Non-linear behaviour of the pulsating white dwarf G29-38 — III. Relative amplitudes of the cross-frequencies

Abstract

In each season when the DA pulsating white dwarf G29–38 has been observed, its period spectrum appears very different, but it always contains a forest of harmonics and cross-frequencies. The ratio of the amplitude of these non-linear frequencies A_c to the product of the amplitudes of the corresponding parent modes A₁A₂ has been measured. The results are compared with the predictions given by three existing theoretical models. Our analysis shows that the non-linear frequencies present in the period spectrum of G29–38 owe their presence mostly to the inelastic response of the stellar medium to the perturbation travelling through it, rather than to the non-linear response of the emergent luminous flux to the surface temperature variation. This analysis also confirms that most identified modes are ℓ=1, as previously asserted by Kleinman et al.

stars: individual: G29-38, stars: oscillations, stars: variables: other, white dwarfs

Introduction

Pulsating white dwarfs experience periodic luminosity variations that can be observed with high-speed photometry, for example. The light curve obtained in this way can then be Fourier transformed. The resulting amplitude spectrum will show numerous peaks that indicate the different periods at which the star is pulsating simultaneously. Some of these peaks correspond to normal modes, i.e. to oscillation eigenfrequencies of the star, while others do not, and are called non-linear frequencies. The latter are either harmonics or combination frequencies (another term for cross-frequencies) in the sense that their frequencies are either sums or differences of frequencies corresponding to excited normal modes.

These non-linear frequencies do not exist on their own, nor can they be described by a set of indices (k, ℓ, m), in the same way as normal modes are (Unno et al. 1989), because they are not related in any way to the actual pulsation mechanism. They owe their presence to the non-linear response of the star to the normal mode pulsations. In other words, part of the power originally contained in the eigenmodes, or parent modes, gets naturally redistributed into these combination frequencies. This phenomenon is called harmonic distortion, or the ‘pulse shape effect’, and may have two main physical origins.

First is the inelastic response of the stellar medium to the perturbation travelling through it: a normal mode driven to a finite amplitude at the base of the convection zone will see its corresponding sinusoidal wave being distorted on its way to the surface, similarly to the way in which an acoustic wave behaves in water, for example (Rudenko & Soluyan 1977). Second is the non-linear response of the outgoing Eddington flux to the surface temperature variations, which is directly related to the temperature via the blackbody relation L∝T⁴. Whatever the origin of the harmonic distortion is, its effect is to spread the energy contained in the original frequency amongst harmonics and, in the case in which more than one normal mode is excited simultaneously, combination frequencies.

As the non-linear frequencies bear no relation to the pulsation mechanism, they cannot reveal anything about the mode selection and driving mechanisms of the star. However, their actual amplitudes directly depend on the amount of non-linear response experienced by the star, which itself depends on the physical conditions in the stellar interior. Therefore the study of these pulse shape effects can potentially lead to constraints on some atmospheric parameters, as done, for instance, in the case of the ZZ Ceti star G117-B15A (Brassard et al. 1993; Fontaine & Brassard 1994).

The ZZ Ceti star G29-38 has been extensively observed over the 1985–94 period, when about 40 photometric campaigns targeted this star, including two Whole Earth Telescope (Nather et al. 1990) runs in 1988 and 1992 (Kleinman et al. 1998). Although the period spectrum of this star has shown tremendous qualitative changes from season to season, numerous combination frequencies could always been identified. Various analyses (Vuille 2000a, b) have led to the conclusion that most of the combination frequencies are due to harmonic distortion and not to resonant mode coupling. However, because none of the different harmonic distortion processes bears a distinctive signature, it is difficult to disentangle them and determine which one, if any, is dominating in G29–38. The purpose of the present paper is precisely to address this question. The comparison of the measured amplitudes of the cross-frequencies with those predicted by the theoretical models should yield clues as to what process can mostly be held responsible for the presence of these non-linear frequencies.

Section 2 describes and compares the theoretical models available to date. Section 3 presents the observational results and compares them with the theoretical predictions from Brickhill and from Brassard et al. Section 4 compares the predictions from the model of Brickhill with those from the model of Wu. Section 5 concludes the analysis.

Theoretical models

Three different models exist that treat harmonic distortion effects. They are by Brickhill (1991a, b, 1992a, b), by Brassard, Fontaine & Wesemael (1995) and by Wu (1998). Each of these models treats only one of the harmonic distortion processes. Fortunately, the models make different enough predictions that they can be tested with existing data. The model of Wu has already been tested, precisely in the case of G29-38, where good agreement was obtained (Wu 1998). We will therefore concentrate our comparative analysis on the models of Brassard et al. and of Brickhill.

Because these models are built on the linear adiabatic theory, they cannot yield any information as to the stability of the modes. No rules concerning the mode selection mechanism (which preferentially excites some normal modes and not others) or the driving mechanism (which determines the stability and amplitudes of the excited modes) (Winget & Fontaine 1982) are included. However, these models can predict the frequencies, relative amplitudes and phases of the combination frequencies, given the observed properties of the excited eigenmodes.

All these models consider only the case of small-amplitude pulsators, where the perturbation can be expressed as a Taylor series around the equilibrium position. To second order of perturbation, first harmonics and cross-frequencies will naturally appear in the synthetic spectrum with amplitudes A_c proportional to the product A₁A₂ of their parent modes (or ∝ A² in the case of pure harmonics). This provides a direct way of comparing theory and observation, as the ratio A_cA₁A₂ is then free of any absolute amplitude dependence and can be readily measured from the observed period spectra.

The above models are derived for small-amplitude pulsators (in which second-order perturbation theory correctly describes the non-linear pulsations), so it is not clear to what extent they can successfully be applied to a large-amplitude variable like G29-38, in which higher order effects might be significant. In particular, if modes with the same geometry are compared (i.e. same ℓ and m indices), their cross-frequencies are expected to have the same relative amplitudes, i.e. the same ratios A_cA₁A₂. This is not the case if third- (or higher) order effects are present, as the latter will directly affect the amplitudes of the normal modes (Vuille & Brassard 2000).

The model of Brickhill

The theoretical model proposed by Brickhill treats the non-linear response of the stellar medium, more specifically of the convection zone, to the oscillatory perturbation, but neglects the non-linear relation between the emergent flux and the surface temperature. His discussion is based on a numerical model of the outer layer of the ZZ Ceti stars extending down to the base of the driving zone, which is below the convection zone (Brickhill 1991a). Assuming that the linear theory (which implies an incoming sinusoidal pulse) provides an adequate representation of the spatial and temporal behaviour of both the horizontal displacement and the pressure changes (Brickhill 1992a), this model numerically follows the evolution of the temperature wave along its way up to the surface of the star.

From the non-linear (i.e. non-sinusoidal) behaviour of the temperature variations at the stellar surface thus obtained, the emergent flux is derived (Brickhill 1992b) following the prescription given by Robinson, Kepler & Nather (1982, RKN hereafter). This very important work showed that the luminosity variations can be accounted for exclusively by surface temperature changes, and that other factors such as geometric distortions or gravity variations are negligible. To second order of perturbation, the local emergent flux in Brickhill’s model is given by (Brickhill 1992b)

 

(1)

where the Y_i are the spherical harmonics of degree ℓ_i and azimuthal index m_i, while a, b and c are model-dependent constants. For models corresponding to log g=8, their respective values are a≈4.5, b≈9 and c≈8 (Brickhill 1992b). The four lines in the above equation describe the flux perturbation at each point on the stellar surface generated respectively by the normal modes of amplitude A_i, frequency σ _i and phase ψ _i (first line), by their first harmonics (second line), by their sum frequencies (third line), and by their difference frequencies (fourth line). a, b and c thus represent the factors of proportionality, or coupling constants, between these various non-linear components and the original normal modes, so far as the local flux is concerned.

The integration of the local emergent luminosity flux over the visible disc of the star is also carried out following the RKN prescriptions. This assumes a linear relationship between flux and surface temperature variations, which therefore neglects the effects of radiative transfer, but allows an analytical approach. Furthermore, other approximations have been made by RKN for the treatment of the atmosphere, such as use of a linear limb darkening law (RKN equation 14), omission of the temperature dependence of the limb darkening (RKN equation 11), and use of a grey atmosphere, i.e. with the continuous opacity being wavelength-independent (Böhm-Vitense 1989).

The amplitudes of the harmonics and cross-frequencies, as measured in the Fourier spectrum of the light curve, are then given analytically by functions of the form A_cA₁A₂f(Θ). A₁ and A₂ are the amplitudes of the respective parent modes, while f(Θ) is a trigonometric function of the viewing angle Θ, which is the angle between the line of sight and the axis of symmetry of the surface perturbations. The amplitude ratio A_cA₁A₂f(Θ) can then be calculated and is a function of just the viewing angle Θ. Unfortunately, the viewing angle is generally not a known stellar parameter, so that direct comparison between the predicted and measured amplitude ratios is possible only when the function f(Θ) is a constant (see Section 3).

The range of values taken by the amplitude ratio A_cA₁A₂ for all possible (ℓ₁, m₁; ℓ₂, m₂) combinations of normal modes, with ℓ₁,ℓ₂≤2, is given in Tables 1 and 2. Most ratios have a minimum value (corresponding normally to either Θ=0 or Θ=π/2), but no upper limit. This is explained by the fact that the normal modes change their phase over the visible disc, and thus there is always a viewing angle for which they are invisible. For instance, if Θ=90°, the mode ℓ=1, m=0 has a zero visible amplitude A₁, as the opposite brightnesses of the two hemispheres will exactly cancel each other when integrating the light over the stellar disc. However, the corresponding harmonic does not change its phase at all over the visible disc (true for all harmonics of standing waves), and will thus have a finite amplitude A_c whatever the viewing angle is. The corresponding ratio A_cA₁A₂ can thus become infinite.

Table 1

Minima and maxima of the harmonic ratio A_hA² given by the models of Brickhill and BFW respectively, where A_h is the amplitude of the harmonic, and A that of its parent mode.

Table 2.

Table 2. Minima and maxima of the combination ratio A_c=A₁ A₂ given by the models of Brickhill and BFW respectively.

The model of Brassard et al.

The Brassard et al. (1995, hereafter BFW) model supposes that the dominant non-linearities appearing in the period spectrum are due to the highly non-linear relation between the emergent flux and the surface temperature (L∝T⁴). This is expected to be particularly true for ZZ Cetis, because the visible part of their continuous spectrum lies in the highly non-linear region of the blackbody curve, and to a lesser extent for DB variables as their higher effective temperature shifts their visible light into the Rayleigh—Jeans tail. Assuming a sinusoidal surface temperature wave, a distribution referred to by Brassard et al. (1995) as first linearization, we can numerically integrate the exact radiative transfer equations to obtain the emergent Eddington flux.

Analytically, the model is based on the linear, adiabatic theory of stellar pulsation, with the emergent specific intensity being expanded in a Taylor series around the equilibrium temperature T₀. By keeping terms up to second order in ΔT, first-order harmonics and combination frequencies will appear naturally in the synthetic period spectrum. These non-linear peaks will also have amplitudes proportional to the product of the amplitudes of their parent modes, that is again A_c∝A₁A₂ (respectively A_h∝A² in the case of harmonics). The corresponding constants of proportionality are both temperature and surface gravity dependent. The most recent values for the effective temperature and surface gravity of G29-38 are respectively T_eff≈11 820 K and log g≈8.14 (Bergeron et al. 1995). We have calculated, for each ℓ=1 and ℓ=2 combination, the ratios A_cA₁A₂ corresponding to this model. They are listed in the last two columns of Tables 1 and 2. It can be seen that, for any ℓ and m combination, these ratios are between one and two orders of magnitude lower than the predictions from Brickhill’s model.

Comparison with observations

Out of the 40 observing campaigns conducted on G29-38, only five of them are long enough for beating arising from multiplet splitting (Jones et al. 1989) to be resolved. They are described briefly by Vuille (2000a), and in full detail by Kleinman et al. (1998). Altogether, 41 non-linear frequencies have been identified in the period spectra corresponding to these five campaigns (Kleinman et al. 1998); 31 of them are first-order combinations, while nine of them are second-order combinations, i.e. combinations of three modes. Based on the frequency table of Vuille (2000a), we have measured the amplitude ratio A_cA₁A₂ for each of these 31 first-order cases according to the rule that the smallest of the three peaks forming each combination is assumed to be the non-linear frequency, while the other two are the ‘parent’ normal modes. The results are plotted in Fig. 1 as a function of the frequencies of the non-linear modes.

Figure 1.

Ratio A_cA₁A₂ for the 31 first-order combination frequencies. The abscissa indicates the frequencies f_c of the combinations peaks. The circles, crosses and squares represent respectively difference frequencies, sum frequencies and harmonics.

Clearly, most ratios are of the same order of magnitude, although they could, according to Tables 1 and 2, have any values, depending on both the viewing angle Θ and their set of indices (ℓ, m). Because the viewing angle is a global quantity of the star, it is of course the same for all modes. The fact that all the first-order amplitude ratios have the same order of magnitude suggests that the non-linear frequencies observed in G29-38 are combinations of normal modes having mostly the same ℓ and m degrees.

The viewing angle Θ is unfortunately unknown for G29-38, so a direct comparison between the measured ratios A_cA₁A₂ and the ones predicted by the models is not possible. However, in certain specific cases, the above ratio happens not to be dependent on the viewing angle, but is simply a constant (see Tables 1 and 2). This is the case only for those cross-frequencies that are changing phase over the visible disc, like their parent modes do. Indeed, whatever the normal mode considered, a value of the viewing angle Θ always exist for which the bright and dark areas cancel each other geometrically, so that its visible amplitude is actually zero. This is a direct consequence of the fact that the light received in the telescopes is unavoidably integrated over the visible disc, as the stellar surface cannot be resolved. Mathematically, only the combination frequencies that can be described in terms of just one spherical harmonic function rather than in terms of a linear combination of them, change their phase over the visible disc. This is, for instance, always true for harmonics of running waves with equal spherical and azimuthal degrees, ℓ=m, whereas it is never the case for combinations of pure standing waves, i.e. with azimuthal degree m=0 (see Tables 1 and 2).

In order to determine which of the observed amplitude ratios A_cA₁A₂ are independent of the viewing angle Θ, and could thus be used for comparison with the theoretical predictions, a secure mode identification is therefore required. This strengthens the importance of the mode identification, without which hardly any asteroseismology can be performed. According to the work of Kleinman et al. (1998), the majority of the modes observed in G29-38 are thought to be ℓ=1. Assuming, for now, that they are allℓ=1, the various measured ratios of Fig. 1 have been sorted, in Fig. 2, according to whether they are dependent on the viewing angle Θ (bottom panel), or independent of Θ (top panel). This sorting therefore depends exclusively on the identification of the m index of each mode, as the degree ℓ has been provisionally set to be 1. The data points in Fig. 2 have been labelled by the k orders of the respective parent modes that combine to form them, together with the year in which they have been observed. These k values, taken from Kleinman (1995), do not represent a secure identification. Their values should therefore be considered as purely indicative of which frequencies are combined together.

Figure 2.

Ratios A_cA₁A₂ for the combination frequencies that are independent of the viewing angle Θ (top panel), and Θ-dependent (bottom panel). The data points have been labelled by the k orders of the respective parent modes, together with the year in which they have been observed; the ‘+’ and ‘−’ subscripts indicate azimuthal degrees m=±1, while the modes with m=0 are given without subscripts. The abscissa indicates the frequencies f_c of the combination frequencies. The circles, crosses and squares represent respectively difference frequencies, sum frequencies and harmonics. Brickhill’s predictions for harmonics [ℓ, m] and combination frequencies [ℓ₁, m₁; ℓ₂, m₂] are indicated by fixed levels (top panel) and with a corresponding Θ value (bottom panel).

The various levels corresponding to the predictions by Brickhill are also plotted in Fig. 2 for the different ℓ and m combinations. Harmonic levels are indicated by one set of indices [ℓ, m], while the combination frequencies are indicated by two sets [ℓ₁, m₁; ℓ₂, m₂]. In the top panel these levels are fixed, whereas in the bottom panel they are given for the Θ values corresponding to best fits of the different categories of combinations. The corresponding BFW values are too small to appear in the graph.

Provided that the above ℓ=1 assumption is correct for most modes, it is clear, from the top panel of Fig. 2, and according to Tables 1 and 2, that Brickhill’s results are of the right order of magnitude, while BFW’s predictions are not. Prior to drawing any conclusions, a more detailed comparison between the observations and Brickhill’s predictions is required. Although most harmonics and two sum frequencies are well fitted by Brickhill’s model, most combination frequencies identified in 1988 and 1993 are about three times larger than the actual predictions. A detailed step-by-step analysis will possibly help to clarify this issue.

(1) Because the modes k=7,8,9,11,12,14 and 17, individually combined with k=10 in 1993, all show very similar amplitude ratios (Fig. 2, top panel), this strongly suggests that these modes all have the same ℓ and m indices. Their period spacing is in good agreement with an ℓ=1 range (Kleinman et al. 1998), while the fact that these modes never show any multiplet structure suggests (but does not imply) that they are m=0. Note that k=14, observed in 1988 at 1297 μHz, has been observed in 1997 at 1288.6 μHz (Van Kerkwijk, Clemens & Wu 2000). If these are really two observations of the same mode (numerous modes in G29–38 have experienced significant frequency shifts over the years), the above conclusion is in contradiction with that by Clemens, Van Kerkwijk & Wu (2000), who identified this mode as an ℓ=2.

(2) The same remark applies to the k=6, 8 and 18 modes, which are also combined with the k=10 mode; because all show extremely similar difference frequency ratios in 1988 (Fig. 2, top panel), we believe that they must have the same ℓ and m degrees. Their frequencies complement well the ℓ=1 period spacing discussed in point (1) above, which provides additional circumstantial evidence in favour of this identification.

(3) The top panel of Fig. 2 also shows that the difference frequencies have, globally speaking, slightly smaller ratios than the sum frequencies. This is a prediction of Brickhill’s model, provided that modes of the same ℓ and m are compared (see equation 1 in Section 2.1). This is again in good agreement with the assertions previously made in points (1) and (2) above.

(4) The k=10 multiplet shows a very different structure from season to season, experiencing both amplitude and frequency variations (up to 7 μHz), as seen in Fig. 3. Only in the 1993 spectrum does it show a clear multiplet feature. We discuss two possibilities here, although neither one gives fully satisfactory results. (i) If this k=10 peak is an ℓ=1 mode, then the largest component in the 1993 triplet is definitely m=−1 (Fig. 3). It turns out that this is likely also to be true for the largest peaks in the 1985 and 1988 data, as the three corresponding harmonics (2×10₋) not only show the same relative amplitudes, but are well fitted by Brickhill’s model (Fig. 2, top panel). Furthermore, an ℓ=1 identification would be consistent with the above-described period spacing (Kleinman et al. 1998), and with the conclusion of Clemens et al. (2000). However, this would imply that this multiplet has experienced an enormous seasonal shift. The latter could hardly be explained by mode trapping, as this would require G29-38 to have a thin H-layer (Brassard, Fontaine & Wesemael 1992), which is in contradiction with current beliefs (Clemens 1993, 1994; Bradley 1996; Bradley & Kleinman 1997; Kleinman et al. 1998). Furthermore, this ℓ=1. identification also implies that all the 1993 corresponding sum frequency ratios are, as assumed, independent of the viewing angle Θ, with an average value at around 18, which is three times larger than the value predicted by Brickhill’s model. The same remark is true for the difference frequency ratios observed in 1988 (Fig. 2, top panel). This mismatch could possibly be explained by the non-applicability of the model to large-amplitude ZZ Cetis, but the frequency shift of the triplet remains a mystery. Third-order interactions could perhaps be the cause (Vuille & Brassard 2000), but this is not obvious at all. (ii) If this k=10 mode is ℓ=2, then the large peak in the 1993 multiplet is likely to be the m=0 component, to be consistent with the 1985 and 1988 observations, the respective largest peak of which would then correspond to the m=−2 component (Fig. 3). Although the 1993 sum frequency ratios could be fitted with a value of the viewing angle of around Θ=14°, the minimum possible value for the harmonic ratio is 13.6 for Θ=0°(Table 1), which is about three times larger than the observations. Furthermore, the 1988 difference frequency ratios should be 2.6 according to the model (Table 2), whereas the observed ratios are about 13. Therefore trying to make the mode labelled k=10 be an ℓ=2 mode leads to a complete mismatch between theory and observations. We therefore believe that the ℓ=1 identification is correct, because the unexplained features are at least consistent with each other.

Figure 3.

A schematic plot of the multiplet k=10 in three different years.

(5) The mode labelled k=8 shows a peculiar behaviour in 1989. Not only is its harmonic relatively three times larger than all the other ones observed (Fig. 2, top panel), but also its frequency experienced an enormous 12-μHz shift from 1988 to 1989 (Kleinman 1995). This suggests that we might not be seeing the same mode. This 1989 harmonic ratio at 11.4±2.4 (Fig. 2) is consistent with an ℓ=2, m=0 mode, the predicted value of which is 13.7 (Table 1).

(6) If the k=16 mode observed in 1985 is ℓ=1, m=0, then its first harmonic ratio of 7.1 (Fig. 2, bottom panel) corresponds, in Brickhill’s model, to a viewing angle Θ=25°±8°.

(7) If the sum frequencies in the bottom panel of Fig. 2 are indeed combinations of respectively (ℓ=1, m=1) and (ℓ=1, m=−1) modes, then these observed ratios are all too low to fit Brickhill’s model. Indeed, the latter predicts a minimum value of 17.8 corresponding to a viewing angle Θ=0°. However, it has to be stressed that the identification of harmonics of modes belonging to a multiplet is very difficult because they are often blended. For instance, a peak has been observed in 1989 with frequency corresponding to the combination 6₋+6₊, which is impossible to disentangle from the harmonic 2 × 6₀ of the central peak.

(8) Although the two difference frequencies (8–16) and (6–8) (Fig. 2, bottom panel) have very different values, they are not far apart in terms of viewing angle. Provided that the identification ℓ=1, m=0 for the corresponding normal mode is correct, then Θ=60°±3° indeed encompasses both of them. This is explained by the fact that, when Θ is large, the normal modes corresponding to standing waves, that is with m=0, are very small, and actually vanish at Θ=90°. Therefore a small error in the viewing angle may induce a large change in the corresponding amplitude ratio A_cA₁A₂.

(9) In Fig. 1, the ratio at 888 μHz was, with a value at around 85, much larger than any others (it has not been plotted in Fig. 2). This difference frequency, which corresponds to the combination 8–17, was already standing out when a phase analysis was performed (Vuille 2000b), which led to the suspicion that this frequency might be a resonant mode rather than a harmonic distortion peak. The fact that it is also standing out in the present analysis strengthens this suspicion. A possible misidentification is discarded as not only is power definitely present, but also the frequency match is one of the best observed with f_c− ( f₈−f₁₇)=888.150−(2006.587−1118.553)=0.116 μHz (Vuille 2000a).

Comparison with the model of Wu

Wu (1998) developed a model physically very similar to that of Brickhill (1992b). Both of these models treat the non-linear response of the convection zone to the sinusoidal flux perturbation generated at the base of the convective layer, and both neglect the non-linear response of the radiative transfer, in contrast to BFW. Given the amplitude of the normal modes, the model by Wu enables one to calculate analytically the power in the combination frequencies. The model has been tested in the case of G29-38, where the theoretical predictions were compared with a 5-h data set obtained with the Keck II Telescope (Van Kerkwijk et al. 2000): the model predicted the amplitudes of the observed combination frequencies to the right order of magnitude (Wu 1998).

It is of interest to compare directly the predictions of Wu with those of Brickhill, and the observed relative amplitude obtained from long multisite campaigns with those obtained from the single-night run on Keck II. The amplitude ratios A_cA₁A₂ for the non-linear frequencies identified in the Keck II period spectrum are shown in Fig. 4. For each of these frequencies, the corresponding amplitude ratios predicted by the model of Wu are indicated by filled circles. Because this Keck II run is only 5 h long, the measured amplitudes are certainly affected by unresolved beating processes, so that only a global comparison with the predictions of the model is meaningful.

Figure 4.

Amplitude ratios A_cA₁A₂ for the combination frequencies identified in the Keck II data set (Van Kerkwijk et al. 2000). The abscissa indicates the frequencies f_c of the combination frequencies. The open circles, crosses and squares represent respectively difference frequencies, sum frequencies and harmonics. For each of these identified non-linear frequencies, the corresponding predictions from the model of Wu (1998) are indicated by filled circles. Brickhill’s predictions of interest for harmonics [ℓ, m] and combination frequencies [ℓ₁, m₁; ℓ₂, m₂] are indicated by fixed levels.

The comparison of Fig. 4 with the top panel of Fig. 2 (the same scale is used in both figures) shows that the Keck II data are very similar to those obtained with longer runs (the slightly lower amplitude ratios obtained with Keck II can probably be explained by the higher signal-to-noise ratio obtainable with this telescope). A direct comparison of the predictions from Wu and Brickhill is therefore meaningful. Fig. 4 shows clearly that not only do the predictions from Wu match the observations well, as mentioned above, but also they are consistent with the predictions from Brickhill, so far as the order of magnitude is concerned. Both models are frequency-independent, except at very low frequency where the predictions by Wu drop to zero (see Wu 1998, equation 8.14). We cannot give a sound physical interpretation for this behaviour.

Conclusion

The lack of a secure mode identification for G29-38 has made this whole analysis very dependent on initial assumptions. In particular, we have had to assume that the identified eigenmodes are all ℓ=1. Our subsequent analyses show that this assumption is self-consistent, and we conclude that it is correct for the majority of the identified modes. The strongest argument in favour of this ℓ=1 contention arises from the clear clustering of most amplitude ratios A_cA₁A₂, which suggests that the majority of the identified normal modes are of the same spherical degree, most probably ℓ=1, in keeping with the period spacing analysis by Kleinman et al. (1998). However, the mode identified as k=8 in 1989 might be an ℓ=2 mode, although no clear evidence can be put forward. The analysis has also confirmed the probable discovery of a resonant mode at 888 μHz, corresponding to the combination 8–17.

As far as the comparison with the theoretical model is concerned, it can first be determined that the model of Brickhill fits the amplitudes of the non-linear frequencies much better than does the BFW model, the predictions of which are about one order of magnitude too low. This directly suggests that the non-linear response of the convection zone to the oscillatory perturbation travelling through it is a much stronger effect than the non-linear response of the emergent flux to the surface temperature variations. However, as both of these models are built on the same degree of approximation, their analytical predictions are essentially identical, although their physical approach is different. It is therefore surprising that their numerical predictions are so different. It can be determined that the discrepancy lies in the value of the coupling constants a, b and c (equation 1). For instance, a≈4.5 in Brickhill’s model (see Section 2), but it is only about 0.2 in BFW’s model for an 11 500-K star.

While the present work represents the first serious test of Brickhill’s model, BFW’s model has been (successfully) tested in the case of the ZZ Ceti star G117-B15A (Brassard et al. 1993). Built on three eigenmodes only, the latter model reproduced well the 10 frequencies observed in the amplitude spectrum of this star. The discrepancy between the excellent agreement in the case of G117-B15A and the poor matching for G29–38 is surprising. However, G117-B15A is a much lower amplitude pulsator than G29–38, which suggests that third- (and higher) order effects may be too significant in G29–38 for a perturbation theory to be applied. The comparison of the ratios (A_cA₁A₂) of the combination frequencies with that (A_hA²) of the first harmonics supports this consideration. It can be shown, using plain trigonometric arguments, that, to second order of perturbation, these two quantities should satisfy A_cA₁A₂=2A_hA² (compare Tables 1 and 2; see also Brickhill 1992a). This is not, however, what is observed in G29-38 (Fig. 2), suggesting the presence of higher order perturbations.

However, Brickhill’s second-order model gives satisfactory results. The latter could unfortunately not be compared with BFW’s predictions in the case of G117-B15A, because the coupling parameters a, b and c (equation 1), which are not true constants but temperature and surface gravity dependent functions, are provided only in the one case that, by chance, corresponds quite well to G29–38. This truly limits the applicability and testing possibilities of Brickhill’s model.

Although the exact origin of the discrepancy between the two models could not be uncovered, it nevertheless seems that the radiative transfer is really not the dominant non-linear process in G29-38. The model of Wu (1998), which is built on the same assumptions as Brickhill’s model, fully supports this conclusion. This is quite surprising, as the visible part of the emission spectrum of DA white dwarfs lies precisely in the highly non-linear part of the energy distribution curve, where a relation of the type L∝T⁴ applies. The effect of the radiative transfer is thus expected to be the first significant deviation from linearity. As mentioned earlier, this is, however, not necessarily the case for DB variables, for which the visible part of the spectrum lies in the fairly linear Rayleigh—Jeans tail of the energy distribution, where the white light luminosity L is more or less proportional to the temperature T.

Although Brickhill’s predictions are of the right order of magnitude, it has not been possible, even by trying to attribute different ℓ and m indices to the identified modes, to fit the various amplitude ratios observed properly. For this reason, the viewing angle Θ could not be determined, as the different measurements were not consistent with each other. Three possible causes could be at the origin of the mismatch between the predictions of the model and the observations. First, the coupling constantsa, b and c (equation 1) are provided only for a model with log g=8 and T_eff=11 500 K, which does not correspond exactly to the case of G29-38, for which the most recent measurements are log g=8.14 and T_eff=11 820 K (Bergeron et al. 1995). Secondly, Brickhill’s second-order perturbative model might not be appropriate for a large-amplitude pulsator like G29-38, where third- (and possibly higher) order effects might be important. Thirdly, it is not clear whether the model accounts for normal modes involved simultaneously in many combinations. For instance, only one combination frequency involved the mode k=10 in 1985, whereas the same mode generated eight cross-frequencies in 1993 (Fig. 2). Therefore much more of the initial energy of this normal mode has been spread into combination frequencies in 1993 than in 1985, which correspondingly lowers its amplitude and thus increases the amplitude ratio A_cA₁A₂.

Proper comparison between theory and observation will be possible only when a model that encompasses all of the different non-linear processes has been proposed. A first step in this direction would be to combine Brickhill’s and BFW’s models. This could be achieved by using Brickhill’s non-sinusoidal surface temperature distribution as the input for BFW’s model. A ZZ Ceti with a secure mode identification is also crucial.