Abstract

A new method for transforming observed radiances into the radial distribution of the emission of a plasma is described. It is applicable to optically thin plasmas with cylindrical or spherical symmetry, which are often encountered in plasma physics and astrophysics. The observations are introduced as a sequence of n readings on the experimental curve, which are then transformed to a set of values for the emission coefficient. The transformation coefficients are tabulated for n=10, n=20, and, in part, for n=40. The method is more accurate than previously published ones and is well suited for rapid calculation by electronic computers. The sources of errors are discussed and a numerical method for smoothing the readings is suggested.

© 1961 Optical Society of America

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References

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  1. H. Hörman, Z. Physik 97, 539 (1935).
    [CrossRef]
  2. H. Brinkman, “Optische studie van de electrische Lichtboog, Thesis, University of Utrecht, Utrecht, The Netherlands (1937), p. 85.
  3. H. Maecker, Z. Physik 136, 119 (1953); Z. Physik 139, 448 (1954).
    [CrossRef]
  4. W. J. Pearce, Conference on Extremely High Temperatures, edited by H. Fischer and L. C. Mansur (John Wiley & Sons, Inc., New York, 1958), p. 123.
  5. J. Friedrich, Ann. Physik 3, 327 (1959).
    [CrossRef]

1959 (1)

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

1953 (1)

H. Maecker, Z. Physik 136, 119 (1953); Z. Physik 139, 448 (1954).
[CrossRef]

1935 (1)

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

Brinkman, H.

H. Brinkman, “Optische studie van de electrische Lichtboog, Thesis, University of Utrecht, Utrecht, The Netherlands (1937), p. 85.

Friedrich, J.

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

Hörman, H.

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

Maecker, H.

H. Maecker, Z. Physik 136, 119 (1953); Z. Physik 139, 448 (1954).
[CrossRef]

Pearce, W. J.

W. J. Pearce, Conference on Extremely High Temperatures, edited by H. Fischer and L. C. Mansur (John Wiley & Sons, Inc., New York, 1958), p. 123.

Ann. Physik (1)

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

Z. Physik (2)

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

H. Maecker, Z. Physik 136, 119 (1953); Z. Physik 139, 448 (1954).
[CrossRef]

Other (2)

W. J. Pearce, Conference on Extremely High Temperatures, edited by H. Fischer and L. C. Mansur (John Wiley & Sons, Inc., New York, 1958), p. 123.

H. Brinkman, “Optische studie van de electrische Lichtboog, Thesis, University of Utrecht, Utrecht, The Netherlands (1937), p. 85.

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Figures (3)

Fig. 1
Fig. 1

A disk of plasma, circularly symmetric with respect to the z axis, which is normal to the paper.

Fig. 2
Fig. 2

Errors in j due to finite n values. Mean values for three N(x)-functions with Nmax=1 and r0=1.

Fig. 3
Fig. 3

Standard deviations in j corresponding to a maximum error in the Nk readings of 0.002 Nmax and with Nmax=r0=1.

Tables (4)

Tables Icon

Table I Coefficients ajk for transforming observed radiances into radial distribution. n=10. If k<j−1 then ajk=0. The letter s signifies ( k a j k 2 ) 1 2.

Tables Icon

Table II Coefficients ajk for transforming observed radiances into radial distribution. n=20. If k<j−1 then ajk=0. The letter s signifies ( k a j k 2 ) 1 2.

Tables Icon

Table III Coefficients ajk for transforming observed radiances into radial distribution, n=40. If k<j−1 then ajk=0. The letter s signifies ( k a j k 2 ) 1 2.

Tables Icon

Table IV Table of N(x)=1600(1−x2) and ( r ) = 3200 π - 1 ( 1 - r 2 ) 1 2.

Equations (18)

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N ( x ) Δ x Δ z = - y 0 + y 0 ( r ) Δ x Δ y Δ z .
N ( x ) = 2 0 y 0 ( r ) d y .
N ( x ) = 2 x r 0 ( r ) r d r ( r 2 - x 2 ) 1 2 .
( r ) = - 1 π r r 0 N ( x ) d x ( x 2 - r 2 ) 1 2 ,
j = r 0 - 1 k a j k N k ,
j = - 1 π k = j n - 1 x k x k + 1 P k ( x ) d x ( x 2 - r j 2 ) 1 2 ,
j = - n π r 0 k = j n - 1 k k + 1 P k ( t ) d t ( t 2 - j 2 ) 1 2 .
a j k = n ( 12 π ) - 1 [ f ( j , k + 2 ) - 4 f ( j , k + 1 ) + 6 f ( j , k ) - 4 f ( j , k - 1 ) + f ( j , k - 2 ) ] ,
f ( j , k ) = ( 3 j 2 + 6 k 2 - 2 ) ln [ k + ( k 2 - j 2 ) 1 2 ] - 9 k ( k 2 - j 2 ) 1 2 ,
N ( x ) = 1600 ( 1 - x 2 ) ,
( r ) = 3200 π - 1 ( 1 - r 2 ) 1 2 .
N ( x ) = 1 - x 4 ,
N ( x ) = 1 - 3 x 2 + 8 x 4 - 6 x 6 ,
N ( x ) = 0.67 ( 1 - x 2 ) [ 0.25 + ( 1 + 4 x 2 ) ( 1 - x 2 ) 1 2 ] .
σ = 0.002 N max ( r 0 3 ) - 1 ( k a j k 2 ) 1 2 .
21 N ¯ k = - 2 N k - 3 + 3 N k - 2 + 6 N k - 1 + 7 N k + 6 N k + 1 + 3 N k + 2 - 2 N k + 3 .
42 N ¯ n - 2 = N n - 6 - 4 N n - 5 + 2 N n - 4 + 12 N n - 3 + 19 N n - 2 + 16 N n - 1
42 N ¯ n - 1 = 4 N n - 6 - 7 N n - 5 - 4 N n - 4 + 6 N n - 3 + 16 N n - 2 + 19 N n - 1 .