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Astron. Astrophys. 329, 792-798 (1998)

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4. Extrapolation method for the contribution of the first three angular momenta to the total profile

Feautrier et al. (1976) and Feautrier and Tran Minh (1977) used two methods to determine the contribution of the first angular momenta ([FORMULA], 1 and 2), excluded by the exact resonance approximation. The first, based on the program of Seaton and Wilson (1972), consisted of solving the coupled integro-differential equation; and the second leaned on a semi-empirical approach. A rapid and accurate method is still needed for the calculation of this contribution. The present extrapolation procedure consists of continuing the curve of the contribution of the angular momenta to the total profile ([FORMULA]) such as to find the contributions relative to [FORMULA], 1 and 2 with the minimum error. A good choice for this extrapolation is obtained by completing the original curve by a straight line connecting the point [FORMULA] to the point [FORMULA] (Fig.1). The contribution of the first three angular momenta is thus found to be:

[FIGURE] Fig. 1. Extrapolation of the contribution of the first three angular momenta to the total profile for [FORMULA] Å . Data from the exact quantum calculations are presented in Feautrier et al. (1976)

[EQUATION]

while the sum of errors due to extrapolation of the three points is about [FORMULA]. On the other hand, since there is a balancing of the [FORMULA] 's in the sum around the extrapolation line, the actual error is about [FORMULA]. Hence, the total profile, normalized to [FORMULA] in the dipole approximation and taking into account each of the [FORMULA] 's contribution is:

[EQUATION]

Numerically, the error in the so-determined total profile is found to vary from about less than [FORMULA] in the near line wings to about less than [FORMULA] the far wings. Further, it decreases with increasing temperature. The present extrapolation procedure, which has been checked to provide results in good agreement with the semi-empirical method of Feautrier and Tran Minh (1977) is advantageous for its rapidity and its simplicity.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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