Abstract

A method for computing the distribution of emitters from the observed projected intensity profile is described. The method is applicable to optically thin volume sources with cylindrical symmetry. Unlike previous methods, this method yields results which are relatively insensitive to small random errors in the data.

© 1962 Optical Society of America

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References

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  1. R. N. Bracewell, Australian J. Phys. 9, 198 (1956). [This paper includes an interesting list of functions I(y) together with the corresponding inverse functions f(r).]
    [Crossref]
  2. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
    [Crossref]
  3. O. H. Nester and H. N. Olsen, SIAM Review 2, 200 (1960).
    [Crossref]
  4. W. J. Pearce, Optical Spectrometric Measurements of High Temperatures, edited by P. Dickerman (University of Chicago Press, Chicago, 1960), p. 125.
  5. W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949), p. 141.
  6. H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 507.
  7. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (The Macmillan Company, New York, 1948), p. 229.

1961 (1)

1960 (1)

O. H. Nester and H. N. Olsen, SIAM Review 2, 200 (1960).
[Crossref]

1956 (1)

R. N. Bracewell, Australian J. Phys. 9, 198 (1956). [This paper includes an interesting list of functions I(y) together with the corresponding inverse functions f(r).]
[Crossref]

Bockasten, K.

Bracewell, R. N.

R. N. Bracewell, Australian J. Phys. 9, 198 (1956). [This paper includes an interesting list of functions I(y) together with the corresponding inverse functions f(r).]
[Crossref]

Magnus, W.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949), p. 141.

Margenau, H.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 507.

Murphy, G. M.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 507.

Nester, O. H.

O. H. Nester and H. N. Olsen, SIAM Review 2, 200 (1960).
[Crossref]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949), p. 141.

Olsen, H. N.

O. H. Nester and H. N. Olsen, SIAM Review 2, 200 (1960).
[Crossref]

Pearce, W. J.

W. J. Pearce, Optical Spectrometric Measurements of High Temperatures, edited by P. Dickerman (University of Chicago Press, Chicago, 1960), p. 125.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (The Macmillan Company, New York, 1948), p. 229.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (The Macmillan Company, New York, 1948), p. 229.

Australian J. Phys. (1)

R. N. Bracewell, Australian J. Phys. 9, 198 (1956). [This paper includes an interesting list of functions I(y) together with the corresponding inverse functions f(r).]
[Crossref]

J. Opt. Soc. Am. (1)

SIAM Review (1)

O. H. Nester and H. N. Olsen, SIAM Review 2, 200 (1960).
[Crossref]

Other (4)

W. J. Pearce, Optical Spectrometric Measurements of High Temperatures, edited by P. Dickerman (University of Chicago Press, Chicago, 1960), p. 125.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949), p. 141.

H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1943), p. 507.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (The Macmillan Company, New York, 1948), p. 229.

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Figures (2)

Fig. 1
Fig. 1

Illustration of the geometrical relationships between the variables.

Fig. 2
Fig. 2

Comparison of the effect that an “error” in In has on the fk, when fk is calculated (a) by use of Eq. (2), and (b) by use of the present method. The effects of slight round-off errors are also evident in (a). I is plotted on a reduced scale.

Tables (1)

Tables Icon

Table I The coefficients βkn × (−104).

Equations (13)

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I ( y ) l δ y = l δ y X X f ( r ) d x ,
I ( y ) = 2 y R f ( r ) r d r ( r 2 y 2 ) 1 2 .
f ( r ) = 1 π r R ( d I / d y ) d y ( y 2 r 2 ) 1 2 .
f ( r ) = 1 2 π r d F ( r ) d r ,
F ( r ) = 2 r R I ( y ) y d y ( y 2 r 2 ) 1 2 .
y n = n Δ , r k = k Δ , R = N Δ ,
I ( y ) = a n + b n y 2 .
F k = Δ n = k N α k n I n ,
α k k = 4 3 ( 2 k + 1 ) 1 2 , n = k , α k n = 4 3 { [ ( n + 1 ) 2 k 2 ] 3 2 2 n + 1 4 n [ n 2 k 2 ] 3 2 4 n 2 1 + [ ( n 1 ) 2 k 2 ] 3 2 2 n 1 } , n > k .
F ( k ) = ( A k + B k k 2 + C k k 4 ) Δ .
f k = 1 2 π Δ 2 k d F ( k ) d k = 1 π Δ ( B k + 2 C k k 2 ) .
B k + 2 C k k 2 = n = k 2 N β k n I n , k 2 , B k + 2 C k k 2 = n = 0 N β k n I n , k < 2 ,
f k = 1 π Δ n = k 2 N β k n I n , k > 2 , f k = 1 π Δ n = 0 N β k n I n , k 2.