Astron. Astrophys. 357, 587-596 (2000)

2. Model atmospheres and input physics

The general procedures used for the calculation of theoretical atmosphere models and synthetic spectra have been described in some detail by Finley et al. (1997). The most important changes compared to the input physics used e.g. by Koester et al. (1982) or Zeidler-K.T. et al. (1986) are

• use of the Hummer-Mihalas-Däppen occupation probability formalism as described in Hummer & Mihalas (1988), Mihalas et al. (1988), Däppen et al. (1988), and Mihalas et al. (1990). We use small modifications for the critical field strength and the neutral radii as suggested by Bergeron et al. (1991) and Bergeron (1993)

• Pressure ionization is treated exactly as in Mihalas et al. (1988); specifically, we use their values for the "free" parameters

• inclusion of and photodissociation and free-free absorption coefficients after Stancil (1994)

• use of new hydrogen Stark broadening tables for many Lyman, Balmer, and Paschen lines calculated by Lemke (1997) using the theory and codes of Vidal et al. (1973)

• improved calculation of L broadening by charged and neutral perturbers according to the work of Allard and collaborators (Allard & Kielkopf 1991, Allard & Koester 1992, Allard et al. 1994, 1998)

• new Stark broadening tables for HeI lines, calculated and provided by Beauchamp et al. (1997, priv. comm.)

2.1. Equation of state and ionization fraction

Atmospheres for very cool white dwarfs with zero metallicity have been studied in great detail by Bergeron et al. (1995). Their prefered EOS is a modified version of the EOS of Saumon & Chabrier (1994) and Saumon et al. (1995). This model predicts much lower ionization, and pressure ionization at much higher pressures, than earlier work (e.g. Fontaine et al. 1977), a result supported by experimental evidence (Nellis et al. 1984). Even for effective temperatures as low as 4500 K, the electron density given by this model is close to the ideal gas model (using only the unmodified Saha equation) up to pressures of 10¹² dynes/cm². In our present study we consider only much hotter objects, and moreover, the atmospheres are not pure helium but contain traces of elements with much lower ionization potentials. As a consequence the atmospheric pressures are much lower and never exceed 10¹¹ dynes/cm² even at the bottom of our models at Rosseland optical depths 1000. Following the discussion and the results of Bergeron et al. (1995) non-ideal effects should be very small.

We use in our standard models (see above) a version of the Hummer-Mihalas EOS which predicts larger non-ideal effects and larger ionization than the Saumon-Chabrier EOS. To test the sensitivity of our results to this uncertainty of the input physics, we have adopted the following procedure: For the analysis we use our standard models; we then repeat the analysis with models, where all non-ideal effects are switched off and the ionization fraction is only determined by thermal effects using the unmodified Saha equation. For the objects of this study at effective temperatures of about 8000 K both calculations give exactly the same results, i. e. the emergent flux differs by less than 0.2% if models calculated with and without non-ideal effects are compared. This demonstrates that these effects are indeed negligible.

2.2. Lyman wing

The presence of photospheric hydrogen in L 745-46A and Ross 640 has been discovered through the observation of Balmer lines by Koester (1987) and Liebert (1977), respectively. It has to be expected, therefore, that L as the resonance line originating from the ground state is also very strong, and indeed it was obvious from the first test calculations of the UV spectrum that the predicted UV flux is much too high, if L is not included. However, model calculations of this line using the classical van-der-Waals broadening by neutral helium result in a L wing extending far into the optical region, which is not observed. The same is true for profiles calculated with the quasi-molecular broadening theory by Allard (priv. comm.).

In order to be able to analyze the metal lines on the wing of L between 2000 and 3000 Å, Koester & Allard (1995) have weakened the line wing by an exponential function so that the shape of the observed spectrum can be roughly reproduced. Since this artificial procedure is rather unsatisfactory, we have for this study made a new attempt to obtain a more realistic physical description. This is based on the realization that the failure of the quasi-molecular calculation, which has been so successful in DA white dwarfs, is due to the neglect of two effects in the first calculations by Allard (priv. comm.): the variation of the transition probability with the distance between emitter and perturber and the effect of the strongly repulsive potential between H and He atoms at close distances, which are important for the far line wings.

Since we are only interested in the very far wings (the line center, even out to 1500 Å or more, is completely saturated in the spectra), we have used the quasistatic limit of the theory for the multiperturber autocorrelation function as given by Eq. (59) in the review by Allard & Kielkopf (1982):

The integration is over the distance R between emitter and perturber, implicitely assuming that the perturber density is homogenous (not influenced by the interaction with the emitter) and that the transition probability for the transition considered is also independent of R and equal to the transition probability at large distances; is the difference potential between upper and lower state of the transition, which is a function of perturber distance. In this simple picture it is rather obvious, how this equation has to be modified to be more physically realistic, if very high particle densities and close collisions are important:

where is the transition probability and the interaction energy between emitter in the lower state (considering L absorption) and the perturber. The Boltzmann factor here takes into account that in thermal equilibrium the density of perturbers very close to the H atom will be lower because of the strongly repulsive potential. Molecular potential curves and transition probabilities for the H-He system were obtained from Theodorakopoulos et al. (1987). The line profile is finally obtained in the usual way through the Fourier transform of the autocorrelation function (see Allard & Kielkopf 1982).

Somewhat to our surprise, this extremely simple procedure results in theoretical line profiles in our models, which agree very well with the observed flux distribution in both objects studied, at the same hydrogen abundances that are infered from the analysis of H (or H). Fig. 1 shows the new L wing and a comparison with the observation of L 745-46A. Since the shape of the observed line and the overall ultraviolet and optical flux distribution (see Sects. 4 and 5) are reproduced very well, we are quite confident that this is a realistic description of the L profiles under the extreme conditions of our cool atmospheres, where perturber densities may be as high as cm^-3.

Fig. 1. L 745-46A: L line. The metal lines are described in Fig. 5

© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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