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Opacity distribution functions and absorption in Schumann-Runge bands of molecular oxygen
Planet.
Space Sci. 1974,
OPACITY
Vol. 22,
pp.413
to 425.
Pergamon
Press.
Printed
in Northern
Ireland
DISTRIBUTION FUNCTIONS AND ABSORPTION IN SCHUMANN-RUNGE BANDS OF MOLECULAR OXYGEN
T.-M. FANG, S. C. WOFSY* and A. DALGARNO Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts, U.S.A. (Received 17 September 1973) Abstract-Absorption cross wavelength over the domain spectrum has been divided have been constructed for CO%, HeO, HzOz, NOz and
sections have been calculated for molecular oxygen as a function of 1 = 1750-2050 8, for temperatures between 190 and 400°K. The into 19 wavelength intervals and opacity distribution functions each interval. Atmospheric photodissociation rates of Oz, 0,, HNOI are presented. 1. INTRODUCTION
In the wavelength region between 1750 A and 2050 A, the principal source of opacity in the atmosphere of the Earth is provided by the discrete lines of the Schumann-Runge band system (B3x;c X31;) of molecular oxygen. Because of the rapid variation of the absorption cross section, averaged cross sections lead to inaccuracies and tedious and expensive line-by-line calculations are apparently required. Kockarts (1971) has carried out such calculations at a frequency interval of 0.5 cm-l and has given a graphical presentation of the results. Using different molecular constants, Hudson and Mahle (1972) have carried out similar calculations at higher resolution and they report their results in a lengthy table which allows the calculation of transmission and photodissociation in 19 wavelength intervals. Another study has been made by Brinkmann (1971) using a bandmodel approximation but his results appear to be model dependent. In this paper, we discuss the application of opacity distribution functions to the calculation of the transmission of solar radiation in the atmosphere of the Earth. We discuss also the selection of molecular constants and the construction of the 0, cross sections in the Schumann-Runge bands and adjoining continua and we apply the results to the calculation of the photodissociation rates for a number of constituents of the mesosphere. 2. OPACITY
DISTRIBUTION
FUNCTIONS
Opacity distribution functions have broad applicability to radiative transfer problems where the absorber exhibits a sharp line spectrum. The opacity distribution function method has been applied to stellar atmospheres (Avrett and Krook, 1963; Strom and Kurucz, 1966; Carbon, 1972) and to infra-red radiative transfer in planetary atmospheres (Arking and Grossman, 1972). To calculate the transmission of light in the ith wavelength interval we must evaluate the integral
Ti=
YiflI(v) exp (- o(~)iV) dv svi
(1)
where v is the frequency in cm- l, I(v,) is the unattenuated intensity in photons cm-2 (cm-l)-l se&, cr is the absorption cross section in cm2, and N is the column number density of absorbers in cm- 2. If a(v) is a slowly varying function of v, the integral may be * National Academy of Sciences Research Associate at the Smithsonian Astrophysical Observatory. 6
413
414
T.-M. FANG,
S. C. WOFSY and A. DALGARNO
approximated by taking the interval Avi = v~+~- vi sufficiently small such that u(v) and I(v) may be approximated by their average values oi and K) in the interval. The transmission then becomes Ti= 1%) exp (--0, N)Av, (2) which can be evaluated without integration for any N. However, if a(v) varies rapidly with v, this procedure amounts to a ‘line-by-line’ calculation which can be quite tedious. In the case of the Schumann-Runge bands, the lines may be as narrow as O-2 cm-l and the bands extend for about 10,000 cm- l. Use of an average cross section requires at least 5 intervals per line, which yields the formidable sum of 2.5 x lo5 intervals (for each N). The difficulty may be avoided by introducing a normalized distribution function &(o) such that S,(u) da gives the probability that u lies between u and u + du in the interval Avi, subject to co S,(u) da = 1. (3) s0 Using S,(u) we may change the variables in (1) (Arking and Grossman, Ti = I(Y,)Av,
1972) to obtain
*msx exp {-u N}S$(u) da. s %iIl
In (4) the exponential contains no rapid function of the integration variable. However S,(O) is not a smooth function and its construction may present some numericalcomplexities. The task can be greatly simplified by the definition of a new function u(f), the opacity distribution function (ODF), such that
f= US
da’.
(5)
um is implicitly defined such that a given fraction f of the interval is occupied by cross sections less than u(J), and (1 -f> by cross-sections greater than u(f). Clearly S(a) da = df
(6)
and (4) becomes Ti = I(Y,)Av, :exp {-u(f)N} s
dJ
(7)
The opacity distribution function u(f) is a smooth function and In u(f) may be accurately reproduced by a polynomial in f,The construction of u(f) merely requires the calculation of oj({vJ) for a sufficiently large set of {vi}, and an ordering of the a) which we choose to be ascending. If n values of u are calculated, the value which ranks ith in the ordered set corresponds to f = (k- l)/(n - 1). The photoexcitation rate Ji of the absorber due to photons in the ith interval is given Ji = I(yi)
s
olexp {-Q)N}u
(f)
dJ
(8)
For the predissociated bands of 0, (8) may be used when summed over i to obtain the photodissociation rate of 0,. The construction of the ODFs may be achieved economically by the use of random selection of frequencies in each interval. Although slowly convergent, the method gives good results with relatively few points, and the ‘noise’ produced by random selection is
OPACITY
OF MOLECULAR
OXYGEN
415
filtered by the polynomial fitting off. Tests showed that about 500 points per frequency interval yield 5 per cent accuracy or better. At each point in the bands, contributions from all lines within &75 cm-l were included. Small corrections were then applied to take account of the quasi-continuum due to all lines lying between 75 and 200 cm-l from the given frequencies. Table 1 gives the fits to the ODFs, and Fig. 1 shows several ODFs and their polynomial representations. Table 1 also gives the ‘excess’ continuum-absorption discussed below. For wavenumbers less than about 50,700 cm-r (A > 1972 A) the ODF is adequately represented by a single term, although correlations between the coefficients in Table 1 obscure this fact. Average cross sections are thus adequate for representing oxygen absorption for 1> 1972 A. 3. ABSORPTION
CROSS SECTIONS
FOR 0,
Accurate molecular constants are not available for all bands of the Schumann-Runge system. Although the rotational constants are fairly well determined spectroscopically (Brix and Herzberg, 1954; Ackerman and Biaume, 1970), centrifugal distortion and spin-splitting were not completely determined by Brix and Herzberg and no values for fine structure were presented by Ackerman and Biaume. The spin-splitting is comparable to, or larger than, the width of the rotational lines and knowledge of it is required to compute line positions accurately and to interpret photographic line widths. Bergeman and Wofsy (1972) derived from the spectroscopic data a complete set of molecular constants for vibrational levels u’ = 9-19, and also obtained some information about v’ = O-3. Because of limitations in the data, some of the results are not highly accurate but the molecular constants of Bergeman and Wofsy (1972) were adopted as the best available. Where they made no determination, the rotational constants were adopted from Ackerman and Biaume (1970) and the centrifugal distortion and spin-splitting constants extrapolated from Bergeman and Wofsy (1972). Table 2 gives the molecular constants used in the present calculations. The photodissociation halt-widths that enter into the assumed Voigt lineshapes, were taken from the photoelectrically determined values given by Hudson and Mahle (1972). These line widths are different from those used by Ackerman er al. (1970), who adopted photographic widths that include appreciable contributions from fine structure and other effects. The oscillator strengths were calculated for each rotational line by Allison (1972) from the same potentials and transition dipole employed by Allison et al. (1971), which gave an excellent representation of the available measurements. The line oscillator strengths vary by as much as a factor of two as a function of rotational quantum number N for lines which are important at 300°K. The variation of the line oscillator strengths with N are shown in Table 3 for several bands. N = 45 was the largest used in the present calculations forv” =OandN=30forv” - 1 and 2. Since all fine structure components were included, the total number of lines was about 9000. The calculation of Jarmain and Nicholls (1967) was fitted and used to calculate the cross section for absorption in the Herzberg continuum of 0,, an absorption process that is crucial to the photodissociation of 0, in the stratosphere. The theoretical cross sections agree very well with the experimental values (cf. Hudson, 1971). The Schumann-Runge continuum contribution was calculated from the potentials and transition dipole of Allison, Dalgarno and Pasachoff (1971). In order to consider temperatures above 300”K, the calculation had to be extended to a” = 3. At elevated temperatures the long-wavelength edge of the Schumann-Runge continuum is shifted into
-5.2351 -52329 -5.2315 -5.2308 -5.2300
- 5.2429 -5.2418 -5.2414 -5.2414 -5.2421
-5*0836 - 5.0543 -5.0291 -50106 -4.8915
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
500009-49382~7
4938Z7-48780.5
55054+-54626.9
101 +01 +Ol +Ol +Ol
+ 01 +Ol +Ol +Ol +Ol
+Ol +Ol +Ol +Ol +Ol
i-01 -i-O1 +Ol +01 +Ol
i-01 $01 +Ol i-01 $01
+Ol f01 +Ol +Ol +Ol
is interpreted as 10+J1.
-5.2303 -5.2281 -5.2269 -5.2265 -5.2270
190 230 270 300 400
50377%-50000*0
* The notation “$01”
-5.2297 -5.2301 -5.2309 -5.2317 -5.2323
190 230 270 300 400
50715.1-50377.8
%
-5.2366 --5.2357 -5‘2353 -5.2352 -5.2350
190 230 210
51361*1-50715.1
CO
Temperature (“K)
Wavelength interval
4.2281 1.6404 6.7890 1.5378 2.5826
-3.3653 -3.2792 -3.1768 -3.1053 -3.0166
-4.5780 -5GO81 -5.4151 -5.6630 -6.2994
-5.1694 -56448 -5.9595 -6G945 -6G919
-4.7523 -3.9221 -3*2085 -2,7035 -2.1083
-7.9789 -6.5344 -5.1164 -4.1662 -3*9084
Cl
+00 +00 -01 +00 +OO
+00 $00 +00 $00 $00
$00 +OO $00 +00 +00
+00 +OO +00 +00 +00
100 +00 $00 +00 $00
-01 -01 -01 -01 -01
6.9940 4.1386 5.8739 5.3253 2.2293
3.8361 3.7386 3.6254 375467 3.4606
5.1778 5.6624 6.1199 6.4003 7.1333
5.8372 6.3962 6.7761 6.9487 7.0501
5.4895 4.5696 3.7902 3.2502 2.6579
8.6120 7.6399 6.4221 5.6427 8.8626
cz
+00 ;Of +Ol $01 +Ol
$01 +Ol $01 +Ol +Ol
+Ol +Ol +Ol +Ol +01
+Ol +Ol +Ol +Ol $01
+Ol +Ol +01 $01 +Ol
_tOO +00 +00 +00 +00
-5.8024 -2GO81 -2.7897 -2.6326 -1.3285
- 1.7092 -1.6654 -1.6160 -15819 -1.5488
-2.2976 -2.5142 -2.7182 --2.8441 -3.1717
-2.6060 -2.8664 -3*0488 -3.1361 -3.2192
-2.4904 -2.0816 -1.7382 -1.5049 -1.2605
-3~5.562 -3.3675 -29096 -2+598 -4*OlOl
C3
3.5378 34467 3.3468 3.2780 3.2208
+ 02 to2 +02 +02 +02
1.8003 4.4370 5.9283 5.6842 3.2712
4.7374 5.1870 5.6103 5.8729 6.5797
1-02 j-02 +02 +02 +02
$01 1-02 j-02 +02 $02
54085 5.9728 6.3812 6.5861 6.8337
5.2602 4-4288 3.7397 3.2809 2.8365
+02 f02 +02 i-02 +02 102 $02 +02 -to2 +02
6.8490 6.9883 6.3670 5.7554 8%09
101 101 -t-O1 $01 +Ol
C4
TABLB 1. OPA~ITYDISTRIBUTIONFUNCTIONSFORMOLECULAROXYGEN*
-f-O2 +02 i-02 +02 -l-02
102 +02 -t-O2 +02 102
$02 -t-O2 +02 1-02 -l-O2
j-02 -+ 02 4-02 “f-02 102
+02 -i-O2 to2 -+02 -1-02
101 -f-O1 $-01 3-01 -t-O1
-2-2289 -4.4894 -5.7880 -5.5949 -3.5216
-3*4102 -3.3220 -3.2280 -3.1634 -3.1193
-4.5508 -4.9856 -5.3946 -5.6500 -6.3478
-52296 - 5.8000 -6.2265 -64494 -6.7621
-5.1751 -4-3916 -3.7545 -3.3397 -2.9747
-6.2437 -6.5852 -6.1563 -5.6216 -7.8162
c5
102 j-02 $02 $02 +02
1-02 +02 $02 j-02 102
j-02 -i-O2 $02 1-02 -+02
+02 + 02 +02 102 1-02
-to2 402 1-02 -j-O2 102
+Ol $01 +Ol +Ol +Ol
9.8978 1.7206 2.1430 2.0810 1.4025
1,2369 1.2049 1.1716 1.1488 1.1368
l-6454 I.8036 1.9524 2-0459 2.3052
l-9034 2.1206 2.2883 2.3793 2.5219
l-9461 1.6680 14470 1.3064 1.1939
2.6814 2.7700 2.6056 2.3966 2.9568
C6
*r
+Ol I+-21 1-02 + 02 +02 -I-02
+02 +02 +02 102 102
102 $02 -t-O2 +02 +02
+02 102 3-02 +- 02 $02
102 +02 +02 -t-O2 $02
$01 +Ol +Ol +Ol +-01
5
-5.2351 -5.2349 -5.2360 -5.2364 -5.2314
-4.5783 -4.5321 -4.4956 -4.4726 -4.4101
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
53665.3-53129.3
53129.3-52565.2
52565.2-51975.1
51975.1-51361.1
-5.2437 -5.2379 -5.2336 -5.2273 -5.2058
-5.2213 -5.2040 -5.1876 -5.1787 -5.1603
-5.1616 -5~1444 -5.1174 -5.1053 -5.0910
-5.0855 -5.0599 -5.0469 -5.0394 -4.9991
190 230 270 300 400
54165.3-53665.3
-5.0662 -5a475 -50436 -5.0401 -4.9697
190 230 270 300 400
54626.9-54165.3
+ 01 $01 +Ol i 01 +01
+Ol +01 +Ol +Ol +Ol
+01 +01 t-01 +Ol t-01
+Ol +Ol CO1 +Ol +Ol
+01 +01 $01 $01 +01
$01 +01 $01 $01 +01
$01 to1 $01 $01 $01
+00 -02 +00 +00 +OO
+00 -01 t-00 +oO -01
-02
+oo
+OO
$00
+Oo
+Oo $00 +OO +00 $00
-02 -01 + 00 +00 +01
-01 -01 $00 i-00 +Ol
6.0965 $00 3.9096 +00 2.3651 100 1.5431 $00 5.0789 -02
-1.1521 -7.9130 1.8215 3.1788 4.8856
- 1.8416 -3.0855 1.6833 1.5655 1.4243
-1.3053 -2.5927 -4.6871 -4.9117 -7.3118
-2.9359 2.3613 2.3058 2.6477 9.6179
7.6764 -2.5303 2.2997 4.7781 1.0375
-6.5302 -3.9945 4.7125 8.3241 1.0420
3.5537 1.0773 1.5600 I.7648 l-8537
1.6303 5.8213 -1.3992 -2.7857 -3.8153
3.0770 1.8515 2.8527 7.1970 3.3470
3.3147 4.5371 7.2196 8.0653 5.1643
5.2544 -8.8134 -1.7065 -1.7960 -6.3188
2.6717 3.1746 1.3213 -7.2212 -5.7907
4.2230 4.1558 -3.8496 -34445 -5.5181
+Oo +01 +Ol $01 $01
+Ol +00 +Ol f01 +Ol
$01 +Ol +00 +Oo +01
$01 $01 +Ol $01 +Ol
+Ol +00 +Ol t-01 +Ol
+Ol +Ol +Ol +OO to1
+Ol +Ol +OO +Ol +Ol
-5.5819 -6.7895 -7.6252 -7.8748 -7.2157
-74033 -2.5497 6.3981 1.2573 1.6492
-1.3432 -8.9894 -3.1604 -4.8743 -1.5091
-1.5506 -1.8879 -2.9461 -3.3371 -2.4371
-2.1043 6.0831 1.2238 1.2892 2.7151
-5.9480 -7.4087 - 1.2786 5.8121 2.3140
-1.7156 -1.6330 9.6358 1.1956 1.9304
+Ol $01 101 +01 $01
+01 + 01 101 +02 $02
+02 +Ol +Ol $01 +02
+02 +02 +02 $02 +02
+02 +Ol +02 +02 +02
+Ol +Ol +Ol +Ol +02
+02 t02 +OO +02 +02
1.2707 1.3327 1.3864 1.3834 1.1931
1.6272 7.1362 - 1.0388 -2.2460 -2.9387
3.0848 2.4119 l-4044 1.6913 3.3246
3.7646 4.1834 6.0437 6.7566 5.1640
4.5051 -8.5832 -2.3893 -2.6123 -4.9899
7.2616 8.6922 - 1.6645 -1.3796 -42732
3.5206 3.2591 1.0886 -1.8102 - 3.2240
+02 +02 t02 +02 +02
$02 $01 +02 +02 +02
+02 +02 +02 +02 +02
+02 +02 $02 +02 +02
+02 +Ol +02 +02 $02
+Ol $01 + 01 +02 t-02
+02 t02 +01 +02 + 02
-1.1776 -1.1364 -1.1240 -1.0939 -9m95
-1.4837 -7.3401 8.0419 1.8657 24078
-3.1303 -2.7069 -1.9055 -2.1310 -3.2943
-4.0607 -4.3152 -5.8428 -6.4249 -4.9938
-4.4333 3.8895 1.9454 2.2128 4.1569
-4.9187 -5.2850 3.2466 1.3273 3.6761
-3.4173 -3.1200 -4.0855 1.1977 2.5708
+02 +02 t02 +02 +01
+02 $01 +Ol f-02 +02
+02 +02 t02 $02 +02
+ 02 $02 $02 + 02 $02
+02 $01 +02 $02 +02
+Ol t-01 +Ol +02 +02
+02 +02 +Ol f02 +02
4.1425 3.7565 3.5566 3.3830 2.6805
5.0407 2.7562 -2.2683 -5.7421 -7.3176
1.1735 1.0847 8.4325 9.0981 1.2086
1.6043 1.6657 2.1421 2.3189 1.8190
1.6188 4.1370 -5.5777 -6.6383 - 1.2768
1.7007 1.5926 -1.1372 -4.3422 -1.1778
1.2829 1.1659 2.7662 -2.4115 -7.5849
$01 $01 f01 401 +Ol
$01 +Ol $01 +Ol +Ol
+02 +02 +Ol $01 + 02
1-02 +02 t-02 +02 $02
+02 -01 +01 $01 +02
+01 +Ol 401 +Ol +02
f02 +02 +Ol +Ol +Ol
45-20
2.5-22
P
f01 +Ol + 01 +Ol +Ol +Ol $01 +Ol $01 + 01 +Ol $01 + 01 +Ol +Ol
-4.9574 -4.8731 -4.7823 -4.7200 -4.5619
-49637 -49086 -4.8305 -4.7696 -4.6028
-4.8977 -4.8756 -4.8321 -4.7920 -4.6557
-4.9747 -4.9389 -49060 -4.8800 -4.7664
190 230 270 300 400
190 230 270
190 230 270 300 400
190 230 270 300 400
56350.7-56097.8
56097.8-55784.9
55784.9-55444.7
55444.7-55054.0
z
+01 +Ol + 01 +Ol $01
+01 +01 t-01 +Ol $01
-4.7797 -4.7133 -4.6539 -4.6141 -4.5059
190 230 270 300 400
56541.9-56350.7
1.2336 1.2680 1.1456 9.7865 2.9198
4.2830 6.4831 6.2280 6.4682 5.9816
1.9163 2.7111 -5.9451 -6.6144 -1.3723
3.3071 2.5160 7.3604 -3.6818 -2.4236
-6.5833 -6.1774 -5.3500 -4.6018 -2.5175
2.0107 1.7931 1.4820 I.2481 6.5901
+ 01 i-01 +Ol + 01 +Ol
-4.7076 -4.6596 -4.6119 -4.5782 -4.4819
230 270 300 400
190
56715.1-56541.9
Cl
Temperature (“K)
co
Wavelength interval
+Ol +Ol +Ol +00 +00
$00 +00 +00 $00 +OO
+00 -01 -01 +01 +00
+00 +00 -01 -01 +00
+00 +00 +00 $00 +00
$01 $01 +Ol +Ol +00
-2.7108 -2.7535 -1.5938 -2.3412 3.8303
-6.9983 -2.9154 -3.3834 -4.3468 -4,7253
3.3412 4.1646 3.8559 3.2728 2.7976
6.6499 5.7338 5.3100 5.0919 4.7928
1.2459 1.0129 7.9505 6.5200 3.3487
-1.1956 -1.0476 -8.3674 -6.7935 -2.9636
c2
+Ol $01 $01 +00 $01
+00 SO1 +Ol +Ol +Ol
+Ol +Ol +Ol f01 $01
$01 +Ol $01 +Ol +Ol
+02 +02 +Ol +Ol $01
+02 f02 +Ol +Ol $01
Table 1 cond.
6.1872 54497 3.5729 -5.1208 -1.9261
5.0746 1.3584 1.5638 1.9740 2.0363
-1.7373 -2.0137 -1.8625 -1.6318 -1.3645
-3.7439 -3.2659 -2.8825 -2.6609 -2.2815
-4.6753 -36043 -2.7223 -2.1898 -1.1019
3.9423 3.3314 2.5542 1.9952 6.8210
c3
t-01 +01 t-00 $01 $02
$01 +02 t02 +02 +02
+02 +02 +02 +02 +02
+02 +02 +02 $01 +02
+02 +02 + 02 +02 $02
+02 +02 +02 +02 +Ol
+02 +02 +02 $02 $01
-8.6655 -6.8116 3.0918 1.3514 3.8601
-1.3951 -3GO15 -34058 -4.1837 -4.1773
3.9345 4.3881 4.0988 3.6701 3.0264
8.4958 7.5292 6.5990 6.0222 4.9128
+Ol +01 +Ol $02 1-02
to2 +02 +02 102 +02
$02 $02 $02 $02 +02
+02 +02 +02 $02 $02
8.2167 +02 6.1209 +02 4.5270 d-02 3.6176 $02 1~8320 +02
-6.4162 -5.2113 -3.8115 -2.8387 -6.3130
c4
66055 4.9892 -3.7803 -1.3005 -3.4329
1-5912 3.0513 3.4349 4.1225 40162
-4.0281 -4.3664 -4.1030 -3.7236 -3.0788
-8.5731 -7.6889 -6.7392 -6.1188 - 4.8426
-6.9722 -5.0826 -3.7194 -2.9731 -1.5298
4.9895 39012 2.7143 1.9119 1.4260
CS
+Ol +Ol +Ol $02 +02
$02 +02 to2 $02 +02
t-02 +02 $02 +02 +02
$02 +02 _t 02 $02 + 02
+ 02 +02 $02 +02 +02
$02 +02 $02 +02 +Ol
-1.8838 -1.4226 l-4585 4.5173 l-1391
-5.9939 -1.1081 -1.2487 - 1.4793 -1.4130
1.5654 1.6551 1.5613 1.4329 1.2020
3.2191 2.9140 2.5611 2.3219 1.8082
2.3224 1.6790 1.2304 9.9211 5.3010
-1.4598 --I+)964 -7.1742 -4.6658 7.2977
C6
8.8-21
+Ol +Ol -1-01 $01 +02
$01 +02 t02 + 02 $02
$02 $02 $02 +02 +02
1.8-12
+02 4G-21 +02 $02 $02 $02
+02 $02 t02 $01 +01
f02 2%20 $02 +Ol +Ol +00
Qex
OPACITY
OF MOLECULAR
0.2
0
0.6
0.4
OXYGEN
0.8
419
I.0
f
FIG.1. OPAC~ DISTRIEKJTIONFUNCTIONSFOR O,, T= 300°K. Curve a, 53129.3 cm-52565.2 cm-t; curve b, 56715.1-56541.9 cm-l. Curve b should be scaled by the factor 105. The dotted curves are the computed ODF’s and the solid curves are their polynomial representations. TABLET. MO~~ARCO~~A~
X”IZ,f a” ZZ
FOR
0, SCH~A~N-~~~EBA~~*
0 1 2
1.43768 I.42198 14068
4.913 -6 4.825 -6 3.6 -6
1.9848 1.9848 1.9848
-0+)0843 -O+lO843 -0*00843
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 19 203 218
0.8127 0.8001 0.7852 0*7699 0.7537 0.7372 0.7194 0.6997 0.6771 O-6531 0.6279 0.5990 0.5621 0.5244 0.4832 O-4396 0.3945 0.3471 0.2872 0.2649 omitted omitted
5.06 661 510 4.54 356 5.71 5~71 6.96 6.71 8.60 1.26 2.14 1.29 1.67 2.09 264 3.3 4.0 55 6.0
1.63 1.49 l-45 1.50 1.50 1.50 1.50 1.50 1.50 I.63 1.60 1.81 2.15 254 283 3.36 4.04 5.18 6.51 7.63
-o+xi7 -0*008 -0.009 -0*010 -0~010 -O*Oll -0*012 -0-014 -0-015 -0*022 -0.028 -0.050 -0.060 -0.094 -0.137 -0.179 -0.272 -0.349 -0.494 -0.604
B%VP =
-6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -5 -5 -S -5 -5 -5 -5 -5 -5 -5
2.78 -10 3.31 -9 2.03 -8 8.63 -8 2.83 -6 7.85 -6 1.83 -6 3.72 -6 6.65 -6 1.07 -5 158 -5 2.13 -5 2.66 -5 3.06 -5 3.24 -5 3.16 -5 2.91 -5 254 -5 2.07 -5 la.58 -5
O@l 0*002 0.34 1.25 3.30 2.20 1.70 2-25 2-21 O-72 0.34 1.80 0.48 0.08 0.06 0.20 o-25 040 040 0.40
49358.2 50045.7 50710.8 51352.3 51969.8 52561.4 53122.8 536563 541563 54622.3 55050.9 55438.9 55784.5 56085.2 56339.9 56549-7 56718-l 56850.2 56951.6 57025.8
* All quantities given in cm-l except for oscillator strengthsf. The notation follows Bergeman and Wofsy (1972). t Oscillator strength for rotationless state (J = 0) from Allison (1972), for (0’. 0) band. $ AvllB denotes Lorentzian full width at half maximum, used in Voigt profile lineshape (Hudson and Mahle, 1972). 8 Brix and Herzberg (1954) assigned lines to (21,O) and (20,O) bands, but no molecular constants are available for the apparently perturbed bands.
T.-M. FANG, S. C. WOFSY and A. DALGARNO
420
TABLE 3. OSCILLATORSTRENGTHS(R-BRANCH):VARIATIONWITH ROTATIONALQUANTUM NUMBER* IN THE SCHUMANN-RLINGEBANDS
N=
1 11 21 31
(17,O)
(8,O)
(0, 0)
(17,l)
(8,l)
(0,l )
(17,2)
(872)
25-5 2.2-5 1.6-5 l.l-5t
6.7-6 6.46 5.7-6 4.8-6
2.8-10 2.7-10 2.410 2.1-10
2.1-4 1.8-4 1.44 9%-57
8.8-5 8.5-5 7.7-5 6.6-5
7.5-9 7.3-9 6.7-9 5.9-9
6.a 5.9-4 46-4 3*4-4t
5.1-4 4.9-4 4.5-4 3.94
* Allison (1972) 7 Given for N = 27, the highest N-transition in the R-branch for v’ = 17.
the region of the Schumann-Runge bands. Calculations at these temperatures required careful evaluation of the rotational dependence of the continuum cross section. For rotational levels up to iV = 50 an excellent representation of the rotational dependence is given by (T(Y,iv) = c(l&
N = 0)
(9)
where Yeft = Y - 0.561 N(N + 1);
(10)
the coefficient of N(N + 1) is an empirically determined value roughly equal to the difference in rotational constants in the B and X ground vibrational levels. The calculated cross section was used from about 1350 A to threshold. Extrapolation to N = 80 was necessary for the highest temperatures (900°K). The measured value (cf. Ackerman, 1971) o(1215.7 A) = 1-O x 1O-2ocm2 was used for the oxygen absorption cross section at Lyman a. COMPARISON OF SYNTHETIC AND
EXPERIMENTAL
CROSS SECTIONS
4.7 Schumann-Runge bands Most of the experimental data on oscillator strengths and line positions have been used in the construction of the synthetic cross sections, either directly or through the work of Bergeman and Wofsy (1972) and Allison et al. (1971). Hudson and Mahle (1972) deduced an independent set of line oscillator strengths in the same data analysis which yielded the predissociation linewidths and their oscillator strengths agree very well with those of Allison et al. (1971). Ackerman et al. (1971) obtained linewidths from photographic tracings and from a fit to the oxygen cross sections measured at various silicon emission lines. Using the distinctly different set of half-widths from Hudson and Mahle (1972), we obtained excellent agrement between the synthetic cross section and the observations of silicon emission line absorption. In a few instances small discrepancies could be ascribed to uncertainties in the molecular constants such as a displacement of the R-branch for u’ = 9 by about O-1cm-l. A displacement of O-1 cm-l is well within the uncertainty of the molecular constants. In the a’ = 0 level the difference between the linewidths of Hudson and Mahle (1972) and of Ackerman ef al. (1971) is roughly equal to the fine-structure contribution to the apparent line-with. The procedure used by Hudson and Mahle (1971), as described by Hudson (1973), utilizes the absorption in the wings of the line. Hudson’s procedure is thus relatively insensitive to uncertainties in molecular constants, although it is sensitive to the assumed Lorentzian shape of the line wings.
OPACITY
OF MOLECULAR
OXYGEN
421
4.2 Schumann-Runge continuum The work of Allison et al. (1971), which was used in constructing the present synthetic cross section, was based on experimental data for 0, absorption in the discrete bands and in the continuum for 2 < 1750 A. A significant test of the cross section is provided by a comparison with the continuum data of Hudson, Carter and Stein (1966) for 2 > 1750A. Excellent agreement is obtained between the calculated continuum absorption and the measurements at 600” and 900°K as shown in Table 4. Agreement is also obtained with room-temperature data outside the range 1750-1900 A (Hudson and Mahle, 1972; Hudson, Carter and Stein, 1966). TABLE 4. CONTINUUMCROSS
Wavelength (A)
Calculated 600” 1.99-19 1.22-19 8.05-20 6.16-20 4.90-20 1.32-20 538-21 1.78-21 4.62-22 1.40-22
1760 1770 1780 1790 1800 1820 1840 1860 1880 1900 1920 1940 * Hudson,
SECTIONSAT 600 AND 900°K(cm*) Observed * 600
900°K 4.35-19 3.32-19 2.56-19 2.09-19 1.70-19 9.65-20 5973-20 2.73-20 1.13-20 5.27-21 244-21 9.64-22
1.85-19 1.20-19 8.4-20 56-20 3.7-20 1.7-20 6.0-21 2.0-21
900°K 4.30-19 3.52-19 2.96-19 2.30-19 1.76-19 94-20 5.0-20 1.5-20 9.0-21 5.0-21 2.0-21
Carter, Stein (1966).
In the range 1750-1900 A at 300”K, the experiments show an excess continuum absorption compared to the calculations. Hudson and Mahle (1972) have ascribed this excess to absorption from the U” = 1 level. The excellent agreement of the calculations with the observations at T = 600” and 900°K, which depend strongly on absorption from v” = 1, make this unlikely. An alternative source is the 3C - 311continuum predicted by Wilkinson and Mulliken (1957). A definite determination is precluded by uncertainties in the molecular constants, the treatment of predissociation lineshapes, and the experiments themselves. Because of this disagreement with experiment, two sets of opacity distribution functions were constructed for 0,. In one set the synthetic cross section was used unchanged, and in the other the ODFs were fitted to the average transmissions given by Hudson and Mahle (1971) by the addition of a slowly varying continuum cross section. The additional cross sections are listed in Table 1. This procedure also greatly improved the agreement between the experimental and calculated continua underlying the Schumann-Runge bands at 300°K. We stress, however, that uncertainties in the molecular constants may be at fault, at least at some wavelengths. In particular, the wavelengths occupied by the (11, 0) band seem to suffer from anomalously large discrepancies. The molecular constants of the (1 I, 0) band are poorly known (Bergeman and Wofsy, 1972). 5. CALCULATION
OF ATMOSPHERIC
PHOTODISSOCIATION
RATES
The atmospheric effects of the sharp-line structure in the Schumann-Runge bands were evaluated by calculating photodissociation rates (J-values) for several constituents using the opacity distribution functions and using average cross sections. The species considered were H,O,, H,O, HNO,, N,O, CO,, 0, and O,, with cross sections taken
422
T.-M. FANG, S. C. WOFSY and A. DALGARNO TABLE
Wavenumber (cm-l) 16923 74074 71429 68966 66667 64516 62500 60606 58824 57143 55556 54054 52632 51282 5ooO0 48780 47619 46512 45454 43478 42553 41667 40816 39216 38835 38095 37383 36697 36036 35398 34783 34188 33613 33058 32520 32000 31496 31008 30534 30075 29630 29197 28777 28369
5.
PHOTODISSOCIATIONCROSS-SECTIONSIN
H&I
1.5 -18 1.2 -18 75 -19 6.2 -19 4.9 -19 4.3 -19 3.8 -19 3.4 -19 3.0 -19 2.4 -19 1.9 -19 1.6 -19 1.3 -19 1.1 -19 8.4 -20 6.5 -20 5.3 -20 3.8 -20 3.0 -20 2.3 -20 1.8 -20 1.4 -20 1.1 -20 8.1 -21 5.8 -21 3.4 -21 2.7 -21 1.9 -21 1.3 -21 6.4 -22 5.1 -22 3.8 -22 2.5 -22 1.3 -22 1.0 -22 7.3 -23
from McConnell and McElroy flux was taken from Ackerman
0,
8.519 -19 8.519 -19 8.519 -19 6.667 -19 5.185 -19 4.074 -19 2.963 -19 3.333 -19 5.926 -19 8.148 -19 1.700 -18 2.9 -18 4.4 -18 6.2 -18 8.1 -18 9.9 -18 1.1 -17 1.1 -17 1.1 -17 1.0 -17 ::4” 4.6 2.7 1.6 1.0 4.7 2.6 1.4 6.8 3.6 1.8 7.1 ;:; 6.9 4.0 2.5
Z:; -18 -18 -18 -18 -19 -19 -19 -20 -20 -20 -21 -21 -21 -22 -22 -22
KG 7.0 3.2 7.0 5.0 1.0 2.2 3.5 4.8 4.3 2.8 7.5 5.0 5.2 6.3 6.3
-18 -18 -19 -19 -18 -18 -18 -18 -18 -18 -19 -20 -21 -22 -23
cm’
HNO,
1.32 -17 9.10 -18 5.50 -18 2.55 -18 9.70 -19 3.28 -19 144 -19 8.51 -20 5.63 -20 3.74 -20 260 -20 2.10 -20 1.95 -20 1.94 -20 1.92 -20 1.85 -20 1.70 -20 1.50 -20 1.27 -20 9.95 -21 7.0 -21 5.0 -21 3.1 -21 2.1 -21 1.3 -21 7.0 -22 3.0 -22 1.0 -23
NaO 8.9 -17 3.7 -18 5.0 -19 4.0 -18 1-O -18 1.0 -19 3.0 -20 4.0 -20 7.0 -20 9.0 -20 1.1 -19 1.1 -19 1.0 -19 6.0 -20 3.3 -20 1.5 -20 7.0 -21 3.0 -21 1.2 -21 4.1 -22 1.1 -22 3.2 -23 1.6 -23 9.6 -24 6.1 -24 4.6 -24 4.3 -24 4.2 -24 4.9 -24 5.3 -24 5.3 -24 5.0 -24 4.6 -24 4.5 -24 4.5 -24 4.1 -24 3.3 -24 2.0 -24
co, 6.7 -19 7.8 -18 6.1 -19 5.9 -19 5.4 -19 3.6 -19 1.8 -19 7.3 -20 2.2 -20 6.0 -21 1.4 -21 2.8 -22 7.0 -23 1.9 -23 2.5 -24 8.0 -25
(1973) and references cited therein (see Table 5). The solar (1971) and the atmosphere from U.S. Standard Atmosphere
Supplements (1966) for 30”N lat (July). Calculations were carried out for zenith angles a such that cos ct = 1.0, O-5, and O-25. Ozone densities were taken from Hilsenrath (1972) and from Hering and Borden (1966). Photodissociation by Ly a was included because of the 0, window at 1215.7 A. The results for H,O, COz and 0, are given in Fig. 2. The results for H,O are in fairly
423
% PHOTODISSOCIATION
RATE (se&)
424
T.-M. FANG,
S. C. WOFSY
and A. DALGARNO
good agreement with the calculation of Kockarts (1971) for cos a = 1.0. The agreement is better where the excess absorption is taken into account. Detailed comparison was made for cos a = O-5 between J-values calculated with the opacity distribution functions and those calculated with cross sections averaged over 50 A segments. The differences were inconsequential for constituents with substantial photodissociation cross-sections for 3L> 1950A such as H,O,, HNO,, N,O and 0,. Photodissociation of these species is adequately described by a model in which shielding is computed using 50 A average cross sections for 0, and 0,. Moreover, for altitudes less than 50 km light with 1 < 2000A is very substantially shielded. Hence at these altitudes photodissociation rates for OS, CO, and H,O may be calculated using 50 A average cross sections for 0, and 0,. The J-values for CO, and H,O are negligibly small in this height regime, and absorption by 0, is dominated by the Herzberg continuum. Below 50 km the J-values for all constituents except CO, and H,O are very sensitive to the 0, distribution. Typical J-values for H,O,, O,, HNO, and N,O at cos a = 0.5 are presented in Table 6. TABLE 6.* PHOTODISSOCIATION RATES (see-l)t
Altitude (km) 120 70 65 60 55 50 45
HNO,
Hz01 1.1 1.1 1.1 1.0 9.5 8.1 5.2
-4 -4 -4 -4 -5 -5 -5
1.5 1.4 1.4 1.3 1.2 1.1 8.3
-4 -4 -4 -4 -4 -4 -5
N,O 3.7 9.1 8.4 7.8 7.1 6,3 5.0
-6 -7 -7 -7 -7 -7 -7
0, 9.2 9.2 9.1 8.8 8.1 6.5 3.4
-3 -3 -3 -3 -3 -3 -3
* Signed number is power of 10. t Cos a = 0.5.
For altitudes between 100 and 50 km the J-values for 0,, H,O, and CO, are not well represented by a model using average cross sections for 0,. For 0, the discrepancy is no more than 20 per cent, but for CO, and HZ0 the average-cross section model gives J-values too small by more than a factor of 2. When the ODFs are adjusted for the ‘excess absorption’ discussed above, significant modifications occur for CO,, H,O, and 0, in the height range 100-50 km. The results are shown in Fig. 2 for CO, and H,O at cos a = 0.5. The magnitude of the effect may be regarded as the minimum estimate of the uncertainty in atmospheric photodissociation rates due to uncertainties in the 0, absorption spectrum. CONCLUSIONS The
applicability
of opacity
distribution
functions
to atmospheric
absorption
by 0,
has been demonstrated, and our current best estimates of the ODFs have been presented. For constituents with substantial photodissociation for wavelengths 1750 A < 1 < 1950 A, the ODFs give significantly different J-values from those of an average cross section model in the height-regime 100-50 km. would like to thank Dr. R. D. Hudson for helpful discussions of his work, Dr. A. C. Allison for making available his unpublished calculations, and Dr. K. Docken and Ms. E. Laviana for technical assistance. This work has been partially supported by the National Science Foundation Atmospheric Science division. Acknowledgements-We
OPACITY OF MOLECULAR
OXYGEN
425
REFERENCES ACKERMAN,M. (1971). Ultraviolet solar radiation related to mesospheric processes. Mesosphevic Models and Related Experiments (Ed. G. Fiocco), pp. 149-159. Reidel, Dordrecht, Holland. ACKERMAN,M. and BIAUME,F. (1970). Structure of the Schumann-Runge bands from the 0-O to the 13-O band. J. molec. Spectrosc. 35, 73-82. ACKERMAN,M., BIAUME,F. and KOCKARTS,G. (1970). Absorption cross sections of the Schumann-Runge bands of molecular oxygen. Planet. Space Sci. 18,1639-1651. ALLISON,A. (1972). Unpublished calculations. ALLISON,A., DALGARNO,A. and PASACHOFF,N. W. (1971). Absorption by vibrationally-excited molecular oxygen in the Schumann-Runge continuum. Planet. Space Sci. 19, 1463-1473. ARKING, A. and GROSSMAN,K. (1972). The influence of line shape and band structure on temperature in planetary atmospheres. J. atmos. Sci. 29, 931-949. AVRET~, E. H. and KROOK, M. (1963). The temperature distribution in a stellar atmosphere. Astrophys. J. 137, 874-880. BERGEMAN,T. H. and WOFSY,S. C. (1972). The fine structure of 0, (B3 C,-). Chem. Phys. Lett. 15,104-107. BRINKMAN,R. T. (1971). Photochemistry and the escape efficiency of terrestrial hydrogen. Mesospheric Models and Related Experiments (Ed. G. Fiocco), pp. 89-102. Reidel, Dordrecht, Holland. BRIX, P. and HERZBERG,G. (1954). Fine structure of the Schumann-Runge bands near the convergence limit and the dissociation of the oxygen molecule. Can. J. Phys. 32, 110-135. CARBON, D. F. (1972). A model atmosphere opacity code for the cyanogen red system. Smithsonian Report 202-131. HUDSON, R. D. (1971). Critical review of ultraviolet photoabsorption cross-sections for molecules of astrophysical and aeronomic interest. Rev. Geophys. Space Phys. 9, 305-407. HUDSON, R. D. (1973). Private communication. HUDSON,R. D., CARTER,V. L. and STEIN,J. A. (1966). An investigation of the effect on temperature on the Schumann-Runge continuum of oxygen, 158&1950 A. J. geophys. Res. 71,2295-2298. HUDSON, R. D. and MAHLE, S. H. (1972). Photodissociation rates of molecular oxygen in the mesosphere and lower thermosphere. J. geophys. Res. 77, 2902-2914. JARMAIN,W. and NICHOLLS,R. W. (1967). A theoretical study of the v” = 0,1,2 progressions of bands and adjoining photodissociation continua of the 0, Herzberg I system. Proc. phys. Sot. 90, 545-553. KOCKARTS,G. (1971). Penetration of solar radiation in the Schumann-Runge bands of molecular oxygen. Mesospheric Models and Related Experiments (Ed. G. Fiocco), p. 160. Reidel, Dordrecht, Holland. MCCONNELL,J. C. and MCELROY, M. B. (1974). Odd nitrogen in the atmosphere. J. atmos. Sci. To be published. STROM,S. E. and K~RUCZ, R. L. (1966). A statistical procedure for computing line-blanketed stellar model atmospheres. J. guant. spectrosc. Radiat. Transfer 6, 591-607. WILKINSON,P. G. and MULLIKEN,R. S. (1957). Dissociation process in oxygen above 1750 A. Astrophys. J. 125, 594-600.
Space Sci. 1974,
OPACITY
Vol. 22,
pp.413
to 425.
Pergamon
Press.
Printed
in Northern
Ireland
DISTRIBUTION FUNCTIONS AND ABSORPTION IN SCHUMANN-RUNGE BANDS OF MOLECULAR OXYGEN
T.-M. FANG, S. C. WOFSY* and A. DALGARNO Harvard College Observatory and Smithsonian Astrophysical Observatory, Cambridge, Massachusetts, U.S.A. (Received 17 September 1973) Abstract-Absorption cross wavelength over the domain spectrum has been divided have been constructed for CO%, HeO, HzOz, NOz and
sections have been calculated for molecular oxygen as a function of 1 = 1750-2050 8, for temperatures between 190 and 400°K. The into 19 wavelength intervals and opacity distribution functions each interval. Atmospheric photodissociation rates of Oz, 0,, HNOI are presented. 1. INTRODUCTION
In the wavelength region between 1750 A and 2050 A, the principal source of opacity in the atmosphere of the Earth is provided by the discrete lines of the Schumann-Runge band system (B3x;c X31;) of molecular oxygen. Because of the rapid variation of the absorption cross section, averaged cross sections lead to inaccuracies and tedious and expensive line-by-line calculations are apparently required. Kockarts (1971) has carried out such calculations at a frequency interval of 0.5 cm-l and has given a graphical presentation of the results. Using different molecular constants, Hudson and Mahle (1972) have carried out similar calculations at higher resolution and they report their results in a lengthy table which allows the calculation of transmission and photodissociation in 19 wavelength intervals. Another study has been made by Brinkmann (1971) using a bandmodel approximation but his results appear to be model dependent. In this paper, we discuss the application of opacity distribution functions to the calculation of the transmission of solar radiation in the atmosphere of the Earth. We discuss also the selection of molecular constants and the construction of the 0, cross sections in the Schumann-Runge bands and adjoining continua and we apply the results to the calculation of the photodissociation rates for a number of constituents of the mesosphere. 2. OPACITY
DISTRIBUTION
FUNCTIONS
Opacity distribution functions have broad applicability to radiative transfer problems where the absorber exhibits a sharp line spectrum. The opacity distribution function method has been applied to stellar atmospheres (Avrett and Krook, 1963; Strom and Kurucz, 1966; Carbon, 1972) and to infra-red radiative transfer in planetary atmospheres (Arking and Grossman, 1972). To calculate the transmission of light in the ith wavelength interval we must evaluate the integral
Ti=
YiflI(v) exp (- o(~)iV) dv svi
(1)
where v is the frequency in cm- l, I(v,) is the unattenuated intensity in photons cm-2 (cm-l)-l se&, cr is the absorption cross section in cm2, and N is the column number density of absorbers in cm- 2. If a(v) is a slowly varying function of v, the integral may be * National Academy of Sciences Research Associate at the Smithsonian Astrophysical Observatory. 6
413
414
T.-M. FANG,
S. C. WOFSY and A. DALGARNO
approximated by taking the interval Avi = v~+~- vi sufficiently small such that u(v) and I(v) may be approximated by their average values oi and K) in the interval. The transmission then becomes Ti= 1%) exp (--0, N)Av, (2) which can be evaluated without integration for any N. However, if a(v) varies rapidly with v, this procedure amounts to a ‘line-by-line’ calculation which can be quite tedious. In the case of the Schumann-Runge bands, the lines may be as narrow as O-2 cm-l and the bands extend for about 10,000 cm- l. Use of an average cross section requires at least 5 intervals per line, which yields the formidable sum of 2.5 x lo5 intervals (for each N). The difficulty may be avoided by introducing a normalized distribution function &(o) such that S,(u) da gives the probability that u lies between u and u + du in the interval Avi, subject to co S,(u) da = 1. (3) s0 Using S,(u) we may change the variables in (1) (Arking and Grossman, Ti = I(Y,)Av,
1972) to obtain
*msx exp {-u N}S$(u) da. s %iIl
In (4) the exponential contains no rapid function of the integration variable. However S,(O) is not a smooth function and its construction may present some numericalcomplexities. The task can be greatly simplified by the definition of a new function u(f), the opacity distribution function (ODF), such that
f= US
da’.
(5)
um is implicitly defined such that a given fraction f of the interval is occupied by cross sections less than u(J), and (1 -f> by cross-sections greater than u(f). Clearly S(a) da = df
(6)
and (4) becomes Ti = I(Y,)Av, :exp {-u(f)N} s
dJ
(7)
The opacity distribution function u(f) is a smooth function and In u(f) may be accurately reproduced by a polynomial in f,The construction of u(f) merely requires the calculation of oj({vJ) for a sufficiently large set of {vi}, and an ordering of the a) which we choose to be ascending. If n values of u are calculated, the value which ranks ith in the ordered set corresponds to f = (k- l)/(n - 1). The photoexcitation rate Ji of the absorber due to photons in the ith interval is given Ji = I(yi)
s
olexp {-Q)N}u
(f)
dJ
(8)
For the predissociated bands of 0, (8) may be used when summed over i to obtain the photodissociation rate of 0,. The construction of the ODFs may be achieved economically by the use of random selection of frequencies in each interval. Although slowly convergent, the method gives good results with relatively few points, and the ‘noise’ produced by random selection is
OPACITY
OF MOLECULAR
OXYGEN
415
filtered by the polynomial fitting off. Tests showed that about 500 points per frequency interval yield 5 per cent accuracy or better. At each point in the bands, contributions from all lines within &75 cm-l were included. Small corrections were then applied to take account of the quasi-continuum due to all lines lying between 75 and 200 cm-l from the given frequencies. Table 1 gives the fits to the ODFs, and Fig. 1 shows several ODFs and their polynomial representations. Table 1 also gives the ‘excess’ continuum-absorption discussed below. For wavenumbers less than about 50,700 cm-r (A > 1972 A) the ODF is adequately represented by a single term, although correlations between the coefficients in Table 1 obscure this fact. Average cross sections are thus adequate for representing oxygen absorption for 1> 1972 A. 3. ABSORPTION
CROSS SECTIONS
FOR 0,
Accurate molecular constants are not available for all bands of the Schumann-Runge system. Although the rotational constants are fairly well determined spectroscopically (Brix and Herzberg, 1954; Ackerman and Biaume, 1970), centrifugal distortion and spin-splitting were not completely determined by Brix and Herzberg and no values for fine structure were presented by Ackerman and Biaume. The spin-splitting is comparable to, or larger than, the width of the rotational lines and knowledge of it is required to compute line positions accurately and to interpret photographic line widths. Bergeman and Wofsy (1972) derived from the spectroscopic data a complete set of molecular constants for vibrational levels u’ = 9-19, and also obtained some information about v’ = O-3. Because of limitations in the data, some of the results are not highly accurate but the molecular constants of Bergeman and Wofsy (1972) were adopted as the best available. Where they made no determination, the rotational constants were adopted from Ackerman and Biaume (1970) and the centrifugal distortion and spin-splitting constants extrapolated from Bergeman and Wofsy (1972). Table 2 gives the molecular constants used in the present calculations. The photodissociation halt-widths that enter into the assumed Voigt lineshapes, were taken from the photoelectrically determined values given by Hudson and Mahle (1972). These line widths are different from those used by Ackerman er al. (1970), who adopted photographic widths that include appreciable contributions from fine structure and other effects. The oscillator strengths were calculated for each rotational line by Allison (1972) from the same potentials and transition dipole employed by Allison et al. (1971), which gave an excellent representation of the available measurements. The line oscillator strengths vary by as much as a factor of two as a function of rotational quantum number N for lines which are important at 300°K. The variation of the line oscillator strengths with N are shown in Table 3 for several bands. N = 45 was the largest used in the present calculations forv” =OandN=30forv” - 1 and 2. Since all fine structure components were included, the total number of lines was about 9000. The calculation of Jarmain and Nicholls (1967) was fitted and used to calculate the cross section for absorption in the Herzberg continuum of 0,, an absorption process that is crucial to the photodissociation of 0, in the stratosphere. The theoretical cross sections agree very well with the experimental values (cf. Hudson, 1971). The Schumann-Runge continuum contribution was calculated from the potentials and transition dipole of Allison, Dalgarno and Pasachoff (1971). In order to consider temperatures above 300”K, the calculation had to be extended to a” = 3. At elevated temperatures the long-wavelength edge of the Schumann-Runge continuum is shifted into
-5.2351 -52329 -5.2315 -5.2308 -5.2300
- 5.2429 -5.2418 -5.2414 -5.2414 -5.2421
-5*0836 - 5.0543 -5.0291 -50106 -4.8915
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
500009-49382~7
4938Z7-48780.5
55054+-54626.9
101 +01 +Ol +Ol +Ol
+ 01 +Ol +Ol +Ol +Ol
+Ol +Ol +Ol +Ol +Ol
i-01 -i-O1 +Ol +01 +Ol
i-01 $01 +Ol i-01 $01
+Ol f01 +Ol +Ol +Ol
is interpreted as 10+J1.
-5.2303 -5.2281 -5.2269 -5.2265 -5.2270
190 230 270 300 400
50377%-50000*0
* The notation “$01”
-5.2297 -5.2301 -5.2309 -5.2317 -5.2323
190 230 270 300 400
50715.1-50377.8
%
-5.2366 --5.2357 -5‘2353 -5.2352 -5.2350
190 230 210
51361*1-50715.1
CO
Temperature (“K)
Wavelength interval
4.2281 1.6404 6.7890 1.5378 2.5826
-3.3653 -3.2792 -3.1768 -3.1053 -3.0166
-4.5780 -5GO81 -5.4151 -5.6630 -6.2994
-5.1694 -56448 -5.9595 -6G945 -6G919
-4.7523 -3.9221 -3*2085 -2,7035 -2.1083
-7.9789 -6.5344 -5.1164 -4.1662 -3*9084
Cl
+00 +00 -01 +00 +OO
+00 $00 +00 $00 $00
$00 +OO $00 +00 +00
+00 +OO +00 +00 +00
100 +00 $00 +00 $00
-01 -01 -01 -01 -01
6.9940 4.1386 5.8739 5.3253 2.2293
3.8361 3.7386 3.6254 375467 3.4606
5.1778 5.6624 6.1199 6.4003 7.1333
5.8372 6.3962 6.7761 6.9487 7.0501
5.4895 4.5696 3.7902 3.2502 2.6579
8.6120 7.6399 6.4221 5.6427 8.8626
cz
+00 ;Of +Ol $01 +Ol
$01 +Ol $01 +Ol +Ol
+Ol +Ol +Ol +Ol +01
+Ol +Ol +Ol +Ol $01
+Ol +Ol +01 $01 +Ol
_tOO +00 +00 +00 +00
-5.8024 -2GO81 -2.7897 -2.6326 -1.3285
- 1.7092 -1.6654 -1.6160 -15819 -1.5488
-2.2976 -2.5142 -2.7182 --2.8441 -3.1717
-2.6060 -2.8664 -3*0488 -3.1361 -3.2192
-2.4904 -2.0816 -1.7382 -1.5049 -1.2605
-3~5.562 -3.3675 -29096 -2+598 -4*OlOl
C3
3.5378 34467 3.3468 3.2780 3.2208
+ 02 to2 +02 +02 +02
1.8003 4.4370 5.9283 5.6842 3.2712
4.7374 5.1870 5.6103 5.8729 6.5797
1-02 j-02 +02 +02 +02
$01 1-02 j-02 +02 $02
54085 5.9728 6.3812 6.5861 6.8337
5.2602 4-4288 3.7397 3.2809 2.8365
+02 f02 +02 i-02 +02 102 $02 +02 -to2 +02
6.8490 6.9883 6.3670 5.7554 8%09
101 101 -t-O1 $01 +Ol
C4
TABLB 1. OPA~ITYDISTRIBUTIONFUNCTIONSFORMOLECULAROXYGEN*
-f-O2 +02 i-02 +02 -l-02
102 +02 -t-O2 +02 102
$02 -t-O2 +02 1-02 -l-O2
j-02 -+ 02 4-02 “f-02 102
+02 -i-O2 to2 -+02 -1-02
101 -f-O1 $-01 3-01 -t-O1
-2-2289 -4.4894 -5.7880 -5.5949 -3.5216
-3*4102 -3.3220 -3.2280 -3.1634 -3.1193
-4.5508 -4.9856 -5.3946 -5.6500 -6.3478
-52296 - 5.8000 -6.2265 -64494 -6.7621
-5.1751 -4-3916 -3.7545 -3.3397 -2.9747
-6.2437 -6.5852 -6.1563 -5.6216 -7.8162
c5
102 j-02 $02 $02 +02
1-02 +02 $02 j-02 102
j-02 -i-O2 $02 1-02 -+02
+02 + 02 +02 102 1-02
-to2 402 1-02 -j-O2 102
+Ol $01 +Ol +Ol +Ol
9.8978 1.7206 2.1430 2.0810 1.4025
1,2369 1.2049 1.1716 1.1488 1.1368
l-6454 I.8036 1.9524 2-0459 2.3052
l-9034 2.1206 2.2883 2.3793 2.5219
l-9461 1.6680 14470 1.3064 1.1939
2.6814 2.7700 2.6056 2.3966 2.9568
C6
*r
+Ol I+-21 1-02 + 02 +02 -I-02
+02 +02 +02 102 102
102 $02 -t-O2 +02 +02
+02 102 3-02 +- 02 $02
102 +02 +02 -t-O2 $02
$01 +Ol +Ol +Ol +-01
5
-5.2351 -5.2349 -5.2360 -5.2364 -5.2314
-4.5783 -4.5321 -4.4956 -4.4726 -4.4101
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
190 230 270 300 400
53665.3-53129.3
53129.3-52565.2
52565.2-51975.1
51975.1-51361.1
-5.2437 -5.2379 -5.2336 -5.2273 -5.2058
-5.2213 -5.2040 -5.1876 -5.1787 -5.1603
-5.1616 -5~1444 -5.1174 -5.1053 -5.0910
-5.0855 -5.0599 -5.0469 -5.0394 -4.9991
190 230 270 300 400
54165.3-53665.3
-5.0662 -5a475 -50436 -5.0401 -4.9697
190 230 270 300 400
54626.9-54165.3
+ 01 $01 +Ol i 01 +01
+Ol +01 +Ol +Ol +Ol
+01 +01 t-01 +Ol t-01
+Ol +Ol CO1 +Ol +Ol
+01 +01 $01 $01 +01
$01 +01 $01 $01 +01
$01 to1 $01 $01 $01
+00 -02 +00 +00 +OO
+00 -01 t-00 +oO -01
-02
+oo
+OO
$00
+Oo
+Oo $00 +OO +00 $00
-02 -01 + 00 +00 +01
-01 -01 $00 i-00 +Ol
6.0965 $00 3.9096 +00 2.3651 100 1.5431 $00 5.0789 -02
-1.1521 -7.9130 1.8215 3.1788 4.8856
- 1.8416 -3.0855 1.6833 1.5655 1.4243
-1.3053 -2.5927 -4.6871 -4.9117 -7.3118
-2.9359 2.3613 2.3058 2.6477 9.6179
7.6764 -2.5303 2.2997 4.7781 1.0375
-6.5302 -3.9945 4.7125 8.3241 1.0420
3.5537 1.0773 1.5600 I.7648 l-8537
1.6303 5.8213 -1.3992 -2.7857 -3.8153
3.0770 1.8515 2.8527 7.1970 3.3470
3.3147 4.5371 7.2196 8.0653 5.1643
5.2544 -8.8134 -1.7065 -1.7960 -6.3188
2.6717 3.1746 1.3213 -7.2212 -5.7907
4.2230 4.1558 -3.8496 -34445 -5.5181
+Oo +01 +Ol $01 $01
+Ol +00 +Ol f01 +Ol
$01 +Ol +00 +Oo +01
$01 $01 +Ol $01 +Ol
+Ol +00 +Ol t-01 +Ol
+Ol +Ol +Ol +OO to1
+Ol +Ol +OO +Ol +Ol
-5.5819 -6.7895 -7.6252 -7.8748 -7.2157
-74033 -2.5497 6.3981 1.2573 1.6492
-1.3432 -8.9894 -3.1604 -4.8743 -1.5091
-1.5506 -1.8879 -2.9461 -3.3371 -2.4371
-2.1043 6.0831 1.2238 1.2892 2.7151
-5.9480 -7.4087 - 1.2786 5.8121 2.3140
-1.7156 -1.6330 9.6358 1.1956 1.9304
+Ol $01 101 +01 $01
+01 + 01 101 +02 $02
+02 +Ol +Ol $01 +02
+02 +02 +02 $02 +02
+02 +Ol +02 +02 +02
+Ol +Ol +Ol +Ol +02
+02 t02 +OO +02 +02
1.2707 1.3327 1.3864 1.3834 1.1931
1.6272 7.1362 - 1.0388 -2.2460 -2.9387
3.0848 2.4119 l-4044 1.6913 3.3246
3.7646 4.1834 6.0437 6.7566 5.1640
4.5051 -8.5832 -2.3893 -2.6123 -4.9899
7.2616 8.6922 - 1.6645 -1.3796 -42732
3.5206 3.2591 1.0886 -1.8102 - 3.2240
+02 +02 t02 +02 +02
$02 $01 +02 +02 +02
+02 +02 +02 +02 +02
+02 +02 $02 +02 +02
+02 +Ol +02 +02 $02
+Ol $01 + 01 +02 t-02
+02 t02 +01 +02 + 02
-1.1776 -1.1364 -1.1240 -1.0939 -9m95
-1.4837 -7.3401 8.0419 1.8657 24078
-3.1303 -2.7069 -1.9055 -2.1310 -3.2943
-4.0607 -4.3152 -5.8428 -6.4249 -4.9938
-4.4333 3.8895 1.9454 2.2128 4.1569
-4.9187 -5.2850 3.2466 1.3273 3.6761
-3.4173 -3.1200 -4.0855 1.1977 2.5708
+02 +02 t02 +02 +01
+02 $01 +Ol f-02 +02
+02 +02 t02 $02 +02
+ 02 $02 $02 + 02 $02
+02 $01 +02 $02 +02
+Ol t-01 +Ol +02 +02
+02 +02 +Ol f02 +02
4.1425 3.7565 3.5566 3.3830 2.6805
5.0407 2.7562 -2.2683 -5.7421 -7.3176
1.1735 1.0847 8.4325 9.0981 1.2086
1.6043 1.6657 2.1421 2.3189 1.8190
1.6188 4.1370 -5.5777 -6.6383 - 1.2768
1.7007 1.5926 -1.1372 -4.3422 -1.1778
1.2829 1.1659 2.7662 -2.4115 -7.5849
$01 $01 f01 401 +Ol
$01 +Ol $01 +Ol +Ol
+02 +02 +Ol $01 + 02
1-02 +02 t-02 +02 $02
+02 -01 +01 $01 +02
+01 +Ol 401 +Ol +02
f02 +02 +Ol +Ol +Ol
45-20
2.5-22
P
f01 +Ol + 01 +Ol +Ol +Ol $01 +Ol $01 + 01 +Ol $01 + 01 +Ol +Ol
-4.9574 -4.8731 -4.7823 -4.7200 -4.5619
-49637 -49086 -4.8305 -4.7696 -4.6028
-4.8977 -4.8756 -4.8321 -4.7920 -4.6557
-4.9747 -4.9389 -49060 -4.8800 -4.7664
190 230 270 300 400
190 230 270
190 230 270 300 400
190 230 270 300 400
56350.7-56097.8
56097.8-55784.9
55784.9-55444.7
55444.7-55054.0
z
+01 +Ol + 01 +Ol $01
+01 +01 t-01 +Ol $01
-4.7797 -4.7133 -4.6539 -4.6141 -4.5059
190 230 270 300 400
56541.9-56350.7
1.2336 1.2680 1.1456 9.7865 2.9198
4.2830 6.4831 6.2280 6.4682 5.9816
1.9163 2.7111 -5.9451 -6.6144 -1.3723
3.3071 2.5160 7.3604 -3.6818 -2.4236
-6.5833 -6.1774 -5.3500 -4.6018 -2.5175
2.0107 1.7931 1.4820 I.2481 6.5901
+ 01 i-01 +Ol + 01 +Ol
-4.7076 -4.6596 -4.6119 -4.5782 -4.4819
230 270 300 400
190
56715.1-56541.9
Cl
Temperature (“K)
co
Wavelength interval
+Ol +Ol +Ol +00 +00
$00 +00 +00 $00 +OO
+00 -01 -01 +01 +00
+00 +00 -01 -01 +00
+00 +00 +00 $00 +00
$01 $01 +Ol +Ol +00
-2.7108 -2.7535 -1.5938 -2.3412 3.8303
-6.9983 -2.9154 -3.3834 -4.3468 -4,7253
3.3412 4.1646 3.8559 3.2728 2.7976
6.6499 5.7338 5.3100 5.0919 4.7928
1.2459 1.0129 7.9505 6.5200 3.3487
-1.1956 -1.0476 -8.3674 -6.7935 -2.9636
c2
+Ol $01 $01 +00 $01
+00 SO1 +Ol +Ol +Ol
+Ol +Ol +Ol f01 $01
$01 +Ol $01 +Ol +Ol
+02 +02 +Ol +Ol $01
+02 f02 +Ol +Ol $01
Table 1 cond.
6.1872 54497 3.5729 -5.1208 -1.9261
5.0746 1.3584 1.5638 1.9740 2.0363
-1.7373 -2.0137 -1.8625 -1.6318 -1.3645
-3.7439 -3.2659 -2.8825 -2.6609 -2.2815
-4.6753 -36043 -2.7223 -2.1898 -1.1019
3.9423 3.3314 2.5542 1.9952 6.8210
c3
t-01 +01 t-00 $01 $02
$01 +02 t02 +02 +02
+02 +02 +02 +02 +02
+02 +02 +02 $01 +02
+02 +02 + 02 +02 $02
+02 +02 +02 +02 +Ol
+02 +02 +02 $02 $01
-8.6655 -6.8116 3.0918 1.3514 3.8601
-1.3951 -3GO15 -34058 -4.1837 -4.1773
3.9345 4.3881 4.0988 3.6701 3.0264
8.4958 7.5292 6.5990 6.0222 4.9128
+Ol +01 +Ol $02 1-02
to2 +02 +02 102 +02
$02 $02 $02 $02 +02
+02 +02 +02 $02 $02
8.2167 +02 6.1209 +02 4.5270 d-02 3.6176 $02 1~8320 +02
-6.4162 -5.2113 -3.8115 -2.8387 -6.3130
c4
66055 4.9892 -3.7803 -1.3005 -3.4329
1-5912 3.0513 3.4349 4.1225 40162
-4.0281 -4.3664 -4.1030 -3.7236 -3.0788
-8.5731 -7.6889 -6.7392 -6.1188 - 4.8426
-6.9722 -5.0826 -3.7194 -2.9731 -1.5298
4.9895 39012 2.7143 1.9119 1.4260
CS
+Ol +Ol +Ol $02 +02
$02 +02 to2 $02 +02
t-02 +02 $02 +02 +02
$02 +02 _t 02 $02 + 02
+ 02 +02 $02 +02 +02
$02 +02 $02 +02 +Ol
-1.8838 -1.4226 l-4585 4.5173 l-1391
-5.9939 -1.1081 -1.2487 - 1.4793 -1.4130
1.5654 1.6551 1.5613 1.4329 1.2020
3.2191 2.9140 2.5611 2.3219 1.8082
2.3224 1.6790 1.2304 9.9211 5.3010
-1.4598 --I+)964 -7.1742 -4.6658 7.2977
C6
8.8-21
+Ol +Ol -1-01 $01 +02
$01 +02 t02 + 02 $02
$02 $02 $02 +02 +02
1.8-12
+02 4G-21 +02 $02 $02 $02
+02 $02 t02 $01 +01
f02 2%20 $02 +Ol +Ol +00
Qex
OPACITY
OF MOLECULAR
0.2
0
0.6
0.4
OXYGEN
0.8
419
I.0
f
FIG.1. OPAC~ DISTRIEKJTIONFUNCTIONSFOR O,, T= 300°K. Curve a, 53129.3 cm-52565.2 cm-t; curve b, 56715.1-56541.9 cm-l. Curve b should be scaled by the factor 105. The dotted curves are the computed ODF’s and the solid curves are their polynomial representations. TABLET. MO~~ARCO~~A~
X”IZ,f a” ZZ
FOR
0, SCH~A~N-~~~EBA~~*
0 1 2
1.43768 I.42198 14068
4.913 -6 4.825 -6 3.6 -6
1.9848 1.9848 1.9848
-0+)0843 -O+lO843 -0*00843
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 19 203 218
0.8127 0.8001 0.7852 0*7699 0.7537 0.7372 0.7194 0.6997 0.6771 O-6531 0.6279 0.5990 0.5621 0.5244 0.4832 O-4396 0.3945 0.3471 0.2872 0.2649 omitted omitted
5.06 661 510 4.54 356 5.71 5~71 6.96 6.71 8.60 1.26 2.14 1.29 1.67 2.09 264 3.3 4.0 55 6.0
1.63 1.49 l-45 1.50 1.50 1.50 1.50 1.50 1.50 I.63 1.60 1.81 2.15 254 283 3.36 4.04 5.18 6.51 7.63
-o+xi7 -0*008 -0.009 -0*010 -0~010 -O*Oll -0*012 -0-014 -0-015 -0*022 -0.028 -0.050 -0.060 -0.094 -0.137 -0.179 -0.272 -0.349 -0.494 -0.604
B%VP =
-6 -6 -6 -6 -6 -6 -6 -6 -6 -6 -5 -5 -S -5 -5 -5 -5 -5 -5 -5
2.78 -10 3.31 -9 2.03 -8 8.63 -8 2.83 -6 7.85 -6 1.83 -6 3.72 -6 6.65 -6 1.07 -5 158 -5 2.13 -5 2.66 -5 3.06 -5 3.24 -5 3.16 -5 2.91 -5 254 -5 2.07 -5 la.58 -5
O@l 0*002 0.34 1.25 3.30 2.20 1.70 2-25 2-21 O-72 0.34 1.80 0.48 0.08 0.06 0.20 o-25 040 040 0.40
49358.2 50045.7 50710.8 51352.3 51969.8 52561.4 53122.8 536563 541563 54622.3 55050.9 55438.9 55784.5 56085.2 56339.9 56549-7 56718-l 56850.2 56951.6 57025.8
* All quantities given in cm-l except for oscillator strengthsf. The notation follows Bergeman and Wofsy (1972). t Oscillator strength for rotationless state (J = 0) from Allison (1972), for (0’. 0) band. $ AvllB denotes Lorentzian full width at half maximum, used in Voigt profile lineshape (Hudson and Mahle, 1972). 8 Brix and Herzberg (1954) assigned lines to (21,O) and (20,O) bands, but no molecular constants are available for the apparently perturbed bands.
T.-M. FANG, S. C. WOFSY and A. DALGARNO
420
TABLE 3. OSCILLATORSTRENGTHS(R-BRANCH):VARIATIONWITH ROTATIONALQUANTUM NUMBER* IN THE SCHUMANN-RLINGEBANDS
N=
1 11 21 31
(17,O)
(8,O)
(0, 0)
(17,l)
(8,l)
(0,l )
(17,2)
(872)
25-5 2.2-5 1.6-5 l.l-5t
6.7-6 6.46 5.7-6 4.8-6
2.8-10 2.7-10 2.410 2.1-10
2.1-4 1.8-4 1.44 9%-57
8.8-5 8.5-5 7.7-5 6.6-5
7.5-9 7.3-9 6.7-9 5.9-9
6.a 5.9-4 46-4 3*4-4t
5.1-4 4.9-4 4.5-4 3.94
* Allison (1972) 7 Given for N = 27, the highest N-transition in the R-branch for v’ = 17.
the region of the Schumann-Runge bands. Calculations at these temperatures required careful evaluation of the rotational dependence of the continuum cross section. For rotational levels up to iV = 50 an excellent representation of the rotational dependence is given by (T(Y,iv) = c(l&
N = 0)
(9)
where Yeft = Y - 0.561 N(N + 1);
(10)
the coefficient of N(N + 1) is an empirically determined value roughly equal to the difference in rotational constants in the B and X ground vibrational levels. The calculated cross section was used from about 1350 A to threshold. Extrapolation to N = 80 was necessary for the highest temperatures (900°K). The measured value (cf. Ackerman, 1971) o(1215.7 A) = 1-O x 1O-2ocm2 was used for the oxygen absorption cross section at Lyman a. COMPARISON OF SYNTHETIC AND
EXPERIMENTAL
CROSS SECTIONS
4.7 Schumann-Runge bands Most of the experimental data on oscillator strengths and line positions have been used in the construction of the synthetic cross sections, either directly or through the work of Bergeman and Wofsy (1972) and Allison et al. (1971). Hudson and Mahle (1972) deduced an independent set of line oscillator strengths in the same data analysis which yielded the predissociation linewidths and their oscillator strengths agree very well with those of Allison et al. (1971). Ackerman et al. (1971) obtained linewidths from photographic tracings and from a fit to the oxygen cross sections measured at various silicon emission lines. Using the distinctly different set of half-widths from Hudson and Mahle (1972), we obtained excellent agrement between the synthetic cross section and the observations of silicon emission line absorption. In a few instances small discrepancies could be ascribed to uncertainties in the molecular constants such as a displacement of the R-branch for u’ = 9 by about O-1cm-l. A displacement of O-1 cm-l is well within the uncertainty of the molecular constants. In the a’ = 0 level the difference between the linewidths of Hudson and Mahle (1972) and of Ackerman ef al. (1971) is roughly equal to the fine-structure contribution to the apparent line-with. The procedure used by Hudson and Mahle (1971), as described by Hudson (1973), utilizes the absorption in the wings of the line. Hudson’s procedure is thus relatively insensitive to uncertainties in molecular constants, although it is sensitive to the assumed Lorentzian shape of the line wings.
OPACITY
OF MOLECULAR
OXYGEN
421
4.2 Schumann-Runge continuum The work of Allison et al. (1971), which was used in constructing the present synthetic cross section, was based on experimental data for 0, absorption in the discrete bands and in the continuum for 2 < 1750 A. A significant test of the cross section is provided by a comparison with the continuum data of Hudson, Carter and Stein (1966) for 2 > 1750A. Excellent agreement is obtained between the calculated continuum absorption and the measurements at 600” and 900°K as shown in Table 4. Agreement is also obtained with room-temperature data outside the range 1750-1900 A (Hudson and Mahle, 1972; Hudson, Carter and Stein, 1966). TABLE 4. CONTINUUMCROSS
Wavelength (A)
Calculated 600” 1.99-19 1.22-19 8.05-20 6.16-20 4.90-20 1.32-20 538-21 1.78-21 4.62-22 1.40-22
1760 1770 1780 1790 1800 1820 1840 1860 1880 1900 1920 1940 * Hudson,
SECTIONSAT 600 AND 900°K(cm*) Observed * 600
900°K 4.35-19 3.32-19 2.56-19 2.09-19 1.70-19 9.65-20 5973-20 2.73-20 1.13-20 5.27-21 244-21 9.64-22
1.85-19 1.20-19 8.4-20 56-20 3.7-20 1.7-20 6.0-21 2.0-21
900°K 4.30-19 3.52-19 2.96-19 2.30-19 1.76-19 94-20 5.0-20 1.5-20 9.0-21 5.0-21 2.0-21
Carter, Stein (1966).
In the range 1750-1900 A at 300”K, the experiments show an excess continuum absorption compared to the calculations. Hudson and Mahle (1972) have ascribed this excess to absorption from the U” = 1 level. The excellent agreement of the calculations with the observations at T = 600” and 900°K, which depend strongly on absorption from v” = 1, make this unlikely. An alternative source is the 3C - 311continuum predicted by Wilkinson and Mulliken (1957). A definite determination is precluded by uncertainties in the molecular constants, the treatment of predissociation lineshapes, and the experiments themselves. Because of this disagreement with experiment, two sets of opacity distribution functions were constructed for 0,. In one set the synthetic cross section was used unchanged, and in the other the ODFs were fitted to the average transmissions given by Hudson and Mahle (1971) by the addition of a slowly varying continuum cross section. The additional cross sections are listed in Table 1. This procedure also greatly improved the agreement between the experimental and calculated continua underlying the Schumann-Runge bands at 300°K. We stress, however, that uncertainties in the molecular constants may be at fault, at least at some wavelengths. In particular, the wavelengths occupied by the (11, 0) band seem to suffer from anomalously large discrepancies. The molecular constants of the (1 I, 0) band are poorly known (Bergeman and Wofsy, 1972). 5. CALCULATION
OF ATMOSPHERIC
PHOTODISSOCIATION
RATES
The atmospheric effects of the sharp-line structure in the Schumann-Runge bands were evaluated by calculating photodissociation rates (J-values) for several constituents using the opacity distribution functions and using average cross sections. The species considered were H,O,, H,O, HNO,, N,O, CO,, 0, and O,, with cross sections taken
422
T.-M. FANG, S. C. WOFSY and A. DALGARNO TABLE
Wavenumber (cm-l) 16923 74074 71429 68966 66667 64516 62500 60606 58824 57143 55556 54054 52632 51282 5ooO0 48780 47619 46512 45454 43478 42553 41667 40816 39216 38835 38095 37383 36697 36036 35398 34783 34188 33613 33058 32520 32000 31496 31008 30534 30075 29630 29197 28777 28369
5.
PHOTODISSOCIATIONCROSS-SECTIONSIN
H&I
1.5 -18 1.2 -18 75 -19 6.2 -19 4.9 -19 4.3 -19 3.8 -19 3.4 -19 3.0 -19 2.4 -19 1.9 -19 1.6 -19 1.3 -19 1.1 -19 8.4 -20 6.5 -20 5.3 -20 3.8 -20 3.0 -20 2.3 -20 1.8 -20 1.4 -20 1.1 -20 8.1 -21 5.8 -21 3.4 -21 2.7 -21 1.9 -21 1.3 -21 6.4 -22 5.1 -22 3.8 -22 2.5 -22 1.3 -22 1.0 -22 7.3 -23
from McConnell and McElroy flux was taken from Ackerman
0,
8.519 -19 8.519 -19 8.519 -19 6.667 -19 5.185 -19 4.074 -19 2.963 -19 3.333 -19 5.926 -19 8.148 -19 1.700 -18 2.9 -18 4.4 -18 6.2 -18 8.1 -18 9.9 -18 1.1 -17 1.1 -17 1.1 -17 1.0 -17 ::4” 4.6 2.7 1.6 1.0 4.7 2.6 1.4 6.8 3.6 1.8 7.1 ;:; 6.9 4.0 2.5
Z:; -18 -18 -18 -18 -19 -19 -19 -20 -20 -20 -21 -21 -21 -22 -22 -22
KG 7.0 3.2 7.0 5.0 1.0 2.2 3.5 4.8 4.3 2.8 7.5 5.0 5.2 6.3 6.3
-18 -18 -19 -19 -18 -18 -18 -18 -18 -18 -19 -20 -21 -22 -23
cm’
HNO,
1.32 -17 9.10 -18 5.50 -18 2.55 -18 9.70 -19 3.28 -19 144 -19 8.51 -20 5.63 -20 3.74 -20 260 -20 2.10 -20 1.95 -20 1.94 -20 1.92 -20 1.85 -20 1.70 -20 1.50 -20 1.27 -20 9.95 -21 7.0 -21 5.0 -21 3.1 -21 2.1 -21 1.3 -21 7.0 -22 3.0 -22 1.0 -23
NaO 8.9 -17 3.7 -18 5.0 -19 4.0 -18 1-O -18 1.0 -19 3.0 -20 4.0 -20 7.0 -20 9.0 -20 1.1 -19 1.1 -19 1.0 -19 6.0 -20 3.3 -20 1.5 -20 7.0 -21 3.0 -21 1.2 -21 4.1 -22 1.1 -22 3.2 -23 1.6 -23 9.6 -24 6.1 -24 4.6 -24 4.3 -24 4.2 -24 4.9 -24 5.3 -24 5.3 -24 5.0 -24 4.6 -24 4.5 -24 4.5 -24 4.1 -24 3.3 -24 2.0 -24
co, 6.7 -19 7.8 -18 6.1 -19 5.9 -19 5.4 -19 3.6 -19 1.8 -19 7.3 -20 2.2 -20 6.0 -21 1.4 -21 2.8 -22 7.0 -23 1.9 -23 2.5 -24 8.0 -25
(1973) and references cited therein (see Table 5). The solar (1971) and the atmosphere from U.S. Standard Atmosphere
Supplements (1966) for 30”N lat (July). Calculations were carried out for zenith angles a such that cos ct = 1.0, O-5, and O-25. Ozone densities were taken from Hilsenrath (1972) and from Hering and Borden (1966). Photodissociation by Ly a was included because of the 0, window at 1215.7 A. The results for H,O, COz and 0, are given in Fig. 2. The results for H,O are in fairly
423
% PHOTODISSOCIATION
RATE (se&)
424
T.-M. FANG,
S. C. WOFSY
and A. DALGARNO
good agreement with the calculation of Kockarts (1971) for cos a = 1.0. The agreement is better where the excess absorption is taken into account. Detailed comparison was made for cos a = O-5 between J-values calculated with the opacity distribution functions and those calculated with cross sections averaged over 50 A segments. The differences were inconsequential for constituents with substantial photodissociation cross-sections for 3L> 1950A such as H,O,, HNO,, N,O and 0,. Photodissociation of these species is adequately described by a model in which shielding is computed using 50 A average cross sections for 0, and 0,. Moreover, for altitudes less than 50 km light with 1 < 2000A is very substantially shielded. Hence at these altitudes photodissociation rates for OS, CO, and H,O may be calculated using 50 A average cross sections for 0, and 0,. The J-values for CO, and H,O are negligibly small in this height regime, and absorption by 0, is dominated by the Herzberg continuum. Below 50 km the J-values for all constituents except CO, and H,O are very sensitive to the 0, distribution. Typical J-values for H,O,, O,, HNO, and N,O at cos a = 0.5 are presented in Table 6. TABLE 6.* PHOTODISSOCIATION RATES (see-l)t
Altitude (km) 120 70 65 60 55 50 45
HNO,
Hz01 1.1 1.1 1.1 1.0 9.5 8.1 5.2
-4 -4 -4 -4 -5 -5 -5
1.5 1.4 1.4 1.3 1.2 1.1 8.3
-4 -4 -4 -4 -4 -4 -5
N,O 3.7 9.1 8.4 7.8 7.1 6,3 5.0
-6 -7 -7 -7 -7 -7 -7
0, 9.2 9.2 9.1 8.8 8.1 6.5 3.4
-3 -3 -3 -3 -3 -3 -3
* Signed number is power of 10. t Cos a = 0.5.
For altitudes between 100 and 50 km the J-values for 0,, H,O, and CO, are not well represented by a model using average cross sections for 0,. For 0, the discrepancy is no more than 20 per cent, but for CO, and HZ0 the average-cross section model gives J-values too small by more than a factor of 2. When the ODFs are adjusted for the ‘excess absorption’ discussed above, significant modifications occur for CO,, H,O, and 0, in the height range 100-50 km. The results are shown in Fig. 2 for CO, and H,O at cos a = 0.5. The magnitude of the effect may be regarded as the minimum estimate of the uncertainty in atmospheric photodissociation rates due to uncertainties in the 0, absorption spectrum. CONCLUSIONS The
applicability
of opacity
distribution
functions
to atmospheric
absorption
by 0,
has been demonstrated, and our current best estimates of the ODFs have been presented. For constituents with substantial photodissociation for wavelengths 1750 A < 1 < 1950 A, the ODFs give significantly different J-values from those of an average cross section model in the height-regime 100-50 km. would like to thank Dr. R. D. Hudson for helpful discussions of his work, Dr. A. C. Allison for making available his unpublished calculations, and Dr. K. Docken and Ms. E. Laviana for technical assistance. This work has been partially supported by the National Science Foundation Atmospheric Science division. Acknowledgements-We
OPACITY OF MOLECULAR
OXYGEN
425
REFERENCES ACKERMAN,M. (1971). Ultraviolet solar radiation related to mesospheric processes. Mesosphevic Models and Related Experiments (Ed. G. Fiocco), pp. 149-159. Reidel, Dordrecht, Holland. ACKERMAN,M. and BIAUME,F. (1970). Structure of the Schumann-Runge bands from the 0-O to the 13-O band. J. molec. Spectrosc. 35, 73-82. ACKERMAN,M., BIAUME,F. and KOCKARTS,G. (1970). Absorption cross sections of the Schumann-Runge bands of molecular oxygen. Planet. Space Sci. 18,1639-1651. ALLISON,A. (1972). Unpublished calculations. ALLISON,A., DALGARNO,A. and PASACHOFF,N. W. (1971). Absorption by vibrationally-excited molecular oxygen in the Schumann-Runge continuum. Planet. Space Sci. 19, 1463-1473. ARKING, A. and GROSSMAN,K. (1972). The influence of line shape and band structure on temperature in planetary atmospheres. J. atmos. Sci. 29, 931-949. AVRET~, E. H. and KROOK, M. (1963). The temperature distribution in a stellar atmosphere. Astrophys. J. 137, 874-880. BERGEMAN,T. H. and WOFSY,S. C. (1972). The fine structure of 0, (B3 C,-). Chem. Phys. Lett. 15,104-107. BRINKMAN,R. T. (1971). Photochemistry and the escape efficiency of terrestrial hydrogen. Mesospheric Models and Related Experiments (Ed. G. Fiocco), pp. 89-102. Reidel, Dordrecht, Holland. BRIX, P. and HERZBERG,G. (1954). Fine structure of the Schumann-Runge bands near the convergence limit and the dissociation of the oxygen molecule. Can. J. Phys. 32, 110-135. CARBON, D. F. (1972). A model atmosphere opacity code for the cyanogen red system. Smithsonian Report 202-131. HUDSON, R. D. (1971). Critical review of ultraviolet photoabsorption cross-sections for molecules of astrophysical and aeronomic interest. Rev. Geophys. Space Phys. 9, 305-407. HUDSON, R. D. (1973). Private communication. HUDSON,R. D., CARTER,V. L. and STEIN,J. A. (1966). An investigation of the effect on temperature on the Schumann-Runge continuum of oxygen, 158&1950 A. J. geophys. Res. 71,2295-2298. HUDSON, R. D. and MAHLE, S. H. (1972). Photodissociation rates of molecular oxygen in the mesosphere and lower thermosphere. J. geophys. Res. 77, 2902-2914. JARMAIN,W. and NICHOLLS,R. W. (1967). A theoretical study of the v” = 0,1,2 progressions of bands and adjoining photodissociation continua of the 0, Herzberg I system. Proc. phys. Sot. 90, 545-553. KOCKARTS,G. (1971). Penetration of solar radiation in the Schumann-Runge bands of molecular oxygen. Mesospheric Models and Related Experiments (Ed. G. Fiocco), p. 160. Reidel, Dordrecht, Holland. MCCONNELL,J. C. and MCELROY, M. B. (1974). Odd nitrogen in the atmosphere. J. atmos. Sci. To be published. STROM,S. E. and K~RUCZ, R. L. (1966). A statistical procedure for computing line-blanketed stellar model atmospheres. J. guant. spectrosc. Radiat. Transfer 6, 591-607. WILKINSON,P. G. and MULLIKEN,R. S. (1957). Dissociation process in oxygen above 1750 A. Astrophys. J. 125, 594-600.