3 Use of the tables

Whatever quantity is chosen as the detuning parameter, the intensity profile must be area normalized. The change from wavelength $\lambda$ to angular frequency $\omega$ is then given by

$$I(\lambda)=I(\omega) \, \vert {\rm d}\omega /{\rm d}\lambda \vert \,\,\,{\rm with} \,\,\, \omega= 2 \pi c / \lambda .$$ (11)

The physical choice of parameter for the the line shape calculation is derived energy detunings (i.e. frequency, angular frequency, wavenumbers, energy...). For Stark broadening, it may also be convenient to normalize this detuning by the normal Holtsmark field F₀defined by

F₀ = e/r₀² (12)

$$F_0/{\rm esu} 1.25~10^{-9} ( N_{\rm e}/ {\rm cm}^{-3} )^{2/3}$$ = (13)

where r₀, closely equal to the mean interelectronic distance is defined by

$$(4 /15) (2 \pi)^{3/2} r_0^3 N_{\rm e} =1.$$ (14)

An appropriate choice for the detuning parameters for Stark broadening calculations would be $\Delta \Omega = \Delta \omega /F_0$ (units: rad s^-1 esu^-1). Despite this, parameters based on wavelength are often preferred for practical applications. Previous tabulations (Vidal et al. 1973; Stehlé 1994a; Stehlé 1996a; Lemke 1997) used reduced wavelength detunings defined as $\alpha$ units (Å esu^-1) where

$$\Delta \alpha /({\mbox \AA ~{\rm esu}}^{-1})= { (\Delta \lambda /{\mbox \AA}) \over (F_0 / {\rm esu}) }\cdot$$ (15)

As functions of energy or angular frequency detunings, the line shapes are symmetrical around the line center, i.e. $I(-\Delta \omega) = I(\Delta \omega)$. The transformation of this relation in wavelength detunings, should allow for the albeit small asymmetry introduced by this choice of detunings, that is we use

$$\Delta\lambda = - 2 \pi c {\Delta \omega \over \omega_0 (\omega_0 + \Delta \omega)}$$ (16)

where $\omega_0$ and $\lambda_0$ are respectively the angular frequency and the wavelength of the unperturbed transition and $\omega_0 = 2\pi c/\lambda_0$. This "trivial'' asymmetry must be distinguished from the "intrinsic'' asymetry, which is not included in this calculation. Hydrogen line "intrinsic'' asymmetry is negligible for modelling stellar atmospheres. It is due to different competitive mechanisms like for example fine structure effects (which affects the line centers at low densities, Stehlé & Feautrier 1985), and quantum and short range effects (see for example, Demura & Sholin 1975; Feautrier et al. 1976; Stehlé 1986; Demura et al. 1990; Döhrn et al. 1996; Günter & Könies 1997) which mainly affect the line wings. These asymmetry sources are not related to those observed in moving media.

In order to be consistent with earlier tabulations, we performed the tabulation using reduced wavelength $\Delta \alpha$ detunings. For each transition, the line shapes tabulations corresponding to different electronic densities are reported in separate files. They are given for positive (wavelength) detunings, before and after Doppler convolution. The line wing parameter $K_{\alpha}$ is also given. It allows the line shape to be extrapolated towards large detunings, the relation

$$I(\Delta \omega) = {K_{\omega} \over (\Delta \omega)^{5/2}}$$ (17)

or in reduced wavelengths,

$$I(\Delta \alpha) = {K_{\alpha} \over (\Delta \alpha)^{5/2}} \times \left( { \lambda_0/F_0 +\Delta \alpha \over \lambda_0 /F_0 } \right) ^{1/2}$$ (18)

$$K_{\alpha} = K_{\omega} \times \left( { \lambda_0 \over 2 \pi c }\right)^3\cdot$$ (19)

Concerning the transitions from a highly excited level n', in "no lower state interaction'', the intensity profiles of the Balmer, Paschen... and Lyman lines, in angular frequency detunings, are (before Doppler convolution) exactly the same. Thus the factor $K_{\omega}$ is invariant.