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

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6. Results and discussion

This section is concerned with the comparison of our calculations with the data of VCS on one hand, and the presentation of our results and their parametrization at various temperatures on the other hand. The data of VCS were compiled with a formalism based on the unified classical-path theory for electrons and the quasistatic approximation for ions (Vidal et al. 1971). These data have been the most reliable in the far wings for a long time although the electron description is not sufficient in the far wings.

6.1. Comparison with the data of Vidal et al.

In Figs. 3 to 6, line profiles normalized to the asymptotic Holtsmark profiles, provided by both VCS and our formalism are reported. The ionic contribution is assumed to be quasistatic and thus is represented by unity. The Holtsmark-normalized profile is [FORMULA]. There is no need here to include the short range effects in our calculations since they are not taken into account by VCS.

[FIGURE] Fig. 3. Comparison of our results (full line) with those of Vidal et al. (VCS : dotted line) at [FORMULA] cm-3 and [FORMULA] K. The total Holtsmark-normalized profile is plotted versus the detunings [FORMULA].

[FIGURE] Fig. 4. Same as Fig. 3 for [FORMULA] cm-3 and [FORMULA] K.

[FIGURE] Fig. 5. Same as Fig. 3 for [FORMULA] cm-3 and [FORMULA] K.

[FIGURE] Fig. 6. Same as Fig. 3 for [FORMULA] cm-3 and [FORMULA] K.

One observes discrepancies in the far wings between our profiles and those of VCS. These discrepancies, which were already found between the semi-classical and the quantum theories (Tran Minh et al. 1975; Feautrier et al. 1976; Feautrier and Tran Minh 1977), imposed the use of quantum calculations in the far wings. At high temperatures - about 40000K - the discrepancies decrease, so that the quantum and semi-classical results are close; furthermore, our profiles are obtained with a better precision (10% in the far wings versus 15% at low temperatures). Stehlé (1994) used a semi-classical description for electrons and ions, so that the profile exhibits the same static limit as VCS in the line wings, leading to the well known [FORMULA] variation of the intensity. In this case, the Holtsmark-normalized profile is taken to be 2 at large detunings [FORMULA].

It is important to mention that at large detunings, short range effects contribute significantly to the profile and have to be taken into account. Nevertheless, not too far in the wings the dipole approximation holds. Hence our calculations are carried out for detunings not exceeding 80Å.

6.2. Results and parametrized profiles at various temperatures

The results are available upon request for electron densities [FORMULA] varying from [FORMULA] to [FORMULA] cm-3, in the form presented in Table 4. They have been parametrized to facilitate their use. In fact, in the one perturber theory, the electronic line profile reads:


Table 4. Holtsmark-normalized wings profile of Hydrogen Lyman- [FORMULA] line at [FORMULA] cm-3 and [FORMULA] K. The error is that made in the contribution of the first angula momenta to the total profile by the extrapolation method.


where the collisonal width [FORMULA] can be expressed from equation (2). Without the [FORMULA] -dependence of [FORMULA], the Holtsmark-normalized profile would scale like [FORMULA]. The decreasing of this profile at large detunings is due to a combination with other powers of [FORMULA]. In any case, the Holtsmark-normalized profile [FORMULA] can be represented for both wings by a polynomial in [FORMULA] at any temperature within less than [FORMULA].

Let us put [FORMULA] for the red wing and [FORMULA] for the blue wing, and let us call [FORMULA] the red wing Holtsmark-normalized profile and the [FORMULA] blue wing one. The Lyman- [FORMULA] profiles can be fitted for [FORMULA] Å by polynomials, so that one has:




The coefficients [FORMULA] and [FORMULA] are reported in Tables 5 to 8 for [FORMULA], 5000, 10000, 20000 and 40000K and the density [FORMULA] cm [FORMULA] In some cases, the profiles are not fitted completely by only one polynomial. A suitable value of x is found such as to fit the profiles by two different polynomials corresponding to the ranges of x below and above this value.

At high temperatures the coefficients corresponding to the same order of get closer as T increases. This is to be accounted for by the decreasing asymmetry of both wings with increasing temperature.


Table 5. Coefficients [FORMULA] and [FORMULA] for [FORMULA] K and 5000K.


Table 6. Coefficients [FORMULA] and [FORMULA] for [FORMULA] K.


Table 7. Coefficients [FORMULA] and [FORMULA] for [FORMULA] K.


Table 8. Coefficients [FORMULA] and [FORMULA] for [FORMULA] K.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997