SPECTROSCOPIC ANALYSIS OF DA WHITE DWARFS: STARK BROADENING OF HYDROGEN LINES INCLUDING NONIDEAL EFFECTS

The Stark effect is defined as the shifting—or splitting—of spectral lines under the action of an electric field. In the following section, we assume that we have in the atmospheric plasma a local electric microfield due to protons that is constant with time. This static approximation is well justified since the characteristic time for the fluctuation of the local microfield due to a change in the proton distribution is much larger than the characteristic time of absorption and emission processes that we are studying (Stehlé & Jacquemot 1993). We also assume that the microfield is constant in space, a good approximation if the perturbers are far from the area occupied by the bound electron orbits. The regime where this approximation fails is discussed further in Section 2.3. The electric microfield, assumed to be along the z-axis, interacts with the dipolar moment of the atom and leads to a perturbation in the Hamiltonian of the form

$$\begin{equation} H_{s}=e\vec{R} \cdot \vec{F}=eFz. \end{equation} \tag{ 5 }$$

It is common to write the amplitude of the microfield in terms of the unitless parameter β such that

$$\begin{equation} \beta =\frac{F}{F_0} \, \end{equation} \tag{ 6 }$$

where F₀ is the characteristic field defined by Equation (3). The amplitude of the microfield can be expressed using a probability distribution P(β). One such distribution that can be computed analytically is the Holtsmark distribution, which takes into account the interactions (vectorial sum) between the different perturbers. We use here the more physical distribution of Hooper (1968) that takes into account the Debye screening effect, and which corresponds also to the formulation included in the work of VCS. HM88 originally used the Holtsmark distribution, but for consistency we use the approach of Nayfonov et al. (1999) for the HM88 equation of state by replacing the Holtsmark distribution with the Hooper distribution. We note that this improved version of the HM88 equation of state is already included in our model atmosphere code, although it has never been properly documented.

We note that the perturbation H_s does not commute with the angular momentum $\vec{L}^2$ since it has a privileged direction in space. The effect of breaking this symmetry is to lift the degeneracy in energy of the hydrogen atomic levels. There are no exact solutions to this problem and we generally use quantum mechanical perturbation theory to determine corrections to the energy and wave functions. The first-order corrections to the energy are linear in relation to the electric microfield that is applied, and represent the basis of the linear Stark effect. The linear corrections are found by diagonalizing the perturbation operator in the degenerate subspaces. For a hydrogen atom, we can write this expression analytically. First of all, knowing that the operator z conserves the azimuthal quantum number m, we can diagonalize the matrix 〈Ψ_k|H_s|Ψ_l〉 (Condon & Shortley 1935) by making use of the quantum numbers (n, m, q), where q = q₁ − q₂ with q₁ + q₂ = n − |m| − 1, and n is the principal quantum number. The shifts in energy are then given by

$$\begin{equation} \Delta E^{1}_{n,m,q}=\frac{3ea_{0}F}{2}nq. \end{equation} \tag{ 7 }$$

When the electric microfield becomes very high, or equivalently, when the Stark splitting is important enough that the Stark components of two levels with a different principal quantum number are crossing, the linear approximation is no longer valid. The second-order corrections to the energy become quadratic in relation to the microfield, and we find

$$\begin{equation} \Delta E^{2}_{n,m,q}=e^2F^2\sum _{n^{\prime }\ne n}\frac{|\left\langle \Psi _{n^{\prime },m^{\prime },q^{\prime }}\right|z\left| \Psi _{n,m,q}\right\rangle |^{2}}{E^{0}_{n,m,q}-E^{0}_{n^{\prime },m^{\prime },q^{\prime }}}. \end{equation} \tag{ 8 }$$

The problem with this perturbation approach to the Stark effect is that the wave functions, using any order of the perturbation theory, are not necessarily normalizable (Friedrich 2006). This is because the perturbing potential goes to −∞ when z goes to −∞. In other words, states that were bound without a microfield are now only metastable states, and there is a finite probability that the atom will be ionized when the microfield has a sufficiently high amplitude. Classically, the sum of the electric potentials of the absorber and the nearby protons only allows bound states in the local potential minima up to a certain energy, called the saddle point. Part of this problem is because we have used a spatially uniform microfield (Equation (5)). In reality, the microfield cannot be uniform beyond a distance of about the interparticle separation, and thus under normal conditions, most of the states will be bound. However, for sufficiently high microfields, some excited states will become unbound and this is where resides the problem discussed further in the next sections.

In addition to static protons, the absorber also interacts with free electrons. These particles being much faster than the protons, a collisional approach is generally used to describe this interaction. Since the collisions are rapid, the interaction is nonadiabatic and a proper quantum mechanical treatment of the collisions accounting for the internal structure of the hydrogen atom is required. Various theories of Stark broadening for hydrogen lines take into account both the electron and proton interactions (Vidal et al. 1971; Seaton 1990; Stehlé & Jacquemot 1993). Here, we summarize the principal aspects of the unified theory of VCS, which even today stands as the most accurate theory for the conditions encountered in white dwarf atmospheres.

The unified theory of VCS uses the quasi-static proton broadening approximation with the distribution of electric microfields described in Section 2.1, assumed to be constant in space and time. The complete Stark profile is then defined as

$$\begin{equation} S(\alpha)=\displaystyle \int ^{\infty }_{0}P(\beta)I(\alpha,\beta)d\beta, \end{equation} \tag{ 9 }$$

where I(α, β) is the electronic broadening profile. Therefore, the problem is reduced to the calculation of electronic broadening profiles for a fixed microfield of amplitude β. The full broadening profile is then the average of the electronic profiles over all possible microfields weighted by the probability distribution P(β). The unified theory is a nonadiabatic quantum mechanical theory that uses the classical path approximation for the electron–absorber interactions, implying that the wave functions are well separable. The interaction potential considers only the first dipole term. The name of the unified theory comes from the fact that the electronic profile is valid at all detunings from the line center. In the asymptotic limit of the line core, the impact approximation is recovered. Furthermore, in the far wings of the lines, the one-electron theory is recovered and the electrons and protons have similar contributions to the broadening.

Now, we want to compute the electronic profile for an initial Stark state n_a to a final state n'_a, where this abbreviated notation is used to designate any of the (n, m, q) states. Mostly to reduce the computing time and to simplify the inversion of the matrix involved (see below), the unified theory uses the no-quenching approximation, which means that there are only collision-induced transitions within the same principal level. The states after the collisions will be denoted by n_b and n'_b for the initial and final state, respectively. The calculations can then be performed by considering only two-level transitions. This last approximation is valid only if the difference in energy between the highest Stark state from level n' and the lowest state of level n' + 1 is large, or in other words, when the different lines are well separated (i.e., when the line wings do not overlap). Only then is the regime of the linear Stark effect consistent with the basic assumptions of the unified theory.

The electronic broadening profile I(α, β) is given by the sum over all states

$$\begin{eqnarray} I(\alpha,\beta)&=&\frac{1}{\pi }\sum _{n_a,n^{\prime }_a,n_b,n^{\prime }_b}\rho (n_a)\,{\rm Im}[\langle n_a|\vec{d}| n^{\prime }_a\rangle \langle n^{\prime }_b|\vec{d}| n_b\rangle\nonumber\\ &&\times \langle n_b|\langle n^{\prime }_b|\mathcal {K}^{-1}(\alpha,\beta)| n^{\prime }_a\rangle | n_a\rangle], \end{eqnarray} \tag{ 10 }$$

where $\vec{d}$ is the dipole operator and ρ is the probability that the atom is in the initial state (Boltzmann factor). The operator $\mathcal {K}$ takes into account the linear Stark effect as well as the free electron–atom interactions (see Vidal et al. 1970, Equation (XII.2) for more details). Since the operator $\mathcal {K}$ is generally not diagonal in the basis of the (n, m, q) states, it induces a coupling between the various Stark components.

One important advance that has provided a better interpretation of the hydrogen lines in DA white dwarfs is a realistic modeling of nonideal effects at high densities. Bergeron et al. (1991) were the first to include the occupation probability formalism of HM88 to determine the populations of the bound states of hydrogen in white dwarf atmospheres. This probabilistic approach considers the perturbations on each atom by charged and neutral particles. One advantage of this statistical interpretation is that there are no discontinuities in the populations and the opacities when the temperature and the pressure vary, in contrast with other formalisms that simply predict the last bound atomic level under given physical conditions.

Briefly, each atomic level n has a probability w_n of being bound and a probability 1 − w_n of being dissociated due to perturbations from other particles in the plasma. The partition function of a given species is then written as

$$\begin{equation} Z=\sum _{n} w_n g_n \exp \left(-\frac{\chi _n}{kT} \right), \end{equation} \tag{ 11 }$$

where χ_n and g_n are, respectively, the excitation energy and multiplicity of the level. The occupation probability w_n is defined as

$$\begin{equation} w_n=\exp \left(-\frac{\partial f /\partial N_n}{kT} \right), \end{equation} \tag{ 12 }$$

where f is the free energy of the nonideal interaction. The different interactions being statistically independent, the total occupation probability can then be calculated simply as the product of the contributions from neutral and charged particles.

The interaction with neutral particles is treated within a hard sphere model. The configurational free energy is derived from the second virial coefficient in the van de Waals equation of state (excluded volume correction). The exact origin of this excluded volume term is the hard sphere equation of state of Carnahan & Starling (1969). The interaction with charged particles is the one closely related to the Stark effect. This interaction suggests that the electric microfields, fluctuating on long timescales due to changes in the spatial distribution of protons, can destabilize a bound state and cause its dissolution into the continuum. Indeed, we already discussed in Section 2.1 that the application of an intense electric field on a bound atom allows for the ionization of the atom. HM88 therefore suggest an occupation probability for proton perturbations of the form

$$\begin{equation} w_n({\rm charged})=\displaystyle \int ^{\beta _{\rm crit}}_{0}P(\beta)\,d\beta, \end{equation} \tag{ 13 }$$

where P(β) is the probability distribution of the electric microfields introduced in Section 2.1. This occupation probability implies that all microfields larger than the critical field β_crit will ionize the electrons in level n (it is implicit here that the critical field depends on the atomic level considered). The difficulty with this formulation is to find the value of the critical microfield, for which slightly different expressions can be found in the literature (HM88; Seaton 1990; Stehlé & Jacquemot 1993). The simplest formulation is that of Seaton (1990) which consists in taking the critical microfield as the point where the energy of the highest Stark state for a given level n crosses the energy of the lowest Stark state of the next level n + 1. For hydrogen, and using the linear Stark effect, we find

$$\begin{equation} \beta _{\rm crit}=\frac{2n+1}{6n^4(n+1)^2}\frac{e}{a_0^2 F_{0}}. \end{equation} \tag{ 14 }$$

The assumption that the crossing of two atomic levels with different principal quantum numbers leads to the dissolution of the lower level is difficult to prove, and it is based in part on laboratory experiments and theoretical considerations (see HM88 for a more detailed discussion). Let us consider one bound electron in one of its many Stark states of the nth level. When the electric microfield, which fluctuates in time, gets to a value such that there is a crossing¹ between this level and the lowest Stark state of the n + 1 level, there is a significant probability that when the microfield goes down, there will be a transition n → n + 1. The incessant fluctuations of the microfield imply that these transitions will continue until the electron becomes unbound. However, the bound electron will, in fact, be in a rather homogeneous superposition of all its accessible Stark states due to electronic collisions² on timescales much shorter than the fluctuation time for the microfields, and also due to fluctuations in the direction of the microfield. That is, as soon as the first Stark state from a level n crosses another one from the level n + 1, the electron is allowed to cascade to the continuum.

The reader might have noticed that Equation (14) is based on the linear Stark effect although this is not necessarily a good approximation when levels are crossing. This is why HM88 considered a formulation of the critical field that takes into account corrections of higher order from the perturbation theory as well as results from experiments.³ They find that for levels with n>3, the linear theory remains valid. For lower levels, however, the critical field is large enough that it must have been created by a single proton very close to the absorber. In this case, the approximation of a uniform electric field in space fails and we must consider the full potential curve of the H⁺₂ system to find the critical field (Stehlé & Jacquemot 1993), and also to compute the broadening profiles of the far line wings (Allard & Kielkopf 1982). Exact calculations show that no crossing is possible between levels n = 1, 2, and 3, which implies that Lα is not affected by nonideal effects due to proton perturbations. Also, the crossing between n = 3 and 4 is close to the classical saddle point value, which means that the linear Stark theory is in error by ∼14%. Therefore, HM88 added a smoothing factor to Equation (14) so that the exact result is recovered for n = 3. For levels with a high value of n, HM88 obtain the same results as Seaton (1990). For the Balmer lines, these different values of β_crit are not an issue since the nonideal effects discussed above are never important for Hα in DA white dwarfs. This is not the case for Lα and Lβ, however, for which nonideal effects should be treated with caution, or simply neglected (for instance in TLUSTY and in our code discussed in Section 4).

We have seen in Section 2.2 that the interaction of hydrogen atoms with rapid electrons was better described with a collisional approach since the quasi-static theory is generally not valid in this case. These perturbations are also considered in the HM88 model. We must keep in mind that because the plasma is in thermal equilibrium, the equation of state already accounts for the inelastic electronic collisions that cause transitions and ionization. However, we expect that if the time between collisions is small, the energy of a bound level will be left undefined by a certain amount given by the uncertainty principle. If this energy uncertainty is of the order of the ionization potential, we expect the state to become unbound after the collision. As stated in HM88, this contribution to the occupation probability is always much lower than the proton contributions and can be neglected in DA atmospheres. However, the electronic perturbations should not be neglected when calculating the line profiles, as discussed in Section 3. We must also remember that the electrons contribute indirectly to the occupation probability due to proton perturbations by mixing the Stark states. The occupation probability for the bound states of hydrogen is displayed in Figure 3 for typical conditions encountered at the photosphere of white dwarf stars. Nonideal effects are shown to be extremely important except for the very lowest transitions. The solution of the HM88 equation of state yields the populations for all states.

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Finally, the opacity calculation must also be modified to take into account the occupation probability formalism, as described by Däppen et al. (1987). The bound–bound (line) opacity for a transition between levels i and j (Equation (1)) is multiplied by w_j/w_i to take into account the fact that these levels can be dissolved when a photon is absorbed. We thus obtain

$$\begin{equation} \kappa _{ij} (\nu)\,d\nu = N_{i}\frac{\pi e^2}{m_{e}}\frac{w_j}{w_i} f_{ij} \phi (\nu)\,d\nu. \end{equation} \tag{ 15 }$$

When the absorption is from a bound state to a final state that has been dissolved by nonideal interactions (with a probability 1 − w_j/w_i), we obtain instead a bound–free opacity since this process is equivalent to the ionization of the atom. We can directly extrapolate, in this case, the bound–free cross section at frequencies below the usual cutoff frequency ν_c. Däppen et al. (1987) have treated this opacity by considering the absorption from a level i of a photon of energy hν that yields a transition to a fictitious upper level n* given by

$$\begin{equation} n^{\ast} =\left(\frac{1}{n_{i}^2}-\frac{h\nu }{\chi ^I}\right)^{-1/2}. \end{equation} \tag{ 16 }$$

The hydrogen bound–free opacity for ν < ν_c—also called the pseudo-continuum opacity—can then be written as

$$\begin{equation} \kappa _{i} (\nu) d\nu = N_{i}\left(1-\frac{w_{n^\ast }}{w_i}\right) \frac{64 \pi ^4 m_{e} e^{10}}{3 \sqrt{3} c h^6} \frac{g_{bf,i}(\nu)}{n_{i}^5 \nu ^3}\,d\nu. \end{equation} \tag{ 17 }$$

In Figure 4, we show the individual contributions of the line and pseudo-continuum opacities at the photosphere of a 20,000 K DA white dwarf in the spectral region of the Balmer lines. We can see how the pseudo-continuum opacity extends to wavelengths longward of the Balmer jump, potentially affecting the opacity between the lines, and in particular the high Balmer lines.

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The preferred way to validate Stark broadening theories for hydrogen has been the comparison with laboratory plasma emissivity measurements from Wiese et al. (1972). These pure hydrogen plasma arc experiments were performed at high densities and temperatures, comparable to the photospheric conditions of cool white dwarfs. The main advantage of this experiment is that it is not under the constraint of radiative equilibrium, unlike a stellar atmosphere. In other words, the emergent flux at one wavelength is not affected by what occurs at other wavelengths. The emissivity can then be simply written as

$$\begin{equation} j_{\lambda }(\lambda)=\frac{2 c}{\lambda ^4}\kappa (\lambda)e^{-hc/\lambda kT}, \end{equation} \tag{ 20 }$$

where κ(λ) represents the total monochromatic opacity. Another advantage, at least from a theoretical point of view, is that it is easier to control the experiments and the results are independent of various sources of uncertainty intrinsic to DA model atmospheres (i.e., convection, contamination from heavy elements, etc.). We use the data for the lowest and highest density experiments (the most extreme positions on the plasma arc) from Wiese et al., which were compared to broadening theories in various studies (Däppen et al. 1987; Seaton 1990; Bergeron 1993; Stehlé & Jacquemot 1993). Wiese et al. estimate the LTE plasma parameters for these two experiments at log T = 4.00, log N_e = 16.26 and log T = 4.12, log N_e = 16.97, respectively. However, there is no simple way to measure the plasma state parameters and it is not clear whether LTE is reached or not. The authors used the total intensity from two lines, together with the intensity at two continuum points, and fitted the data with an approximate plasma model. Since then, slightly different parameters have been used to compare with broadening theories. Instead of choosing approximate values for the parameters like in previous analyses, we performed a χ² fit to the full data sets, using the same input physics as for our white dwarf models. The results with the original VCS calculations and our improved line profiles are displayed in Figures 6 and 7 in terms of absolute predicted fluxes (i.e., without any renormalization).

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For the lower density experiment of Figure 6, our improved profiles provide a much better fit to the laboratory data, in particular in the regions of the high Balmer lines where the wings overlap. The higher density experiment of Figure 7 is more problematic. We find that there is a partial T–N_e degeneracy in the χ² diagram; it is indeed possible to find many acceptable solutions from visual inspection by increasing both T and N_e. However, to provide the best overall fit with our new profiles, we have to increase the plasma state parameters significantly compared to the Wiese et al. values. We see that our profiles provide the best fit for the higher lines, although the red wings of Hβ and Hγ are predicted a bit too weak. All in all, we conclude that the experiments of Wiese et al. do not provide such a stringent constraint on the broadening theories because of the large range of acceptable plasma state parameters, and also because of potential departures from LTE.

While our work represents a significant advance over previous calculations, second-order effects in the theory of Stark broadening could still come into play. Such second-order effects have been studied extensively in the literature in the case of high-density hydrogen plasma experiments (Lee & Oks 1998; Stehlé et al. 2000; Demura et al. 2008). The sources of these second-order effects are plentiful but they all have the same impact on the line profiles: they produce asymmetries and they shift the central wavelengths, the latter being unimportant for the spectroscopic technique used to measure the atmospheric parameters of white dwarfs. The recent study of Demura et al. (2008) reveals that the various second-order effects compete and interact together in a complex way and that they should all be included in the models simultaneously. The most important effects are due to quadrupole proton-absorber interactions and to the quadratic Stark effect, but second-order effects due to quadrupole electron-absorber interactions are noticeable as well. The lifting of the no-quenching approximation could also be considered a second-order effect, although it was shown by Lee & Oks (1998) to have an impact mostly on the line cores. Furthermore, some of these quenching effects have already been accounted for implicitly by including here the HM88 nonideal effects directly inside the line profile calculations. Finally, even if we include better physics for the atomic transitions, we still have to rely on the HM88 equation of state, which is, or may become, the main source of uncertainty.

Our analysis above of the Wiese et al. (1972) experiments, with our improved profiles has already revealed the existence of line profile asymmetries. For the lower density experiment (see Figure 6), it is only a mild effect in the far wings, however. It is not clear if such second-order effects would be significant in a typical DA white dwarf at log g ∼ 8, for which the electronic density at the photosphere is roughly halfway between the two Wiese et al. experiments. Therefore, it is probably premature at this stage to include second-order effects in our calculations, and it is not clear whether such changes would affect our line profiles significantly. We come back to this point in the analysis of the Palomar–Green (PG) spectroscopic sample discussed below.

We now discuss the astrophysical implications of our improved line profiles described in Section 3 on the modeling of DA white dwarfs. Our model atmosphere code is based on the program originally developed by Bergeron et al. (1991, 1995a) and references therein. The main difference is that we have updated several sources of opacity and partition functions (all within the occupation probability formalism). For completeness, we provide in Table 1 the complete list of opacity sources included in our code. We have not included the Lα line broadening due to H₂ collisions (Kowalski & Saumon 2006) since these calculations are still not available, although this has no impact over the range of effective temperature considered here.

We restrain our analysis to effective temperatures above ∼13,000 K to avoid additional uncertainties related to convective energy transport. Indeed, the atmospheric structure of DA white dwarfs below this temperature depends sensitively on the assumed convective efficiency. The most commonly used approach to include convection in model atmosphere calculations, the mixing-length theory, is at best a very crude approximation, and the convective efficiency must be parameterized by carefully adjusting the value of the mixing length (Bergeron et al. 1995b). It is also believed that large amounts of helium can be brought to the surface by convection while remaining spectroscopically invisible (Bergeron et al. 1990b; Tremblay & Bergeron 2008). Hence, it is not even clear whether cool DA stars have hydrogen-rich atmospheres. Another reason for restricting our analysis to hotter stars is that neutral line broadening becomes important for the Balmer lines below T_eff ∼ 10, 000 K, preventing us from performing a direct comparison of the observed and predicted Stark broadened absorption lines. We also set an upper limit of T_eff = 40, 000 K because NLTE effects become important above this temperature. Also, absorption from heavier elements, particularly in the UV, is likely to affect the atmospheric structure of hot white dwarfs.

We thus computed two grids of model atmospheres, one with the VCS profiles and the other with our improved profiles (both calculated with β_crit × 1), covering the range of T_eff = 12,000–45,000 K (with steps of 500 K up to 15,000 K, 1000 K up to 18,000 K, 2000 K up to 30,000 K, and 5000 K above) and of log g = 6.5–9.5 (by steps of 0.5 dex with additional models at 7.75 and 8.25). To compare the model spectra calculated from both sets of line profiles, we simply fitted the VCS spectra with our improved spectra. The results of this exercise are illustrated in Figure 8. In general, our new models yield systematically higher effective temperatures (by 1000–2000 K) and surface gravities (by ∼0.1 dex) over the entire range of atmospheric parameters considered here. One exception is a region at low temperatures and high surface gravities where the trend in temperature is reversed. The strip, inclined in the diagram near T_eff ∼ 15, 000 K, where this separation occurs corresponds to the region where the strength of the Balmer lines reaches its maximum (Bergeron et al. 1995b). Our new models predict Balmer lines that are stronger, hence the temperatures in this region are pushed toward lower values.

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In this section, we measure the implications of our improved Stark profiles on the analysis of Balmer line observations of DA stars by considering the PG sample of Liebert et al. (2005). This sample of 348 DA stars has been studied in great detail, and the range of effective temperatures for these objects corresponds very well to that considered in our analysis. About 250 white dwarfs fall in the appropriate range of T_eff, depending on which model grid is used. The data set and the fitting procedure of the optical spectra are identical to those described at length in Liebert et al. (2005, and references therein). Briefly, we first normalize the observed and model flux from each line to a continuum set to unity at a fixed distance from the line center. The observed profiles are then compared with the predicted profiles, convolved with a Gaussian instrumental profile. The atmospheric parameters are then obtained using the nonlinear least-squared method of Levenberg–Marquardt, fitting simultaneously five lines (Hβ to H8). In some cases where the spectrum is contaminated by a M dwarf companion, one or two lines are excluded from the fit.

As discussed in Section 1, model spectra calculated with standard Stark broadening profiles yield inconsistent atmospheric parameters when different lines are included in the fitting procedure. As shown in the bottom panel of Figure 2, our improved Stark profiles provide an even better internal consistency than the previous calculations displayed in the two upper panels. To better quantify this internal consistency, we perform for each star in the PG sample the same exercise as that shown in Figure 2 using the VCS profiles and our improved profiles. We then compute for the 250 stars in our sample the average absolute deviations in T_eff and log g between the solutions obtained with a different number of lines included in the fit. The results of this exercise are presented as filled circles in Figure 9 together with the mean uncertainties from the fitting procedure as a reference point (the open circles will be discussed in Section 4.3). We can see that the deviations with our improved profiles have been significantly reduced by a factor of ∼1.6 in T_eff and ∼1.8 in log g. Furthermore, these deviations now both lie within the mean uncertainties of the fitting procedure, a result which is extremely reassuring.

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The quality of the fits is also an important aspect of the comparison between model grids. We compare in Figure 10 our best fit to WD 0205+250 (same star as in Figure 2) using the VCS profiles and our improved Stark profiles. This comparison reveals that while the atmospheric parameters are significantly different, the quality of the fits is similar. We thus conclude that the quality of the fits cannot help to discriminate between both sets of Stark profiles.

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We can also explore the global properties of the PG sample with our two model grids. We first convert the log g values into mass using evolutionary models appropriate for white dwarfs with thick hydrogen layers (see Liebert et al. 2005 for details). Our results are presented in Figure 11 in a mass versus effective temperature diagram. As expected from the previous comparison displayed in Figure 10, our improved line profiles yield significantly larger masses and higher effective temperatures. We note that the results displayed in the bottom panel are qualitatively similar to the early spectroscopic determination of the mass distribution of DA stars by Bergeron et al. (1990a), which was based on the VCS profiles without any modification of the critical field. In this case, the mass distribution has a mean value near 0.53 M_☉, which is uncomfortably low compared to what is expected from earlier phases of stellar evolution (see the discussion in Bergeron et al. 1990a).

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Finally, we have also investigated whether second-order effects in the Stark broadening theory, such as those discussed in Section 3.2, could be detected in our analysis. In particular, if such effects are present, we would expect to observe differences in the fits of the blue and red wings of the Balmer lines. We have thus reanalyzed the PG sample by including in the fit (1) only the red wing of the Balmer lines and (2) only the lines cores (this is accomplished by fitting only half of the wavelength range we normally use for each line). We find that the atmospheric parameters obtained in this manner are entirely consistent, and we thus conclude that second-order effects can be safely neglected in DA white dwarfs.

The spectroscopic determination of the mass distribution by BSL92 was based on model spectra that include the solution proposed by Bergeron (1993) to mimic the nonideal effects, namely by taking twice the value of the critical field (β_crit × 2) in the HM88 formalism. This ad hoc procedure had the effect of artificially reducing the pseudo-continuum opacity, and thus the opacity in the wings of the high Balmer lines. The internal consistency of the solutions obtained from different lines was consequently improved, as can be judged from the results displayed in Figures 2 and 9 (open circles). Since this ad hoc solution has been adopted by BSL92 and in all white dwarf models used in the literature, it is important to evaluate the differences between our revised atmospheric parameters and those published in the literature based on these approximate model spectra. For this purpose, we have also calculated another model atmosphere grid using the original VCS profiles from Lemke (1997) but with twice the value of the critical field. This grid is similar to that used by the Montreal group in the past 10 years or so, and which has been applied to several studies of DA white dwarfs using the spectroscopic technique.

Once again, we rely on the PG sample of DA stars analyzed above. A comparison similar to that shown in Figure 11 is not very instructive here since the differences in T_eff and log g are considerably smaller. Instead, we use the representation displayed in Figure 12 where the differences ΔT_eff and Δlog g are shown as a function of effective temperature. This comparison reveals that our improved line profiles yield higher effective temperatures and surface gravities, with an important correlation with T_eff. In particular, the log g values are about 0.05 dex larger above 20,000 K but can be as much as 0.1 dex larger near 15,000 K. Similarly, the differences in temperatures reach a maximum near 25,000 K but decrease at both lower and higher effective temperatures. The corresponding mass distributions, displayed in Figure 13, indicate that the mean mass is shifted by +0.034 M_☉ when our new models are used. The shape of the mass distributions is statistically equivalent, however, and the dispersion remains the same. We must also point out that according to the results of Tremblay & Bergeron (2008), 15% of the DA white dwarfs in the temperature range considered here probably have thin hydrogen layers (M_H/M_tot < 10⁻⁸). Consequently, the typical values for the mean mass are probably ∼0.005–0.01 M_☉ lower than the values reported in Figure 13.

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As discussed in Section 1, the spectroscopic technique provides very accurate measurements of the atmospheric parameters, allowing for a relative comparison of these parameters among individual stars. However, the absolute values of the atmospheric parameters may suffer from an offset due to uncertainties in the physics included in the model calculations. Hence, it is very important to compare the results with independent methods. The most reliable independent observational constraint for DA white dwarfs comes from trigonometric parallax measurements. Holberg et al. (2008a) have shown using the recent photometric calibrations of Holberg & Bergeron (2006) that there exists a very good correlation between spectroscopically based photometric distance estimates and those derived from trigonometric parallaxes. Here, we compare absolute visual magnitudes instead of distances. We first combine trigonometric parallax measurements with V magnitudes to derive M_V(π) values. We then use the calibration of Holberg & Bergeron (2006) to obtain M_V(spec) from spectroscopic measurements of T_eff and log g. We selected 92 DA stars with known parallaxes from the sample of Bergeron et al. (2007) (the uncertainties on the parallaxes must be less than 30%). The M_V values obtained from both model grids described in this section are compared in Figure 14. The agreement is very good within the parallax uncertainties for both model grids. Hence, despite the fact that our new models yield higher values of T_eff and log g (i.e., smaller radii), these two effects almost cancel each other and the predicted luminosities (or M_V) remain unchanged, and so are the conclusions of Holberg et al. (2008a). Another important constraint is provided by the bright white dwarf 40 Eri B for which a very precise trigonometric parallax and visual magnitude have been measured by Hipparcos. These measurements yield M_V = 11.01 ± 0.01, while we predict M_V = 11.02 ± 0.07 and 10.97 ± 0.07 based on our new and VCS models, respectively. Although both determinations agree within the uncertainties with the observed value, our new grid provides an exact match to the measured M_V value.

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We also point out that trigonometric parallax measurements are available mostly for cool white dwarfs (Bergeron et al. 1997) and nearby white dwarfs (Holberg et al. 2008b), which are excluded from our analysis. In contrast with our approach here, Bergeron et al. (1997) and Holberg et al. (2008b) make use of photometric measurements that cover the full spectral energy distributions, and both studies find mean masses near 0.65–0.66 M_☉, consistent with our spectroscopic determinations for the PG sample.