Abstract

The behavior of random band models composed of lines of pure Doppler shape for several irradiance distribution functions is investigated. The particular irradiance distribution functions chosen are: (1) constant irradiance, (2) exponential distribution, (3) truncated S−1 distribution, and (4) the exponential-tailed S−1 distribution. The curves of growth for the first two models are found to have the asymptotic (log)12 behavior characteristic of an isolated Doppler line while the latter two models show a (log)32 dependence.

© 1968 Optical Society of America

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References

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  1. W. Malkmus, J. Opt. Soc. Am. 57, 323 (1967).
    [Crossref]
  2. A. Unsöld, Astrophys. J. 69, 214 (1929).
  3. O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
    [Crossref]
  4. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959), p. 39.
  5. This result could have been obtained directly from the general expression [R. M. Goody, Atmospheric Radiation. I. Theoretical Basis, (Oxford University Press, London, 1964), p. 133] for a random band model consisting of lines of arbitrary normalized contour f(ω) having an exponential irradiance distribution [Eq. (18)] W¯/d=(1/d)∫-∞∞S0f(ω)u[1+S0f(ω)u]-1dω.
  6. This result is also derivable directly from a general expression given by C. D. Rodgers (private communication) for a random band model consisting of lines of arbitary normalized contour f(ω) having an exponential-tailed S−1irradiance distribution [Eq. (34), R large]W¯/d=(1/d lnR)∫-∞∞ln[1+SMf(ω)u]dω.

1967 (1)

1934 (1)

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

1929 (1)

A. Unsöld, Astrophys. J. 69, 214 (1929).

Elvey, C. T.

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

Goody, R. M.

This result could have been obtained directly from the general expression [R. M. Goody, Atmospheric Radiation. I. Theoretical Basis, (Oxford University Press, London, 1964), p. 133] for a random band model consisting of lines of arbitrary normalized contour f(ω) having an exponential irradiance distribution [Eq. (18)] W¯/d=(1/d)∫-∞∞S0f(ω)u[1+S0f(ω)u]-1dω.

Malkmus, W.

Penner, S. S.

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959), p. 39.

Rodgers, C. D.

This result is also derivable directly from a general expression given by C. D. Rodgers (private communication) for a random band model consisting of lines of arbitary normalized contour f(ω) having an exponential-tailed S−1irradiance distribution [Eq. (34), R large]W¯/d=(1/d lnR)∫-∞∞ln[1+SMf(ω)u]dω.

Struve, O.

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

Unsöld, A.

A. Unsöld, Astrophys. J. 69, 214 (1929).

Astrophys. J. (2)

A. Unsöld, Astrophys. J. 69, 214 (1929).

O. Struve and C. T. Elvey, Astrophys. J. 79, 409 (1934).
[Crossref]

J. Opt. Soc. Am. (1)

Other (3)

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959), p. 39.

This result could have been obtained directly from the general expression [R. M. Goody, Atmospheric Radiation. I. Theoretical Basis, (Oxford University Press, London, 1964), p. 133] for a random band model consisting of lines of arbitrary normalized contour f(ω) having an exponential irradiance distribution [Eq. (18)] W¯/d=(1/d)∫-∞∞S0f(ω)u[1+S0f(ω)u]-1dω.

This result is also derivable directly from a general expression given by C. D. Rodgers (private communication) for a random band model consisting of lines of arbitary normalized contour f(ω) having an exponential-tailed S−1irradiance distribution [Eq. (34), R large]W¯/d=(1/d lnR)∫-∞∞ln[1+SMf(ω)u]dω.

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Figures (2)

Fig. 1
Fig. 1

The curves of growth D(y)[Eq. (9)] and E(y) [Eq. (23)] for random band models composed of Doppler lines with probability distributions given by P(S) = δ(SS0) and P(S) = (S0)−1 exp(−S/S0), respectively.

Fig. 2
Fig. 2

The curves of growth G(y) [Eq. (32)] and H(y) [Eq. (37)] for random band models composed of Doppler lines with probability distributions given by P(S) ∝ S−1 (for SSM), P(S) = 0 (for S>SM) and P(S) ∝ S−1 exp(−S/SM), respectively.

Tables (1)

Tables Icon

Table I Curves of growth for random band models composed of pure Doppler lines with four different irradiance distribution functions.

Equations (49)

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k ( ω ) = ( S / b D ) ( ln 2 / π ) 1 2 exp [ - ( ω - ω 0 ) 2 ln 2 / b D 2 ] ,
b D = ω 0 ( 2 k T ln 2 / m c 2 ) 1 2
k ( ω 0 ± b D ) = 1 2 k ( ω 0 ) .
b = ( π / ln 2 ) 1 2 b D = 2.129 b D ,
k ( ω ) = ( S / b ) exp [ - π ( ω - ω 0 ) 2 / ( b ) 2 ] .
W = { 1 - exp [ - k ( ω ) u ] } d ω ,
W ( S ) = b π - 1 2 - { 1 - exp [ - ( S u / b ) × exp ( - ξ 2 ) ] } d ξ ,
W ( S ) = b D ( y ) ,
D ( y ) = π - 1 2 - { 1 - exp [ - y exp ( - ξ 2 ) ] } d ξ
y = S u / b .
D ( y ) = n = 0 ( - 1 ) n y n + 1 / ( n + 1 ) ! ( n + 1 ) 1 2 .
D ( y ) = 2 π - 1 2 ( ln y ) 1 2 [ 1 + 0.2886 ( ln y ) - 1 - 0.2473 ( ln y ) - 2 + 0.3403 ( ln y ) - 3 - ] .
α ¯ = 1 - exp [ - W ¯ / d ] ,
W ¯ = i = 1 N W i / N
W ¯ = 0 P ( S ) W ( S ) d S
W ¯ / d = ( b / d ) D ( y 0 ) ,
y 0 = S 0 u / b .
P ( S ) = S 0 - 1 exp ( - S / S 0 ) ,
W ¯ / d = ( b / S 0 d ) 0 exp ( - S / S 0 ) D ( S u / b ) d S ,
W ¯ / d = ( b / d ) E ( y 0 ) ,
E ( y ) = 0 exp ( - v ) D ( v y ) d v .
E ( y ) = π - 1 2 0 exp ( - v ) - { 1 - exp [ - v y × exp ( - ξ 2 ) ] } d ξ d v .
E ( y ) = π - 1 2 - { y exp ( - ξ 2 ) / [ y exp ( - ξ 2 ) + 1 ] } d ξ .
E ( y ) = n = 0 ( - 1 ) n y n + 1 / ( n + 1 ) 1 2 .
E ( y ) = n = 0 a n [ y / ( y + 1 ) ] n + 1 ,
a n = m = 0 n n ! ( - 1 ) m / m ! ( n - m ) ! ( m + 1 ) 1 2 .
E ( y ) = 2 π - 1 2 ( ln y ) 1 2 [ 1 - ( π 2 / 24 ) ( ln y ) - 2 - ( 7 π 4 / 192 ) ( ln y ) - 4 - ] .
P ( S ) = ( ln R ) - 1 S - 1 S M / R S S M = 0 S < S M / R             or             S > S M ,
W ¯ / d = ( b / d ln R ) y M / R y M y - 1 D ( y ) d y ,
y M = S M u / b .
W ¯ / d = ( b / d ln R ) G ( y M ) ,
G ( y ) = n = 0 ( - 1 ) n y n + 1 / ( n + 1 ) ! ( n + 1 ) 3 2 .
G ( y ) = ( 4 3 ) π - 1 2 ( ln y ) 3 2 [ 1 + 0.8659 ( ln y ) - 1 + 0.7417 ( ln y ) - 2 - 0.3403 ( ln y ) - 3 + ] .
P ( S ) = ( ln R ) - 1 S - 1 × [ exp ( - S / S M ) - exp ( - R S / S M ) ] .
W ¯ / d = ( b / d ln R ) 0 D ( y ) y - 1 exp ( - y / y M ) d y
W ¯ / d = ( b / d ln R ) H ( y M ) ,
H ( y ) = 2 π - 1 2 0 ln ( 1 + y e - ξ 2 ) d ξ .
H ( y ) = n = 0 ( - 1 ) n y n + 1 / ( n + 1 ) 3 2 .
H ( y ) = m = 0 ( - 1 ) m y m + 1 ( m + 1 ) - 3 2 × [ 1 - erf ( { ( m + 1 ) ln ( y + 1 ) } 1 2 ) ] + ( 4 3 ) π - 1 2 [ ln ( y + 1 ) ] 3 2 × { 1 + 1 2 ( y + 1 ) - 1 + ( 3 / 20 ) y ( y + 1 ) - 2 ln ( y + 1 ) + ( 1 / 28 ) ( y 2 - y ) ( y + 1 ) - 3 [ ln ( y + 1 ) ] 2 + ( 1 / 144 ) ( y 3 - 4 y 2 + y ) ( y + 1 ) - 4 × [ ln ( y + 1 ) ] 3 + } ,
erf ( x ) = 2 π - 1 2 0 x exp ( - y 2 ) d y .
H ( y ) = ( 4 3 π - 1 2 ) ( ln y ) 3 2 [ 1 + ( π 2 / 8 ) ( ln y ) - 2 + ( 7 / 640 ) π 4 ( ln y ) - 4 + ] .
k ¯ = S ¯ / d = S E / d E .
k 0 = S / b ,
k 00 = S 0 / b
k 0 M = S M / b
b / d ln R = ( b / S M ) ( S ¯ / d ) = k ¯ / k 0 M .
b / d = ( b / S 0 ) ( S ¯ / d ) = k ¯ / k 00 ,
W¯/d=(1/d)-S0f(ω)u[1+S0f(ω)u]-1dω.
W¯/d=(1/dlnR)-ln[1+SMf(ω)u]dω.