York, 1961);Report of Iowa Space Study Group, 1963 (National Academy of Science);Report of Woods Hole Space Study Group, 1966 (National Academy of Science).

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (John Wiley & Sons, Inc., New York, 1960).

We are indebted to D. Markle of the Perkin-Elmer Corp. for this result.

We are grateful to C. Smith and Mrs. H. Selberg at the Princeton Plasma Physics Laboratory for programming and executing these calculations.

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Figures (4)

F. 1
F. 1

Schematic of reflecting telescope.

F. 2
F. 2

Idealized slab telescope.

F. 3
F. 3

Temperature distribution in slab telescope. The function U(z) is proportional to the temperature difference between the two sides of the telescope; z is the distance from the primary mirror, in units of the diameter, and Z is the tube length in the same units; the earth illuminates one side of the telescope tube down to a distance z1 from the end of the tube.

F. 4
F. 4

Spherical-cap mirror.

Tables (2)

Tables Icon

Table I U(z), which gives difference of emitted flux between hot and cold sides, in units of F 0.

Tables Icon

Table II Relative amplitude of flux variation across mirror. The quantity θ1 is the minimum angle subtended with the optical axis by the incoming external radiation; z1 equals cotθ1. The quantity Z is the ratio of tube length to mirror width for the two-dimensional telescope analyzed.

Equations (86)

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F = K ( d T m / d z ) = fi σ ( T mf 4 T f 4 ) = bi σ ( T mb 4 T b 4 ) ,
T mb = bi T b + fi T f + bi fi A T b bi + fi + bi fi A ,
A = 4 h σ T b 3 / K .
d T d z = T mf T mb h = 4 fi σ T b 3 K ( T f T b ) .
Δ ( 1 / R ) 1 / R T = α ( d T / d z ) ,
Δ F = 2 F 2 α d T d z = 8 α F 2 fi σ T b 3 K ( T f T b ) .
Δ F 1.11 f 2 λ ,
δ F 3 f 2 λ ,
θ = x / R T ,
d R / d z = 1 + z / R T ,
x = x [ 1 + ( z / R T ) ] ,
z = z + ( z 2 x 2 ) / 2 R T .
Δ z = z ( B 1 2 R T ) x 2 = 3 r 0 2 α 128 f 2 d T d z ,
r 0 / z = 8 f .
F dA = I ( r , θ , φ ) cos θ d ω dA ,
I = F / π .
d ω = ( cos θ d z d y ) / r 2 ,
F C ( z ) = 1 π + d y Z + Z d z F H ( z ) cos θ cos θ r 2 .
cos θ = 1 / r .
r 2 = 1 + y 2 + ( z z ) 2 .
F C ( z ) = 1 2 Z + Z F H ( z ) d z [ 1 + ( z z ) 2 ] 3 2 .
F eu ( z ) = { 1 2 F 0 { z 1 [ 1 + z 1 2 ] 1 2 Z z [ 1 + ( Z z ) 2 ] 1 2 } , if Z z < z 1 , 0 , if Z z z 1 ,
z 1 = cot θ 1 .
F e ( z ) = F eu ( z ) + F eu ( z ) ,
F H ( z ) = F e ( z ) + 1 2 Z + Z F C ( z ) d z [ 1 + ( z z ) 2 ] 3 2 .
U ± ( z ) = F H ( z ) ± F C ( z ) ,
U ± ( z ) = F e ( z ) ± 1 2 Z + Z U ± ( z ) d z [ 1 + ( z z ) 2 ] 3 2 .
U + ( z ) = F 0 ,
U a + = F 0 1 ( 1 + z 1 2 ) 1 2 1 + 2 Z ( 4 Z 2 + 1 ) 1 2 .
U ( z ) = 1 Λ ( z ) [ F e ( z ) 1 2 Z + Z U ( z ) U ( z ) [ 1 + ( z z ) 2 ] 3 2 d z ] ,
Λ ( z ) = 1 + Z + z 2 [ 1 + ( Z + z ) 2 ] 1 2 + Z z 2 [ 1 + ( Z z ) 2 ] 1 2 .
U 1 ( z ) = F e ( z ) / Λ ( z ) .
cos θ = ( 1 2 ± x ) / r ,
cos θ = z / r ,
F m ( x ) = 1 2 ( 1 2 + x ) 0 Z z F H ( z ) d z [ z 2 + ( 1 2 + x ) 2 ] 3 2 + 1 2 ( 1 2 x ) 0 Z z F C ( z ) d z [ z 2 + ( 1 2 x ) 2 ] 3 2 .
W ± = F m ( 1 2 ) ± F m ( 1 2 ) = 1 2 U ± ( 0 ) ± 1 2 0 Z z U ± ( z ) d z ( 1 + z 2 ) 3 2 .
Ω = π / 4 Z 2 .
v δ F v / i F i = v Ω I v δ Q / i σ T f 4 .
v δ F v i F i = 3.3 δ Q Z 2 .
w = ( 6 M T / E h 3 ) ( r 0 2 r 2 ) ; M T = α E T z d z ,
σ r r = σ θ θ = σ 0 ( z ) = 1 1 ν ( α E T + N T h + 12 M T h 3 z ) ; N T = α E T d z ,
σ 0 d z = σ 0 z d z = 0 .
σ r r ( r 0 , z ) = σ 0 ( z ) ; σ r z ( r 0 , z ) = σ r z ( r , ± 1 2 h ) = σ z z ( r , ± 1 2 h ) = 0 .
σ r r r + σ r z z + σ r r σ θ θ r = 0 ; σ r z r + σ r z r + σ z z z = 0 ,
r r = 1 E [ σ r r ν ( σ θ θ + σ z z ) ] = u r , z z = 1 E [ σ z z ν ( σ r r + σ θ θ ) ] = w z , θ θ = 1 E [ σ θ θ ν ( σ r r + σ z z ) ] = u r , r z = 1 + ν E σ r z = 1 2 ( u z + w r ) ,
( r θ θ ) / r = r r .
E θ θ = σ θ θ ν ( σ r r + σ z z ) = r ( 1 + ν ) r c r [ ( 1 ν 2 ) σ r r ν ( 1 + ν ) σ z z ] r d r ,
w ( r , ± 1 2 h ) = r c r u z d r + f ( z ) ,
w ( r , ± 1 2 h ) = 1 E r c r r 1 ν r c r 1 [ ( 1 ν 2 ) σ r r z ν ( 1 + ν ) σ z z z ] r 2 ν d r 2 d r 1 + f ( z ) .
w ( r , ± 1 2 h ) = ( 1 ν 2 E ) r c r r 1 ν r c r 1 r 2 ν × σ r r z ( r 2 , ± 1 2 h ) d r 2 d r 1 + f ( z ) .
w ( r , 1 2 h ) = 1 2 h z z z d z + g ( r ) .
w s ( r , ± h / 2 ) = ± 1 2 szz ( r , z ) d z .
σ r r ( r , z ) = { σ 0 ( z ) ( 4 y 3 3 y 4 ) , y = ( r a ) / ( r 0 a ) , for a r r 0 , 0 , for 0 r a ,
σ θ θ = σ 0 ( ζ ) [ ν ( 4 y 3 3 y 4 ) + ( 1 ν 2 ) 5 y 4 3 y 5 5 ( y + A ) ] , σ r z = h ( 1 + A ) 2 r 0 1 ζ σ 0 ( ζ ) d ζ [ 12 ( y 2 y 3 ) + ( 1 ν ) × 4 y 3 3 y 4 ( y + A ) ( 1 ν 2 ) ( 5 y 4 3 y 5 ) 5 ( y + A ) 2 ] , σ z z = 3 h 2 ( 1 + A ) 2 r 0 2 1 ζ 1 ζ σ 0 ( ζ ) d ζ d ζ × [ 2 y 3 y 2 + ( 2 ν ) ( y 2 y 3 ) ( y + A ) ( 1 ν 2 ) ( 4 y 3 3 y 4 ) ( y + A ) 2 + ( 1 ν 2 ) ( 5 y 4 3 y 5 ) 5 ( y + A ) 3 ] ,
A = a / ( r 0 a ) ; ζ = 2 z / h .
2 E π r 0 2 h U A = ( 0.226 A 2 + 0.492 A 3 1.893 A 4 ) σ 0 2 ( ζ ) d ζ + ( h r 0 ) 2 ( 0.89 0.034 A 2 ) [ 1 ζ σ 0 ( ζ ) d ζ ] 2 d ζ + ( 0.103 0.234 A 2 + 0.131 A 3 ) × σ 0 [ 1 ζ 1 ζ σ 0 ( ζ ) d ζ d ζ ] d ζ + ( h r 0 ) 4 ( 3.6 A 2 + 6.6 A + 5.17 ) × [ 1 ζ 1 ζ σ 0 ( ζ ) d ζ d ζ ] 2 d ζ = 0 .
T ( z ) = T 0 ζ 2 , σ 0 = α E T 0 ( 1 3 ζ 2 ) ,
w s ( r 0 , ± 1 2 h ) = ± ( 1 ν 2 ) r 0 2 5 ( 1 + A ) 2 E h d σ 0 d ζ ( 1 ) = ± 3 h 3 ( 1 + A ) 2 4 r 0 2 E × [ 1 + ( 1 ν 2 ) ( 5 A + 3 ) ( A + 1 ) 3 1 ζ 1 ζ σ 0 ( ζ ) d ζ d ζ d ζ ] .
[ y + A y 1 + A ] 1 ν 1 ( 1 ν ) y y 1 A ;
w ( r 0 , ± 1 2 h ) ( 1 ν 2 ) r 0 2 125 E h | d σ 0 d ζ ( ± 1 ) | .
0 T / z | Q 0 | / K .
| Q 0 | K = | d T d z | max 12 M T α E h 3 ,
| T ( z ) T ( h 2 ) | 1 2 h z | T z | d z < 12 M T α E h 3 ( z + h 2 ) ,
M T α E = [ T ( z ) T ( h 2 ) ] z d z < 12 M T α E h 3 × ( z + h 2 ) z d z = M T α E .
| d σ 0 d z | = 1 1 ν | α E d T d z + 12 M T h 3 | α E 1 ν | d T d z | max = E ( 1 ν ) K | Q 0 | ,
| w | ( 1 + ν ) r 0 2 α | Q 0 | / 125 K .
Q 0 = fi σ T f 4 = 5.7 × 10 4 W / cm 2 ,
T = T 0 ( 1 + c 1 R )
σ φ φ 1 = α μ T 0 c 1 ( 3 λ + 2 μ ) 2 ( R 1 3 R 0 3 ) ( λ + 2 μ ) [ 3 R ( R 1 3 R 0 3 ) + 2 ( R 1 4 R 0 4 ) + R 1 3 R 0 3 ( R 1 R 0 ) / R 3 ] = K R 0 [ Z + ( H 2 / 24 ) + ( Z 2 / 2 ) + O ( H 3 ) ] ; K = α E T 0 c 1 / ( 1 ν ) ;
M = 2 π R 0 R 1 R 2 sin β σ φ φ d R / π ( R 0 + R 1 ) sin β = α T 0 c 1 E 2 ( 1 ν ) [ R 0 3 R 1 3 log ( R 1 / R 0 ) ( R 0 + R 1 ) ( R 1 2 + R 0 R 1 + R 0 2 ) ( R 1 2 + R 0 2 ) ( R 1 R 0 ) 12 ] ,
M = K R 0 2 [ ( H 3 / 12 ) + 0 ( H 4 ) ] .
u 1 = α T 0 R 0 [ 1 + 3 4 c 1 ( R 1 4 R 0 4 R 1 3 R 0 3 ) ] = α T 0 R 0 { 1 + c 1 R 0 [ 1 1 2 H + 1 8 H 2 + O ( H 3 ) ] } .
d 2 U d r 2 + 1 r ( 1 r 2 3 R o 2 + ) d U d r [ 1 ( ν + 2 3 ) r 2 R o 2 + ] U r 2 = E h V R 0 , d 2 V d r 2 + 1 r ( 1 r 2 3 R o 2 + ) d V d r [ 1 + ( ν 2 3 ) r 2 R o 2 + ] V r 2 = U R o D ,
L ( U ) = E h V / R 0 , L ( V ) = U / ( R 0 D ) , L d 2 / d r 2 + ( 1 / r ) d / d r ( 1 / r 2 ) ,
L 2 ( V ) = E h V / ( R 0 2 D ) .
V = a 0 r [ 1 + k 192 ( r r 0 ) 4 + O ( r r 0 ) 8 ] + b 0 r 3 [ 1 + k 1152 ( r r 0 ) 4 + O ( r r 0 ) 8 ]
k = E h r 0 4 R o 2 D = 12 ( 1 ν 2 ) r 0 4 R o 2 h 2 ( = 4.27 for R 0 = 400 cm , r 0 = 50 cm , h = 10 cm ) .
U = L ( V ) = 0 at r = r 0
d V / d r + ν V / r = M / D at r = r 0 .
a 0 = M D [ 1 + ν + 5 + ν 192 k 3 k 192 + k ( 3 + ν + 7 + ν 1152 k ) ] 1 b 0 r 0 2 = 3 k a 0 192 + k . ;
u 2 = 1.08 M r 2 2 ( 1 + ν ) D { 1 + k 1152 ( r r 0 ) 4 0.065 ( r r 0 ) 2 × [ 1 4 + k 8064 ( r r 0 ) 4 ] + } .
u 2 = 1.08 M r 2 2 ( 1 + ν ) D = 1.08 6 M ( 1 ν ) r 2 E h 3 ,
u 1 + u 2 = α T 0 R 0 { 1 + c 1 R 0 [ 1 0.54 ( r / R 0 ) 2 ] } .
σ φ φ 2 = 12 M z / h 3 .
σ φ φ 3 = K R 0 [ ( Z + H 2 24 + Z 2 2 ) R R 0 + Z ] = K h 2 8 ( 1 3 4 z 2 h 2 ) ,
u = 0.0088 α T 0 c 1 r 0 2 .