Abstract

An improved method is proposed for applying the Abel integral equation to radiance data gathered from rotationally symmetric sources by side-on observation. The method involves dividing the data into a number of segments going out from the center. A least-squares polynomial is then fitted to each segment so that the inverted Abel integral equation can be integrated exactly to yield the emission coefficient. It is shown, with test curves approximating spectral line shapes from nonhomogeneous sources, that the method becomes superior to the other more common numerical methods of reducing such data as the number of data points becomes large. This superiority increases as the data scatter increases.

© 1966 Optical Society of America

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References

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  1. C. J. Cremers, Ph.D. Thesis, University of Minnesota, 1964.
  2. F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1952).
  3. J. Friedrich, Ann. Phys. 3, 327 (1959).
    [CrossRef]
  4. H. Hörmann, Z. Physik 97, 539 (1935).
    [CrossRef]
  5. O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
    [CrossRef]
  6. H. Maecker, Z. Physik 136, 119 (1953).
    [CrossRef]
  7. R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
    [CrossRef]
  8. W. Frie, Ann. Phys. 10, 332 (1963).
    [CrossRef]
  9. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [CrossRef]
  10. W. B. Brooks, G. E. Reis, Sandia Corporation SC-4938 (RR) (1963).
  11. M. P. Freeman, S. Katz, J. Opt. Soc. Am. 53, 1172 (1963).
    [CrossRef]
  12. W. L. Barr, J. Opt. Soc. Am. 52, 885 (1962).
    [CrossRef]
  13. S. I. Herlitz, Arkiv Fysik 23, 571 (1963).
  14. C. D. Maldonado, A. P. Caron, H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965).
    [CrossRef]

1965 (1)

1963 (3)

S. I. Herlitz, Arkiv Fysik 23, 571 (1963).

W. Frie, Ann. Phys. 10, 332 (1963).
[CrossRef]

M. P. Freeman, S. Katz, J. Opt. Soc. Am. 53, 1172 (1963).
[CrossRef]

1962 (1)

1961 (1)

1960 (1)

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

1959 (1)

J. Friedrich, Ann. Phys. 3, 327 (1959).
[CrossRef]

1953 (1)

H. Maecker, Z. Physik 136, 119 (1953).
[CrossRef]

1948 (1)

R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

1935 (1)

H. Hörmann, Z. Physik 97, 539 (1935).
[CrossRef]

Barr, W. L.

Bockasten, K.

Brooks, W. B.

W. B. Brooks, G. E. Reis, Sandia Corporation SC-4938 (RR) (1963).

Caron, A. P.

Cremers, C. J.

C. J. Cremers, Ph.D. Thesis, University of Minnesota, 1964.

Freeman, M. P.

Frie, W.

W. Frie, Ann. Phys. 10, 332 (1963).
[CrossRef]

Friedrich, J.

J. Friedrich, Ann. Phys. 3, 327 (1959).
[CrossRef]

Herlitz, S. I.

S. I. Herlitz, Arkiv Fysik 23, 571 (1963).

Hildebrand, F. B.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1952).

Hörmann, H.

H. Hörmann, Z. Physik 97, 539 (1935).
[CrossRef]

Katz, S.

Ladenburg, R.

R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Maecker, H.

H. Maecker, Z. Physik 136, 119 (1953).
[CrossRef]

Maldonado, C. D.

Nestor, O. H.

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Olsen, H. N.

Reis, G. E.

W. B. Brooks, G. E. Reis, Sandia Corporation SC-4938 (RR) (1963).

VanVoorhis, C. C.

R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Winckler, J. A.

R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

Ann. Phys. (2)

J. Friedrich, Ann. Phys. 3, 327 (1959).
[CrossRef]

W. Frie, Ann. Phys. 10, 332 (1963).
[CrossRef]

Arkiv Fysik (1)

S. I. Herlitz, Arkiv Fysik 23, 571 (1963).

J. Opt. Soc. Am. (4)

Phys. Rev. (1)

R. Ladenburg, J. A. Winckler, C. C. VanVoorhis, Phys. Rev. 73, 1359 (1948).
[CrossRef]

SIAM Rev. (1)

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Z. Physik (2)

H. Maecker, Z. Physik 136, 119 (1953).
[CrossRef]

H. Hörmann, Z. Physik 97, 539 (1935).
[CrossRef]

Other (3)

C. J. Cremers, Ph.D. Thesis, University of Minnesota, 1964.

F. B. Hildebrand, Methods of Applied Mathematics (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1952).

W. B. Brooks, G. E. Reis, Sandia Corporation SC-4938 (RR) (1963).

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Figures (2)

Fig. 1
Fig. 1

Emission coefficient function from test equations.

Fig. 2
Fig. 2

Radiance function from test equations.

Tables (8)

Tables Icon

Table I Comparison of Six Methods for Numerically Solving the Abel Equation Using, as Input, Thirty-one Accurate Data Points and the Off-Axis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by 1 30

Tables Icon

Table II Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (11) as Test Values

Tables Icon

Table III Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (12) as Test Values

Tables Icon

Table IV Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (13) as Test Values

Tables Icon

Table V Comparison of Six Methods of Numerically Solving the Abel Equation Using, as Input, Thirty-one Data Points Which Are Good to Two Decimal Places, and the Off-Axis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by 1 30

Tables Icon

Table VI Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (11) as Test Values

Tables Icon

Table VII Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (12) as Test Values

Tables Icon

Table VIII Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (13) as Test Values

Equations (16)

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I ( x ) = 2 x R i ( r ) r d r ( r 2 - x 2 ) 1 2 .
i ( r ) = - 1 π r R I ( x ) d x ( x 2 - r 2 ) 1 2 .
i k ( r ) = - 2 π a n = k N - 1 I n ( x ) B k , n ,
i k ( r ) = ( I k ( x ) 2 a - n = k + 1 N i n ( r ) A k , n ) / [ 2 k - 1 ] 1 2
i k ( r ) = 1 k A k , k - H k , k { I k ( x ) a - n = k + 1 N i n ( r ) [ H k , n - 1 - H k , n - ( n - 3 ) A k , n - 1 + n A k , n ] } , k 2 ,
i 1 ( r ) = [ 4 i 2 ( r ) - i 3 ( r ) ] / 3 , k = 1 ,
i ( r ) = i k ( r ) ( r k + 1 2 - r 2 ) + i k + 1 ( r ) ( r 2 - r k 2 ) r k + 1 2 - r k 2 .
i k ( r ) = 1 C k , k { I k ( x ) 2 a - n = k + 1 N - 1 i n ( r ) ( C k , n - C k , n - 1 ) } ,
I ( x ) = A + B x j + C x 2 j + D x 3 j + E x 4 j .
i ( r ) = - 1 π r R m j ( B m x j - 1 + 2 C m x 2 j - 1 + 3 D m x 3 j - 1 + 4 E m x 4 j - 1 ) d x ( x 2 - r 2 ) 1 2 + i ( R m ) ,
i ( r ) = 1 - 3 r 2 + 2 r 3 ,             0 r 1.0.
i ( r ) = 1 - 2 r 2 ,             0 r 0.5 ;
i ( r ) = 2 ( 1 - r ) 2 ,             0.5 < r 1.0.
i ( r ) = 3 4 + 12 r 2 - 32 r 3 , 0 r 0.25 ;
i ( r ) = 1 2 6 7 ( 1 + 6 r - 15 r 2 + 8 r 3 ) , 0.25 < r 1.0.
σ = { n = 1 N [ Δ i n ( r ) ] 2 N } 1 2 .

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