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Integrated absorption of a spectral line with the Voigt profile
L Quant. Spectrosc. Radiat. Transfer. Vol. 14, pp. 319 323. I'cvgamon Press 1974. Printed in Great Britain.
NOTE I N T E G R A T E D ABSORPTION OF A SPECTRAL LINE WITH THE VOIGT PROFILE C. D. RODGERS and A. P. WILLIAMS Clarendon Laboratory, Oxford, U. K. (Received 12 October 1973)
Abstraet--A simple approximation has been found for the integrated absorpton of a spectral line with Voigt profile. It is expressed in terms of the integrated absorption of the Doppler and Lorentz profiles. The maximum error is 8 %. THERE are two main mechanisms which determine the shape of molecular spectral lines. These are collision (Lorentz) broadening and Doppler broadening. In many cases of interest, one of these mechanisms predominates and the integrated absorption (equivalent width) of a spectral line may easily be calculated using the appropriate line shape. However, throughout a large part of the earth's atmosphere and in many astrophysical problems, both mechanisms are significant and the mixed line shape (Voigt profile) must be used. This is simply a convolution of the Lorentz and Doppler shapes, and may be written as k(v) =
S
exp(_t2)dt
y (~o
~,,,~,/2 ,~ -Y-~o~- ¥ (-2:- ty'
(1)
where t = (v - v')lct o, x = (v - Vo)/~o, y = ctt./~D; aL is the Lorentz halfwidth and ~t~ is the Doppler width of the line, S is the line intensity and k(v) is the absorption coefficient. The equivalent width of a single line is cO
W ( S , m, ctL, ~D) = | o._
1 -- e x p [ - k ( v ) m ] dr. oO
where m is an absorber amount. Unfortunately it is not possible to express the integrated absorption directly in terms of simple functions, or even functions of one variable for which efficient numerical approximation may be found, so any radiative transfer calculations must involve a time-consuming numerical integration. For the purposes of comparisons to be performed later, we have calculated W by numerical integration for a wide range of parameters. A contour plot of Wilt o is shown in Fig. 1. A detailed tabulation may be found in JANSSON and KORB.(~) The equivalent widths of both Lorentz and Doppler shapes can be calculated from simple numerical approximations, (see the Appendix for details). With a view to providing a fast method of computing the equivalent width of a mixed line, WM, CURTIS (2) suggested the approximation W u < WL + W D -
319
WL W a / W w ,
(2)
320
C . D . RODGERSand A. P. WILLIAMS
_o O~ 0
"~5
-4
-3
"2
-1
0
1
2
Io<3,0[o~./o~,]
Fig. 1. Contours of equivalent width of a mixed shaped spectral line divided by ~D, the Doppler width.
where WL and WD are the equivalent widths of the Lorentz- and Doppler-broadened lines, respectively, and W w = S m is the weak limit for a line of any shape. This formula has the correct form in all the limits, has a smooth changeover between the limits, and is easily differentiated. However, it over-estimates the true equivalent width, especially when WL --- WD, with errors as great as 50 per cent [see GILLV and ELLINGSON(a)]. The nature of the error is illustrated in Fig. 2. We have found a better approximation to be W u = [ w L 2 + w ~ 2 - ( w L W o / W w ) 2 ] 1/2.
(3)
This approximation also has the correct form in all the limits, and is almost as easily differentiated as Curtis' relation. A contour diagram of the difference between this function and the exact equivalent width is shown in Fig. 3, giving maximum errors of 8 per cent. Exponents differing slightly from 2.0 are found to give improved accuracy, but the algebraic convenience is lost.
0
m
t
I
~
2
Ioglo[ s m/~,]
'L
3
Zig. 2. Equivalent widths of a Lorentz (WL), Doppler (Wo) and nixed (W)shaped spectral line; original Curtis approximation W 1) and modified Curtis approximation (W2) with 0cL/~o= 0.1.
0.5
_o
:3
O
"SF
-~. -S
1
|
-4
-3
-1
Iogl0[o
-2
0
1
2
Fig. 3. Contours of percentage error in the modified Curtis ~pproximation for the equivalent width of a mixed shaped spectral line.
o OI O
3
O
t~
.q-
.=
O
r~
322
C . D . RODGERS and A. P. WILLIAMS REFERENCES
1. P. A. JANSSON and C. L. KORB., JQSRT 8, 1399 (1968). 2. A. R. CURTIS, unpublished. 3. J. C. GILLE and R. G. ELLINGSON. Appl. Optics 7, 471 (1968). APPENDIX T h e e q u i v a l e n t w i d t h o f a L o r e n t z line is g i v e n b y
WL = 27t~LL(Sm/2~c%), where
L(z) is t h e L a d e n b u r g - R e i c h e
function :
L(z) = z e-z[I o (z) + l l ( z ) ] . With a maximum
r e l a t i v e e r r o r o f 5 x 10 - 6 , we c a n p u t , f o r 0 ~< z ~< 1.98, 6
L(z) = z ~" a, z"; o
f o r 1-98 < z,
L(z) = z 1/2 ~ b,,z"; 0
h e r e t h e c o e f f i c i e n t s a . a n d b , a r e g i v e n in T a b l e 1.
Table 1. The coefficients a. and b, in the approximation of the Ladenburg and Reiche function rl
0 1
2 3 4 5 6
an
9'9999 --4.9988 2-4900 -- 1-0095 3"1358 --6"5014 6'4832
7697 2252 5331 6020 0341 4170 5819
bn
674 × 233 × 874 x 660 x 312 × 817 × 427 ×
10 -~ I0 -~ 10 -1 10- t 10 -2 10 -3 10 -4
7.9788 -9.9755 -1.9762 1.0586 1-3246 1.4794 0.0
5095 5137 3661 5022 7496 7878
473 671 059 723 350 333
× x x x x x
1010 -z 10 -2 10 -z 10-1 10-
T h e e q u i v a l e n t w i d t h o f a D o p p l e r line is g i v e n b y
oo
wo = The function
[ - Sm
2 ~ ° Jo
1-exp[~
] exp(-x
o~oD(Sm/~oTr'/2).
2)j d x =
D(z) c a n b e a p p r o x i m a t e d w i t h a m a x i m u m r e l a t i v e e r r o r 5 x 10 - 6 as f o l l o w s 7
for
0~
by
D(z)=zrt 1/2~c.z"; 0 7
for
5 < z,
by
D(z) = [In(z)] 112 ~ d , [ l n ( z ) ] - " ; 0
h e r e t h e c o e f f i c i e n t s c, a n d d, a r e g i v e n in T a b l e 2.
Integrated a b s o r p t i o n of a spectral line with the Voigt profile Table 2. The coefficients c~ a n d dn in the a p p r o x i m a t i o n to the D o p p l e r profile n
cn 9.9999 --3.5350 9.6026 --2-0496 3-4392 --4-2759 3'4220 1"2838 -
-
8291 8187 7807 9011 7368 3051 9457 0804
dn 698 098 976 013 627 557 833 108
× x x x x x z x
10 -1 10 -1 10 -2 lO -2 lO - a 10 -4 10 -5 10 -6
1"9999 5'7749 --5.0536 8"2189 --2"5222 6"1007 --8"5100 4"6535
9898 1987 7549 6973 6724 0274 1627 1167
289 800 898 657 530 810 836 650
x × × × × x × ×
10 +° 10 -1 10 -1 10 -1 10 +° 10 +° 10 ÷° 10 +°
323
NOTE I N T E G R A T E D ABSORPTION OF A SPECTRAL LINE WITH THE VOIGT PROFILE C. D. RODGERS and A. P. WILLIAMS Clarendon Laboratory, Oxford, U. K. (Received 12 October 1973)
Abstraet--A simple approximation has been found for the integrated absorpton of a spectral line with Voigt profile. It is expressed in terms of the integrated absorption of the Doppler and Lorentz profiles. The maximum error is 8 %. THERE are two main mechanisms which determine the shape of molecular spectral lines. These are collision (Lorentz) broadening and Doppler broadening. In many cases of interest, one of these mechanisms predominates and the integrated absorption (equivalent width) of a spectral line may easily be calculated using the appropriate line shape. However, throughout a large part of the earth's atmosphere and in many astrophysical problems, both mechanisms are significant and the mixed line shape (Voigt profile) must be used. This is simply a convolution of the Lorentz and Doppler shapes, and may be written as k(v) =
S
exp(_t2)dt
y (~o
~,,,~,/2 ,~ -Y-~o~- ¥ (-2:- ty'
(1)
where t = (v - v')lct o, x = (v - Vo)/~o, y = ctt./~D; aL is the Lorentz halfwidth and ~t~ is the Doppler width of the line, S is the line intensity and k(v) is the absorption coefficient. The equivalent width of a single line is cO
W ( S , m, ctL, ~D) = | o._
1 -- e x p [ - k ( v ) m ] dr. oO
where m is an absorber amount. Unfortunately it is not possible to express the integrated absorption directly in terms of simple functions, or even functions of one variable for which efficient numerical approximation may be found, so any radiative transfer calculations must involve a time-consuming numerical integration. For the purposes of comparisons to be performed later, we have calculated W by numerical integration for a wide range of parameters. A contour plot of Wilt o is shown in Fig. 1. A detailed tabulation may be found in JANSSON and KORB.(~) The equivalent widths of both Lorentz and Doppler shapes can be calculated from simple numerical approximations, (see the Appendix for details). With a view to providing a fast method of computing the equivalent width of a mixed line, WM, CURTIS (2) suggested the approximation W u < WL + W D -
319
WL W a / W w ,
(2)
320
C . D . RODGERSand A. P. WILLIAMS
_o O~ 0
"~5
-4
-3
"2
-1
0
1
2
Io<3,0[o~./o~,]
Fig. 1. Contours of equivalent width of a mixed shaped spectral line divided by ~D, the Doppler width.
where WL and WD are the equivalent widths of the Lorentz- and Doppler-broadened lines, respectively, and W w = S m is the weak limit for a line of any shape. This formula has the correct form in all the limits, has a smooth changeover between the limits, and is easily differentiated. However, it over-estimates the true equivalent width, especially when WL --- WD, with errors as great as 50 per cent [see GILLV and ELLINGSON(a)]. The nature of the error is illustrated in Fig. 2. We have found a better approximation to be W u = [ w L 2 + w ~ 2 - ( w L W o / W w ) 2 ] 1/2.
(3)
This approximation also has the correct form in all the limits, and is almost as easily differentiated as Curtis' relation. A contour diagram of the difference between this function and the exact equivalent width is shown in Fig. 3, giving maximum errors of 8 per cent. Exponents differing slightly from 2.0 are found to give improved accuracy, but the algebraic convenience is lost.
0
m
t
I
~
2
Ioglo[ s m/~,]
'L
3
Zig. 2. Equivalent widths of a Lorentz (WL), Doppler (Wo) and nixed (W)shaped spectral line; original Curtis approximation W 1) and modified Curtis approximation (W2) with 0cL/~o= 0.1.
0.5
_o
:3
O
"SF
-~. -S
1
|
-4
-3
-1
Iogl0[o
-2
0
1
2
Fig. 3. Contours of percentage error in the modified Curtis ~pproximation for the equivalent width of a mixed shaped spectral line.
o OI O
3
O
t~
.q-
.=
O
r~
322
C . D . RODGERS and A. P. WILLIAMS REFERENCES
1. P. A. JANSSON and C. L. KORB., JQSRT 8, 1399 (1968). 2. A. R. CURTIS, unpublished. 3. J. C. GILLE and R. G. ELLINGSON. Appl. Optics 7, 471 (1968). APPENDIX T h e e q u i v a l e n t w i d t h o f a L o r e n t z line is g i v e n b y
WL = 27t~LL(Sm/2~c%), where
L(z) is t h e L a d e n b u r g - R e i c h e
function :
L(z) = z e-z[I o (z) + l l ( z ) ] . With a maximum
r e l a t i v e e r r o r o f 5 x 10 - 6 , we c a n p u t , f o r 0 ~< z ~< 1.98, 6
L(z) = z ~" a, z"; o
f o r 1-98 < z,
L(z) = z 1/2 ~ b,,z"; 0
h e r e t h e c o e f f i c i e n t s a . a n d b , a r e g i v e n in T a b l e 1.
Table 1. The coefficients a. and b, in the approximation of the Ladenburg and Reiche function rl
0 1
2 3 4 5 6
an
9'9999 --4.9988 2-4900 -- 1-0095 3"1358 --6"5014 6'4832
7697 2252 5331 6020 0341 4170 5819
bn
674 × 233 × 874 x 660 x 312 × 817 × 427 ×
10 -~ I0 -~ 10 -1 10- t 10 -2 10 -3 10 -4
7.9788 -9.9755 -1.9762 1.0586 1-3246 1.4794 0.0
5095 5137 3661 5022 7496 7878
473 671 059 723 350 333
× x x x x x
1010 -z 10 -2 10 -z 10-1 10-
T h e e q u i v a l e n t w i d t h o f a D o p p l e r line is g i v e n b y
oo
wo = The function
[ - Sm
2 ~ ° Jo
1-exp[~
] exp(-x
o~oD(Sm/~oTr'/2).
2)j d x =
D(z) c a n b e a p p r o x i m a t e d w i t h a m a x i m u m r e l a t i v e e r r o r 5 x 10 - 6 as f o l l o w s 7
for
0~
by
D(z)=zrt 1/2~c.z"; 0 7
for
5 < z,
by
D(z) = [In(z)] 112 ~ d , [ l n ( z ) ] - " ; 0
h e r e t h e c o e f f i c i e n t s c, a n d d, a r e g i v e n in T a b l e 2.
Integrated a b s o r p t i o n of a spectral line with the Voigt profile Table 2. The coefficients c~ a n d dn in the a p p r o x i m a t i o n to the D o p p l e r profile n
cn 9.9999 --3.5350 9.6026 --2-0496 3-4392 --4-2759 3'4220 1"2838 -
-
8291 8187 7807 9011 7368 3051 9457 0804
dn 698 098 976 013 627 557 833 108
× x x x x x z x
10 -1 10 -1 10 -2 lO -2 lO - a 10 -4 10 -5 10 -6
1"9999 5'7749 --5.0536 8"2189 --2"5222 6"1007 --8"5100 4"6535
9898 1987 7549 6973 6724 0274 1627 1167
289 800 898 657 530 810 836 650
x × × × × x × ×
10 +° 10 -1 10 -1 10 -1 10 +° 10 +° 10 ÷° 10 +°
323