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J. Quant. Spectrosc. Radiat. Transfer Vol. 39, No. 5, pp. 399-407, 1988 Printed in Great Britain. All rightsreserved

UNIVERSAL FUNCTIONS FOR ESTIMATING TOTAL VIBRATION-ROTATION BAND ABSORPTANCE-III. ASYMMETRIC TOPS A. G. ISHOV and G. M. SHVED Atmospheric Physics Department, Institute of Physics, Leningrad University, Leningrad-Petrodvorets 198904,U.S.S.R. (Received

18 June 1987)

Abstract-The method of universal functions (MUF) for estimating total band absorptances and their derivatives, for nonoverlapping lines of vibration-rotation bands of linear molecules and spherical and symmetric tops, ‘**is extended to asymmetric top bands. The universal functions for symmetric tops are employed. The accuracy of MUF is within a few percent for the 010-000 and 001-000 bands of H,O and 0, and Lorentz line shapes at temperatures of 175-325 K. A simple technique for obtaining an effective Lorentz width for all of the band lines is proposed for application to MUF.

1. INTRODUCTION In Parts I and 11,‘~~we proposed an approximate method for estimating total band absorptances and their derivatives for the vibration-rotation bands of linear molecules and spherical and symmetric tops with nonoverlapping lines and a Boltzmann population of rotational states. The method involves the use of universal functions to describe absorption in band branches for every type of molecule. In this paper, the MUF is extended to asymmetric tops. The universal functions for symmetric tops are employed for asymmetric tops. The method is tested for the Lorentz shape on the 010-000 and 001-000 bands of H,O and 03. These bands are important for radiation transfer in the atmospheres of the Earth, Venus and Mars. Two versions of MUF are proposed. In the version described in Sec. 2, the line-intensity distribution for the asymmetric top band is modelled by the distribution for some symmetric tops.

In this version of MUF, it is impossible to eliminate the fitting of parameters for the total band-absorptance model. However, the version may be called physical since an attempt has been made to model for each band the peculiar features of radiative transfer which are determined by the band line-intensity distribution. Section 3 deals with another version of MUF, which we call fittable: the universal functions for the symmetric top are used as analytical expressions for the total band absorptance, and all of the model parameters are obtained by fitting to the exact expression for the total band absorptance. In Sec. 4, a technique is proposed for determining an effective Lorentz line width in estimating total band absorptances by using MUF for any type of molecule. 2. THE

PHYSICAL

MUF

In Parts I and II, we introduced convenient dimensionless values Mbk (k = 0, 1,2, . . .) that were proportional to the equivalent bandwidth and its derivatives. The exact expression for Mbk is

where si is the ith line intensity, ti the optical path for the line, sb =

1 i

399

si

400

A. G. ISHOV and G. M. SHVED

the band intensity, rb the optical path for the band, m (1 - exp[-a(x)r,]) s -cc 00 ak(x)exp[-a(x)ri]dx Mk(ri) = A s -m M,(Ti) = A

dx, (k= 1,2,...),

(2)

A is a constant, a (x) the dimensionless line profile, and the summation extends over all of the lines.

The rotational energy of asymmetric tops cannot be described by simple analytic terms. There are therefore for asymmetric tops no specific universal functions with which to describe the band absorption. Rotational energy levels for an asymmetric top can be found, in the first approximation, by interpolating betweeen the prolate and oblate symmetric tops. For this reason, we used the universal functions of symmetric tops in estimating the total band absorptances for asymmetric tops. A description of radiation transfer in asymmetric top bands using symmetric top models was first proposed by Penner.3 For molecules with a slight asymmetry, e.g. 03, the line-intensity distribution of the band is sometimes derived by employing modelled symmetric tops.4 Estimates of the equivalent H,O bandwidth for a smeared-out rotation line structure have been obtained also by using a linear-molecule model5 For asymmetric tops, we approximate Mbk(~b) of Eq. (1) by the following formulae used for symmetric tops:*

where the tik(f; b; i) are the universal functions for symmetric tops given in Eq. (11.29); C, is the abundance ratio of &, in the band branch AJ(AJ = - 1, 0 and 1 for the P-, Q- and R-branches, respectively), and .,i_,

cw=1;

(4)

C AJ.m is the maximum C, value for a given band; b and [ are the parameters of modelled symmetric

tops, b = &/wT,

(5)

(prolate top) (oblate top),

(6)

K is Boltzmann’s constant, h Planck’s constant, T the gas kinetic temperature; A,*, B,* and C: are the rotational constants of a symmetric top model which is constructed from the rotational constants A,, B, and C, of an asymmetric top. The value rb for the asymmetric top bands is also written in the form accepted for symmetric tops, viz.,* rb = SbCAJ.mW,f,U/6N> where u is the absorbing gas mass for the ray path, describing the line profile, co s

(7)

f,the maximum value of the function f (v)

_mfWdv = 1,

v the frequency; w,,, is the maximum value of the function given in Eq. (11.19) and characterizing the line-intensity distribution in the band branches for a symmetric-top model; 6 = 1 if the band is modelled by a parallel band, and 6 = 2 if it is modelled by a perpendicular band; N is the number of band-structure repetitions which, for the asymmetric top, is a fitting parameter. For asymmetric tops, ~~ of Eq. (7) is a convenient dimensionless variable, rather than a value of the order of magnitude of the optical path for the most intense line in the band, as is the case for the symmetric top.

Estimating total vibration-rotation

band absorptance-III

401

We verify MUF by using as an example the function Mb, for the Lorentz shape and fundamental v2- and v,-bands of Hz0 and 03. The selection rules for the v,-band are type B, whereas those for the v,-band are type A. The HZ0 molecule is strongly asymmetric (A, = 27.88 cm-‘, B, = 14.52 cm-‘, C, = 9.278 cm-‘),6 while the O3 molecule is slightly asymmetric (A, = 3.553 cm-‘, B,= 0.4453 cm-‘, C, = 0.3948 ~rn-‘).~ The search for an optimal MUF is performed by (a) varying the rotational constants for a symmetric-top model (Tables 1 and 2); (b) testing the models of parallel and perpendicular bands (C_, = C, = i and C, = 0 for the parallel band and C_, = Cl = i and C,, = 4 for the perpendicular band); (c) varying N (N = 1 and 2). We test MUF for the temperature range 175-325 K corresponding to that of the Earth’s atmosphere. The approximate estimates of Mb, in Eq. (3) are compared with exact calculations of Mb1 from Eq. (1). For the Lorentz shape, zi = s,u/lry,, in Eq. (1) where yLi is the Lorentz halfwidth of the ith line. For exact calculations of Mbl, we take sets of si and yLi from the 1980 AFGL line compilation.* For approximate estimates of Mb’, we use a unique halfwidth yt for all lines of the band to define the value offm in Eq. (7),f, = l/~yf. The technique for calculating 7: is presented in Sec. 4. Examples of the MUF tests are given in Tables 1 and 2 and in Fig. 1. Verification of MUF leads to the following results for the H,O and 0, fundamental bands: (1) The use of A :, B: and C: in the form of a and m or (A, + B,)/2 and (B,+ C,)/2 was recommended previously3.’ for modelling the band absorption of asymmetric tops, but does not guarantee a minimum error in total band-absorptance estimates. When using MUF, minimum relative errors can be obtained by employing other simple ways of constructing A ,*, B,*and C,* from A,, B,and C, (see Tables 1 and 2). (2) The errors with MUF are a few percent over the entire z,-axis at temperatures of 175-325 K for some symmetric-top models (Tables 1 and 2). This accuracy is achieved for: (a) the H,O v,-band for the model of the perpendicular band at N = 1 and 2, (b) for the H,O v,-band in the case of models for the perpendicular and parallel bands at N = 1 and 2, respectively, and (c) for the O3 v,-band with models of the perpendicular and parallel bands at N = 1. (3) Besides choosing rotational constants for a symmetric-top model, it is also possible to minimize the MUF errors by approximating Mbk by

where cl + c, = 1, and the fik functions are taken for different symmetric-top models which would yield Mbk with errors differing in the sign when Eq. (3) is used (Fig. 1).

0.1 -

-

+-band

Fig. 1. Relative errors in estimates for the function Mb, in the physical MUF for the H,O fundamental bands and the Lore& line shape. The model of the perpendicular band and N = 1 have been employed. The estimates according to Eq. (3) are obtained for the v,-band and A? = A,, B: = B, (upper curve) and A: = B,, E: = C, (lower curve); also for the v,-band with A: = m, B: = m (upper curve) and A: = A,, B: = E, (lower curve); the q, values are calculated from Eq. (7) for the corresponding top models.

The estimates according to IQ. (8) are obtained for the same symmetric-top models and c, = c2 = 5; TV is taken for the top model, which overestimates

Mb,.

--

Ar

BF

\IdrB,

__

& 0,

(B,+c,)/2

Ar

(Ar+Br)/2

(B&)/2 Br

jx

%

Ar

Ar

BZ

A;

_

.

0.57

0.73

0.78

0.92

1.34

1.40

2.01

SO

-. , .

0.13

2.00

3.76

6.98

7.46

7.48

8.00

03

-

2;

;g

;;

--

175 325

175 325

g

175 325

.

-

0.100 0.064

4.060 4.109

4.097 4.124

4.198 4.221

._ -

~.

0.061

_ -__

0.077 0.230

.,,

4.133 4.154

4.148 4.169

‘4.244 4.261

4.189. 4.209

4.202

4.182

4.117 4.141

Nd

bend bend

0.253

0.108 0.090

0.091 0.074

4.087 4.108

0.049 4.045

0.091

4.140 4.167

0.056

.

-

&I 0.126 0.105

culer

-J*;;;

4*064 4.0%

l&l

HZ1

_ _ _

0.372 0.387

0.730 0.143

0.043 0.054

4.145 -0.121

4.130 4.106

4.106

4.130

4.115 4.090

bend

_.._.”

Table I. Maximum values of the relative errors in estimates for the function Mb, in the physical MUF m. (3)] for the H,O and 0, fundamental bands and the Lore& line shape, using a prolate symmetric top model and C_ , = C, = j, C, = 0 for a parallel hand or C_ , = C, = f , C, = f for a perpendicular band. The maximum relative error varies monotonically between 175 and 325 K. I Method of constructing the rotaParameter 03 =2O tionel con&ante A: aad Bg of a s= 3 _ , v2-band V3-band V3-bend symmetric top model from the F,K ’ BE Model of a perpen- Model of a Yodel of a Model of a rotational oonetents Ar, Br end diculer bend perpendiperellel parallel C, of an esgmmetric top

-

-_.--_-_

_

of

constructing

the

rota-

im

Br

(Ar+Br)/2

__

%

JBrC,

Or+92

-

.

-0.36

-. , .

-0.11

-0.67

-0.79

-0.44

-0.42

-0.87

Br

Ar

-0.80

-0.69

-0.89

03

-0.48

-0.56

5

(A,+B,)/2

-0.67 -0.54

--

%

A*

%O

IParameter

Or

JE

G:.

B:.

rotational constents A*, B, and C, of an asymmetric top

tional constants Bi and Cz of a symmetric top model from the

Method

-

1

.

-

-0.067

--

-0.041

325

-0.203

325 175

-0.1!?4

175

-0.20% -0.222

175 325

-0.308 -0.321

175 325

-0.176

-0.194

175 P5

0.300 0.257

175

-0.279

325

-0.266

175

h&l 325

I

-

I_ -

0.336

0.375

0.150

0.169

0.126

0.154

-0.040

0.033

0.161

0.189

0.183

0.212

0.055

0.075

N=2

-

-__

J&O

.

0.133

0.148

-0.087

-0.071

-0.108

-0.093

-0.222

-0.212

-0.077

-0.062

-0.057

-0.041

-0.174

-0.164

&I

~.

&II

0,

1.301

1.263

0.573

0.550

0.323

0.301

0.058

0.037

0.332

0.311

0.586

0.563

0.070

0.049

_ _ _

I

_ .

Table 2. Maximum values of the relative errors in estimates forthefunction Mb, inthephysical MUF [Eq. (3)] for the H,O and 0, fundamental bands and the Lorentz line shape, using an ablate symmetric top model and C_, = C, = f, C,, = 0 for a parallel band or Cm, = C, = a, C,, = f for a perpendicular band. The maximum relative error varies monotonicallv between 175 and 325 K.

_ .”

I-X-.-“---~l_._.

__,

A. G.

404

ISHOV

3. THE

and G. M.

FITTABLE

SHVED

MUF

The present state of the physical MUF is such that a rotational band analysis is required to test the accuracy. It is therefore reasonable to test the simplest MUF version which employs a known set of si for a band. In this version, it is assumed that A&=&?&),

t;=cu,

(9)

where 7; is a convenient dimensionless variable, and the coefficient c is fitted to minimize errors in the estimates of Mbk over the entire t;-axis. The approximation of the Mb0 and Mb, functions by Eq. (9) yields accurate values at u = 0 (A&, = Mb, = 1). Therefore, it is reasonable to define for these functions the coefficient c in such a way that Eq. (9) yields the precise values given by Eq. (1) in the limit u + co. This definition of c is possible if the first terms of the asymptotic series for Mk (t) and tik(t) are equally dependent on t. This is the case for the Lorentz shape. 2,3,9For the Lorentz shape and any k,

where I (b, {) and D,(b, [) are defined by Eqs. (11.28) and (11.42), respectively. Testing of the fittable MUF is performed for the same samples as in Sec. 3 (Tables 3 and 4 and Fig. 2). The fittable MUF provides satisfactory accuracy for any method of constructing the rotational constants of a symmetric-top model.

4.

EFFECTIVE

WIDTH

FOR

THE

LORENTZ

SHAPE

For molecules of any type, MUF requires an effective line width for all lines of a band. The value of yLi for different lines may differ considerably. Therefore, the problem arises of defining an effective halfwidth 7: to minimize MUF errors over the entire u-axis. We define rt in the limit u + cc by equating Mbk of Eq. (l), with real values of yLi, with Mbk of Eq. (l), but using yf for all lines of the band. We then derive the equation

(11)

Fig. 2. Relative errors in estimates for the function Mb, by the fittable MUF Eqs. (9) and (IO) for the fundamental vl- and v,-bands of H,O and 0, and the Lore& line shape. A model of the prolate symmetric top with A: = A, and B: = C, is employed.

--

.

of

constructing

the

rota-

Br

%

r-Brcr

\IArBr

%

*r

@,+C,M

Or+C,)/2

Ar

(Ar+Brv2

J Br%

Ar

__

%

Ar

_.

B:

A:

Cr of an asymmetric top

-

symmetric top model f!rcun the rotational constants Ar, Br and

tional constants Ai and Bz of a

Ivlethod

0.068 0.059

175 325

.,.-

0.071 0.061

775 325

_

175 325

175 325 0.071 0.061

0.071 0.062

0.064 0.057

0.061

175 325

0.071

175

0.072 0.062

.

gQ-band

325

175 325

P,K

1 %!O

-

.

. _

-0.035 -0.091

0.081 0.084

-0.036 -0.090

-0.036 -0.092

-0.036 -0.092

0.061 O.C44

0.059 0.041

0.057 0.039

0.055 0.037

0.054 0.036

0.036

0.054

4.035 -0.092

0.054 0.036

VS-band

-0.036 -0.092

-0.035 -0.091

_

O3 V2-bsnd

0.080 0.084

0,080 0.084

0.078 0.082

O.&O 0.083

0.084

0.080

0.081 0.084

$-band

Table 3. Maximum values of the relative errors in estimates for the function Mb, in the fittable MUF [Eqs. (9) and (IO)] for the H,O and 0, fundamental bands and the Lore& line shape, usinga prolate symmetric topmodel.The maximum relative errorvaries monotonically between175and 325K.

J BrCr

%

Br

(Br+CrI/2

Br

%

%

%

C;

J-ArBr

(Ar++Br)/2

Ar

(Ar+Br)/2

J AYcBr

Ar

BI

Cr of an asymmetric top

smtric top model from the rotational constants A,, Br and

Method of constructing the rotational constants Bz and Ci of a

0.085

0.081 0.061

0.069

325

175

175

0.079 0.082

325

0.057

0.062 0.065

0.073 0.078

0.071

175

325

0.078

0.071

0.056

0.063

175 325

0.078

0.073

0.056

0.064

325

175

0.074 0.079

0.057

0.065

175 325

0.077

0.070

0.055

0.062

Ys-band

325

175

Y-band

*2O

I

0.057 0.040

-0.036 -0.092 -0.036 -0.090

0.061 0.044

0.054 0.036

-0.037 -0.093

0.032

0.050 -0.094

-0.038

0.054

0.039

0.057

0.031

0.049

0.036

I

ys-band

-0.093

-0.037

-0.092

-0.036

-0.094

-0.038

y2-band

O3

Table4.Maximum values oftherelative errors in estimates for the function Mb, in the fittable MUF [Eqs. (9) and (IO)] for the H,O and 0, fundamental bands and the Lorentz line shape, using an oblate symmetric top model. The maximum relative error varies monotonically between 175 and 325 K.

Estimating total vibration-rotation

band absorptan-III

407

o-0.01

-

-0.02

-

-0.03

-

-0.04

-

-0.05

n -1

0

1

2

3

4

5

6

Fig. 3. Relative errors in the approximate estimates of M,,, from Eq. (I) for the fundamental v,-band of H,O. The effective Lorentz halfwidth yf is employed for all lines of the band. The value of yt is estimated from Eq. (12).

Use in Eq. (11) of the first term of the asymptotic series of &(T)

yields, for any k, the halfwidth

Fork=Oand l,Eq.(ll)holdsforanyvalueofy,*atu = 0. Therefore, we assume that the use of yt from Eq. (12) is reasonable in estimating MM and Mb, . The example presented in Fig. 3 illustrates the utility of this recommendation. Acknowledgement-The

authors express their gratitude to E. L. Valikhan for translating the manuscript into English.

REFERENCES 1. G. M. Shved, A. G. Ishov, and A. A. Kutepov, JQSRT 31, 35 (1984). 2. G. M. Shved and A. G. Ishov, JQSRT 31,47 (1984). 3. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, MA (1959). 4. H. Yamamoto, J. Geomagn. Geoelectr. 29, 153 (1977). 5. S. S. Penner and D. B. Olfe, Radiation and Reentry, Academic Press, New York, NY (1968). 6. E. Kyrii, J. Molec. Spectrosc. 88, 167 (1981). 7. G. Her&erg, Molecular Spectra and Molecular Structure-III. Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, New York, NY (1966). 8. L. S. Rothman, Appl. Opt. 20, 791 (1981). 9. V. V. Ivanov, “Transfer of Radiation in Spectral Lines” (translation edited by D. G. Hummer), National Bureau of Standards Special Publication 385, U.S. Government Printing Office, Washington, DC (1973).