A method for estimating H2O absorption

NOTES AND DISCUSSIONS The Editors of the Journal of Quanritative Spectroscopy and Radiative Transfer welcome the submittal of brief, original contributions on timely topics. These manuscripts will generally be published. promptly and without Editorial review if they are submitted through one of the Associate Editors of JQSRT

_I. Quant.

Spectrosc.Radiat.

Transfer.

Vol. 5. pp. 867-868.

Pergamon Press Ltd., 1965. Printed in Great Britain

NOTE A METHOD

FOR ESTIMATING

Hz0 ABSORPTION

M. S. VARDYA* Joint Institute for Laboratory

Astrophysics,

Boulder, Colorado,

U.S.A.

(Receioed 12 July 1965) Abstract-A

method is given for estimating the absorption

due to the vibration-rotation

bands of HzO.

THE EXPRESSIONS for the energy levels of an asymmetric top molecule, like H,O, are rather complex. Hence an approximate method is needed to include the effect of H,O absorption in calculating mean opacities suitable for late-type stars. We will consider here only the vibration-rotation bands of H,O. Following PENNER(~), we can write the mean opacity over a band, after approximating H,O to a nearly symmetric top molecule, as : i&/

= --

27r3 f-00 v’ _.-L..-

3!1c(BC)l’~

K-IaOI&.v’12~’

[(y,““erfs]

[exp (-:)I

(1)

Here U’ denotes the higher vibrational state, w~,~, the wave number at the band center, N H o the number of Hz0 molecules/cm3 in the ground state, h the Planck constant, c tde velocity of light, u’ = hcl w- ~~.~,l/kT wh ere k is the Boltzmann constant, T the temperature, and )w - w. ,vf1the half of the band width with center at w~,~,, j3 = A/(BC)“2 and y = /~(Bc)l’~/kT, where A, B, C are the three rotational constants for the ground is the matrix element. As state, and Ro,us I&I2

= gmr2;

(2)

fi., Ul

is the oscillator strength, and e and 111the charge and mass of an electron, where fi,,, respectively, we can write equation (1) as: ico,vf=

re2%o fo,vtu' [exp

($)I

[(y)“”

erfg]

4rn~~(BC)l’~ *Present address:

Sterrewacht

“Sonnenborgh”,

Servass Bolwerk 13, Utrecht, 867

The Netherlands.

868

M. S.

VARDYA

AS wo,vP and fo,,, are available (see e.g. ref. l), the main problem that remains in the evaluation of Ko,vPis the evaluation of the band width or 1w - OJ~,~~~. The expression for the rotational energy levels for a nearly symmetric top molecule for a given vibrational state depends not only on J, the rotational quantum number but on 7 or K also, which defines the sublevels of the J-level. However, one can write, following Mecke (2), the average energy E(J), of all levels for an assymmetric top molecule with a certain J as: E(J)/hc

= 2 E(J,)/hc(2J+

1) = 4(A + B+ C)J(J+

1)

(4)

This expression is similar to the expression for the rotational energy of a diatomic molecule with B replaced by (A + B + C)/3 E Beff, say. Then following the method detailed by PENNER(~) for computing the band width for a diatomic molecule, one obtains: 1w - w~,~.I = Awo,,,/2

21 (35.37 Be,fkT/hc)1’2.

(5)

Here Aco~,~,is the band width. This completes the estimate of Ko,vP. Acknowledgement-This work was done during the tenure of a Visiting Fellowship for Laboratory Astrophysics,which is gratefully acknowledged.

of the Joint Institute

REFERENCES 1. S. S. PENNER, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, Massachusetts (1959). 2. H. HERZBERG, Molecular Spectra and Molecular Structure II, Infrared and Raman Spectra of Polyatomic Molecules, p. 49. Van Nostrand, New Jersey (1945).